K_{0} OF PURELY INFINITE SIMPLE REGULAR RINGS

P. ARA, K. R. GOODEARL, AND E. PARDO

Abstract. We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if R is a purely infinite simple ring, then K_{0}(R)^{+} = K_{0}(R), the monoid of isomorphism classes of finitely generated projective R-modules is isomorphic to the monoid obtained from K_{0}(R) by adjoining a new zero element, and K_{1}(R) is the abelianization of the group of units of R. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable abelian group is isomorphic to K_{0} of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.

Introduction

In 1981, Cuntz [12] introduced the concept of a purely infinite simple C*-algebra. This notion has played a central role in the development of the theory of C*-algebras in the last two decades. A large series of contributions, due to Blackadar, Brown, Lin, Pedersen, Phillips, Rørdam and Zhang, among others, reflect the interest in the structure of such algebras. One of the most important advances in the program of classifying separable C*-algebras through K-Theory, proposed by Elliott in the early seventies, was obtained in this context by Kirchberg [16] and Phillips [25], who showed that nuclear separable unital purely infinite simple C*-algebras are classified by Ktheoretic invariants.

In this work, we introduce a suitable definition of a purely infinite simple ring. This notion agrees with that of Cuntz in the case of C*-algebras. Moreover, a number of basic results, known in the case of C*-algebras, also hold in the purely algebraic context. In particular the algebraic K_{0} and K_{1} groups of a purely infinite simple ring follow the same patterns as the corresponding topological K groups of purely infinite simple C*- algebras, found by Cuntz in [12].

Our goal is to use these concepts and ideas in order to advance the knowledge of the K-theory of (von Neumann) regular rings. In particular, we construct examples of purely infinite simple regular rings whose K_{0} groups are cyclic of arbitrary order, and – in the case when K_{0} is finite cyclic – whose finitely generated projective modules are free. By using these constructions, we build examples of purely infinite simple regular rings whose K_{0} is any countable abelian group.

1991 Mathematics Subject Classification. 16E50, 19A49, 19K14.

Key words and phrases. purely infinite simple ring, Grothendieck group, von Neumann regular ring, universal localization.

The first and third authors were partially supported by MEC-DGESIC grant PB98-0873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second author was partially supported by NSF grant DMS-9970159, and the third author by PAI III grant FQM-298.

A key point of our work lies in the development of new techniques for constructing examples. Our algebras are extensions of the algebras V_{1,n}, first considered by Leavitt in [19]. Recall that the Leavitt algebra V_{1,n} over a field k is the k-algebra having a universal right V_{1,n}-module isomorphism V_{1,n}^{n} → V_{1,n}. Our method for getting the extensions of the V_{1,n} works as follows. We consider a local subalgebra of the algebra of noncommutative power series k⟨⟨X⟩⟩, where X = {x_{0}, …, x_{n}}, closed under a certain set of skew derivations, and containing the free algebra k⟨X⟩. Then we construct our extensions of V_{1,n} by universally inverting the right R-module homomorphism R → R^{n+1} given by left multiplication by the column (x_{0}, …, x_{n})^{T}. We prove that K_{0} of these algebras is always the cyclic group of order n. Taking R to be the full algebra of noncommutative power series one obtains what could be thought of as a completion of V_{1,n+1}. Taking the minimal choice for R, which turns to be the so-called algebra of rational series, one obtains a kind of “algebra of fractions” of V_{1,n+1}. The latter algebra is shown to coincide with a construction of Schofield [29], and also with a construction of Rosenmann and Rosset [27]. In particular it enjoys various additional universal properties.

We briefly outline the contents of the paper. In Section 1 we define purely infinite simple rings, and we prove some of their elementary properties. In Section 2 we give the basic patterns for the algebraic K_{0} and K_{1} of purely infinite simple rings, in analogy with Cuntz’s results on the topological K_{0} and K_{1} of purely infinite simple C*-algebras. Section 3 is devoted to introducing the construction of skew polynomial rings with freely independent indeterminates, which will be a fundamental technical device for the rest of the paper. In Section 4 we outline some known results on Leavitt’s algebras, which can be considered as the earliest universal examples in this class, and also as the algebraic relatives of Cuntz’s algebras. (The Cuntz algebra O_{n} is the C*-completion of the Leavitt algebra V_{1,n} over the field of complex numbers.) Section 5 is the core of the paper: we develop here the construction of our examples, we establish their structure, and we compute their K_{0} groups. In Section 6 we present the examples of Schofield [29], as well as those of Rosenmann and Rosset [27], and we show that they are isomorphic. Section 7 is devoted to show that the examples in Section 6 are isomorphic to particular cases of the examples presented in Section 5. As a consequence, we see that these algebras can be obtained in two different ways as universal localizations of free algebras. Finally, we prove in Section 8 that any countable abelian group can be realized as K_{0} of a purely infinite simple regular ring.

Aside from a few noted exceptions, all rings and modules in this paper will be assumed to be unital. In fact, most of our rings will be algebras over a base field that we denote k. Ring and algebra homomorphisms, except for embeddings of ideals, will also be assumed to be unital. We will often write nA for the direct sum of n copies of a module A, although in the case of a ring R we prefer the notation R^{n} for the free right Rmodule of rank n, whose elements we think of as column vectors. The notations k⟨X⟩ and k⟨⟨X⟩⟩, respectively, will stand for the free algebra and the noncommutative formal power series algebra over k on a set X.

1. Purely infinite simple rings

We introduce the concept of a purely infinite simple ring, and sketch some basic results on this topic. These results are analogous to those obtained by Cuntz.

First, recall that if R is a ring and e, f ∈ R are idempotents, we say that e and f are (Murray–von Neumann) equivalent (denoted e ∼ f) provided that there exist elements x ∈ eRf and y ∈ fRe such that xy = e and yx = f. This is equivalent to demanding that eR ≅ fR as right R-modules. Recall also that e and f are said to be orthogonal (denoted e ⊥ f) provided that ef = fe = 0. In that case, e + f is an idempotent, and (e + f)R = eR ⊕ fR.

The following useful means of producing orthogonal decompositions of idempotents is old and well known; we sketch the easy proof for convenience.

Lemma 1.1. Let e be an idempotent in a ring R. If there exist right ideals A_{i} ⊆ R such that eR = A_{1} ⊕ ⋯ ⊕ A_{n}, then there exist pairwise orthogonal idempotents e_{i} ∈ R such that e = e_{1} + ⋯ + e_{n} and A_{i} = e_{i}R for all i.

Proof. Due to the given direct sum decomposition, the endomorphism ring of the module eR contains orthogonal idempotents f_{1}, …, f_{n} such that f_{1} +⋯+f_{n} equals the identity map on eR and each f_{i}(eR) = A_{i}. The desired idempotents e_{i} are the images of the f_{i} under the canonical isomorphism End_{R}(eR) → eRe.

Definitions 1.2. An idempotent e in a ring R is infinite if there exist orthogonal idempotents f, g ∈ R such that e = f + g while e ∼ f and g ≠ 0. In view of Lemma 1.1, e is infinite if and only if eR is isomorphic to a proper direct summand of itself, that is, eR is a directly infinite module.

A simple ring R is said to be purely infinite if every nonzero right ideal of R contains an infinite idempotent. It will follow from Theorem 1.6 that this concept is left-right symmetric.

Examples 1.3. The class of purely infinite simple rings is rather large; we indicate some subclasses here:

(a) Many purely infinite simple C*-algebras are known; for instance, see [1], [11], [12], [16], [17], [18], [21], [25], [26], [37].

(b) If V is an infinite dimensional vector space over k, then End_{k}(V) modulo its unique maximal ideal M is purely infinite. This follows from the fact that if f ∈ End_{k}(V) \ M, then dim_{k} f(V) = dim_{k} V.

(c) More generally, if R is a regular, right self-injective ring without nonzero directly finite central idempotents, and M is any maximal ideal of R, then R/M is purely infinite. To see this, consider any element x ∈ R \ M. Since R/M is simple, ∑_{i=1}^{n} a_{i}xb_{i} − 1 ∈ M for some a_{i}, b_{i} ∈ R, from which we see that R_{R} embeds in n(xR) ⊕ zR for some z ∈ M. By general comparability [14, Corollary 9.15], there is a central idempotent e ∈ R such that ezR ≲ exR and (1 − e)xR ≲ (1 − e)zR; in particular, (1 − e)x ∈ M. Since x ∉ M, we cannot have e ∈ M, and so 1 − e ∈ M. Now eR is isomorphic to exR. For, since e is a directly infinite central idempotent, we have (n + 1)(eR) ≅ eR by [14, Theorem 10.16]. Thus, as eR ≲ (n + 1)(exR) ≲ (n + 1)(eR), we have (n + 1)(eR) ≅ (n + 1)(exR) by [14, Theorem 10.14]. Hence, eR ≅ exR by [14, Theorem 8.16(b)]. Since the class of e in R/M is 1, we conclude that R/M is isomorphic to x(R/M).

(d) If R is a directly infinite regular ring and all (finitely generated) projective right R-modules are free, then R is a purely infinite simple ring. First, if x is a nonzero element of R, then the projective module xR is free, whence R_{R} is isomorphic to a direct summand of xR, and consequently RxR = R. Thus R is simple. Further, since R_{R} is directly infinite, so is xR. Hence, all nonzero idempotents in R are infinite, and therefore R is purely infinite.

(e) If A is any directly infinite simple ring, then the direct limit B = \underrightarrow{lim} M_{2^{n}}(A) (with block diagonal transition maps) is purely infinite. To show this, let z be a nonzero element of B; then z is the image of a nonzero element x ∈ M_{2^{n}}(A) for some n, and we can assume without loss of generality that n = 0. For a suitable m we have that ∑_{i=1}^{2^{m}} a_{i}xb_{i} = 1 for some a_{i}, b_{i} in A. If we use the a_{i} (respectively, the b_{i}) as the first row (respectively, column) of 2^{m} × 2^{m} matrices whose other entries are zero, we obtain a, b ∈ M_{2^{m}}(A) such that ayb = e_{11}, where y is the image of x in M_{2^{m}}(A) and e_{11} is the matrix unit in the top left corner of M_{2^{m}}(A). Since the idempotent 1 ∈ A is infinite, so is the idempotent ayb in M_{2^{m}}(A). But yba is an idempotent equivalent to ayb, so it too is infinite. Its image in B is an infinite idempotent contained in the right ideal zB.

Lemma 1.4. Let R be a simple ring, and let P and Q be finitely generated projective right R-modules. If P is directly infinite, then there exists a nonzero right R-module A such that P ≅ Q ⊕ A.

Proof. By hypothesis, there exists a nonzero right R-module B such that P ≅ P ⊕ B, whence P ≅ P ⊕ mB for all m ∈ ℕ. Since R is simple and B is projective, B is a generator in Mod-R; in particular, there exist n ∈ ℕ and a right R-module C such that nB ≅ Q ⊕ C. Therefore

P ≅ P ⊕ nB ≅ Q ⊕ (P ⊕ C),

and P ⊕ C is nonzero because P ≠ 0.

Proposition 1.5. Suppose that R is a purely infinite simple ring. Then all nonzero finitely generated projective right R-modules are directly infinite; equivalently, all nonzero idempotents in the matrix rings M_{n}(R) are infinite.

Consequently, if P and Q are any nonzero finitely generated projective right Rmodules, then there exists a nonzero right R-module A such that P ≅ Q ⊕ A.

Proof. By hypothesis, there is at least one infinite idempotent e ∈ R, whence eR is a directly infinite, finitely generated projective right R-module. If Q is any nonzero finitely generated projective right R-module, Lemma 1.4 implies that Q is isomorphic to a direct summand of eR, and so Q is isomorphic to some nonzero right ideal I ⊆ R. Now since R is purely infinite, there is an infinite idempotent f ∈ I, whence fR is a directly infinite direct summand of I. It follows that I is directly infinite, whence Q is directly infinite.

The final conclusion of the proposition now follows immediately from Lemma 1.4.

Theorem 1.6. Let R be a simple ring. Then R is purely infinite if and only if

(a) R is not a division ring, and

(b) For every nonzero element a ∈ R, there exist elements x, y ∈ R such that xay = 1.

Proof. (⟹): Assume that R is purely infinite. Then R contains an infinite idempotent, and so R cannot be a division ring.

Now consider a nonzero element a ∈ R. By assumption, there exists an infinite idempotent e ∈ aR, and then Proposition 1.5 implies that eR ≅ R ⊕ A for some A. Now by Lemma 1.1, there exist orthogonal idempotents f, g ∈ R such that e = f +g and fR ≅ R. Then f ∼ 1, and so there are elements α ∈ fR and β ∈ Rf such that αβ = f and βα = 1. Since f ∈ eR ⊆ aR, we also have f = ar for some r ∈ R. Therefore

1 = βαβα = βfα = βa(rα),

and (b) is established.

(⟸): Now assume conditions (a) and (b), and consider a nonzero right ideal I ⊆ R. Since R is not a division ring, I must contain a proper nonzero right ideal, say J. Choose a nonzero element a ∈ J, and apply condition (b): there exist x, y ∈ R such that xay = 1. Now e := ayx is an idempotent lying in aR, whence e ∈ I and e ≠ 1. Since (eay)(xe) = e and (xe)(eay) = 1, we also have e ∼ 1, and so 1 is infinite. But then e, being equivalent to 1, is infinite too, and we have proved that R is purely infinite.

Notice that Theorem 1.6 implies that the definition of a purely infinite simple ring given above is left-right symmetric (this can also be shown by a direct argument). Also, we point out that in view of condition (b) of the theorem, our definition agrees with the definition in current use among C*-algebraists. (For the equivalence of this definition with Cuntz’s original definition, see [6, Proposition 6.11.5].)

Corollary 1.7. The class of purely infinite simple rings is closed under Morita equivalence.

Proof. It suffices to show that if R is a purely infinite simple ring, then M_{2}(R) and eRe have the same properties, where e is an arbitrary nonzero idempotent in R. It is clear that M_{2}(R) and eRe are simple.

Because R contains an infinite idempotent, it cannot be artinian. But R is Morita equivalent to eRe, and thus eRe cannot be a division ring. For any nonzero element a ∈ eRe, Theorem 1.6 provides elements x, y ∈ R such that xay = 1, and then exe and eye are elements of eRe such that (exe)a(eye) = e. Thus, by the theorem, eRe is purely infinite.

By Proposition 1.5, there is an idempotent f ∈ R such that fR ≅ 2R, whence fRf ≅ M_{2}(R). Since fRf is purely infinite by the previous paragraph, we therefore conclude that M_{2}(R) is purely infinite, as desired.

Remarks 1.8.

(a) Zhang ([37, Theorem 1]), and also Brown and Pedersen ([7, Proposition 3.9]) have proved that every purely infinite simple C*-algebra has real rank zero, which by [2, Theorem 7.2] is equivalent to the property of being an exchange ring [34]. Thus, the following question imposes itself:

Is every purely infinite simple ring an exchange ring?

(b) Let R be any directly infinite, simple exchange ring which is separative (see [2]). Then R is necessarily purely infinite, as follows. It follows from simplicity and the exchange criterion of Nicholson and Goodearl that any nonzero right ideal of R contains a nonzero idempotent, say e. Then R_{R} ≲ n(eR) for some n, whence R is isomorphic to a corner of M_{n}(eRe), and so M_{n}(eRe) is directly infinite. By [2, Proposition 2.3], eRe must be directly infinite, that is, the idempotent e is infinite.

(c) If A is a simple exchange ring, then the simple ring B = \underrightarrow{lim} M_{2^{n}}(A) (with block diagonal transition maps) will either have stable rank 1 or be purely infinite. For, notice that, if there is some n such that M_{2^{n}}(A) is directly infinite, then B is purely infinite by Example 1.3(e). Otherwise, A is stably finite, and so it has powercancellation by [5, Proposition 2.1.8]. This in turn implies that B has cancellation, and thus has stable rank 1 by [36, Theorem 9].

(d) If A is a simple QB-ring (see [3]), then it either has stable rank 1 or is purely infinite [24].

2. K_{0} and K_{1} of purely infinite simple rings

Cuntz [12] computed the general patterns for K_{0} and (topological) K_{1} of purely infinite simple C*-algebras. Here we give parallel results for the K_{0} and the (algebraic) K_{1} groups of purely infinite simple rings. We recall some basics of K-Theory for the convenience of the reader.

Given a ring R, we define V(R) to be the set of isomorphism classes (denoted [A]) of finitely generated projective right R-modules, and we endow V(R) with the structure of a commutative monoid by imposing the operation

[A] + [B] := [A ⊕ B]

for any isomorphism classes [A] and [B]. (The notation [A] will also be used for stable isomorphism classes, that is, elements of K_{0}(R)^{+}, but it should be clear from the context which is meant.) Equivalently [6, Chapter 3], V(R) can be viewed as the set of equivalence classes of idempotents in M_{∞}(R) = ⋃_{n=1}^{∞} M_{n}(R) with the operation

[e] + [f] := [( \begin{matrix}e & 0 \\ 0 & f \end{matrix} )]

for idempotents e, f ∈ M_{∞}(R). The group K_{0}(R) is the universal group of V(R), and the image of the canonical monoid homomorphism V(R) → K_{0}(R) is the positive cone of K_{0}(R), denoted K_{0}(R)^{+}.

Observe that V(R) is conical, that is, x + y = 0 in V(R) only if x = y = 0. Consequently, the subset V(R)^{∗} := V(R) \ {0} is closed under addition. There is a natural pre-order ≤ on V(R), where x ≤ y if and only if there exists z ∈ V(R) such that x + z = y. In terms of isomorphism classes of finitely generated projective modules A and B, we have [A] ≤ [B] if and only if A is isomorphic to a direct summand of B. We say that V(R) is simple provided that for each nonzero x, y ∈ V(R), there exists n ∈ ℕ such that y ≤ nx. Observe that if R is a simple ring, then V(R) is a simple monoid (recall that simplicity of R implies that all nonzero projective R-modules are generators in Mod-R). The converse is false (e.g., take R = ℤ).

Lemma 2.1. Let R be a purely infinite simple ring. For any x, y ∈ V(R)^{∗}, there exist a, b ∈ V(R)^{∗} such that x = y + a and y = x + b.

Proof. This is just a restatement of the last part of Proposition 1.5.

Proposition 2.2. If R is a purely infinite simple ring, then V(R)^{∗} is a group.

In particular, any nonzero finitely generated projective R-modules which are stably isomorphic must be isomorphic.

Proof. According to [2, Proposition 2.4], to prove that V(R)^{∗} is a group it suffices to show that V(R) is conical and simple, and that for each x ∈ V(R)^{∗} there exists b ∈ V(R)^{∗} such that x + b = x. However, we already know that V(R) is conical and simple, and the remaining condition follows from Lemma 2.1 (take y = x). Therefore V(R)^{∗} is a group.

The final statement of the proposition now follows.

An interesting consequence of Proposition 2.2 is that if R is a purely infinite simple ring and [P] is the identity element of V(R)^{∗}, then P is nonzero and P ⊕ Q ≅ Q for all finitely generated projective right R-modules Q.

Notice that, aside from the fact that the stable rank of a purely infinite simple ring is ∞, Proposition 2.2 shows that the cancellative behavior of finitely generated projective modules is almost the same as in the case of rings of stable rank 1.

If M is an additive monoid, we write {0}⊔M for the monoid constructed by adjoining a new zero element to M. (To work with this new monoid, one must choose some notation to distinguish between the old and new zero elements.)

Corollary 2.3. If R is a purely infinite simple ring, then V(R) ≅ {0} ⊔ K_{0}(R), and K_{0}(R)^{+} = K_{0}(R).

Proof. Let ϕ : V(R) → K_{0}(R) be the natural monoid homomorphism; thus if A is any finitely generated projective right R-module, ϕ([A]) = [A] is the stable isomorphism class of A. By Proposition 2.2, V(R)^{∗} is a group, and so ϕ restricts to a group homomorphism ϕ^{∗} : V(R)^{∗} → K_{0}(R). Thus the image of ϕ^{∗} is a subgroup of K_{0}(R). In particular, this image contains [0], and so ϕ^{∗}(V(R)^{∗}) = ϕ(V(R)) = K_{0}(R)^{+}. Since K_{0}(R) is generated by K_{0}(R)^{+}, we now see that K_{0}(R)^{+} = K_{0}(R).

If x, y ∈ V(R)^{∗} and ϕ^{∗}(x) = ϕ^{∗}(y), then x + z = y + z for some z ∈ V(R). Since either z = 0 or z lies in the group V(R)^{∗}, it follows that x = y. This shows that ϕ^{∗} is an isomorphism of V(R)^{∗} onto K_{0}(R)^{+} = K_{0}(R), and therefore we conclude that V(R) ≅ {0} ⊔ K_{0}(R).

For completeness, we present the following theorem which shows the parallelism of patterns between algebraic and topological K_{1} in the cases of purely infinite simple rings and C*-algebras. However, K_{1} will not appear elsewhere in the paper.

Theorem 2.4. If R is a purely infinite simple ring then K_{1}(R) = U(R)^{ab}.

Proof. By [23, Remark after 2.3], R is a GE-ring, so the natural map κ : U(R) → K_{1}(R) is surjective. In order to show that κ is injective, we will proceed in two steps. The first one is similar to Cuntz’s argument in [12].

Step 1. Let v be a unit in the kernel of κ. Assume there is a nonzero idempotent e ∈ R such that v = e + (1 − e)v(1 − e). Then v ∈ U(R)′.

Proof of Step 1: Take an idempotent f < e such that f ∼ e. Set r_{1} = 1−e+f, and note that r_{1} ∼ 1. Now let r_{2}, r_{3}, … be orthogonal idempotents such that r_{2} +⋯+r_{i} ≤ e−f and r_{i} ∼ 1 for all i ≥ 2. Notice that (r_{1} + ⋯ + r_{i})R(r_{1} + ⋯ + r_{i}) ≅ M_{i}(R); this isomorphism may be chosen so that its restriction to r_{1}Rr_{1} sends

1 − e ⟼ diag(1 − e, 0, …, 0)  and  f ⟼ diag(e, 0, …, 0).

Hence, the element v_{i} := (1−e)v(1−e)+f +r_{2} +⋯+r_{i} corresponds to diag(v, 1, …, 1) under this isomorphism. Since the image of v in K_{1}(R) is 0, there is some n ≥ 1 such that diag(v, 1, …, 1) ∈ GL_{n}(R)′. Consequently, v_{n} ∈ U((r_{1} + ⋯ + r_{n})R(r_{1} + ⋯ + r_{n}))′, and so v = v_{n} + e − f − (r_{2} + ⋯ + r_{n}) lies in U(R)′, as desired.

Step 2. For any u ∈ U(R) there is a nonzero idempotent e ∈ R and a unit of the form v = e + (1 − e)v(1 − e) such that u ≡ v (mod U(R)′).

Proof of Step 2: Any decomposition 1 = e_{1}+⋯+e_{n} with e_{1} ∼ ⋯ ∼ e_{n−1} and e_{n} ≲ e_{1} gives rise to an isomorphism R ≅ S, where S is the ring of n × n matrices A = (a_{ij}) over T = e_{1}Re_{1} of the following form: all the entries a_{in} are in T f, and all the entries a_{nj} are in fT, where f is the idempotent corresponding to e_{n} under the subequivalence e_{n} ≲ e_{1}. An argument similar to that of Whitehead’s Lemma proves that, for n ≥ 4, the set of elementary matrices relative to the above matrix decomposition of R is contained in U(R)′. Now take n ≥ 4 and a decomposition of 1 of the form indicated. Consider the corresponding matricial representation over T = e_{1}Re_{1}, which is also a purely infinite simple ring. By the argument in [23, Theorem 2.2], the matrix corresponding to u can be transformed by elementary operations to a matrix of the form

(\begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix}) ,

where ∗ is a matrix of size (n − 1) × (n − 1). Since n ≥ 4, the elementary matrices give rise to elements in U(R)′, so that u is congruent mod U(R)′ to a unit of the form e_{1} + (1 − e_{1})v(1 − e_{1}), as desired.

3. Skew polynomial rings with freely independent indeterminates

In this section, we introduce a general construction that turns out to be a keystone for our work. Let R be a ring and (τ_{1}, δ_{1}), …, (τ_{n}, δ_{n}) some right skew derivations on R. Thus, the τ_{i} are ring endomorphisms of R, and the δ_{i} are additive endomorphisms satisfying the rule δ(rs) = δ(r)τ(s) + rδ(s). Let Y = {y_{1}, …, y_{n}} be an n-element alphabet and denote by Y^{∗} the free monoid on Y, with identity element denoted 1. Label words in Y^{∗} in the form y_{I} = y_{i_{1}}y_{i_{2}} ⋯ y_{i_{t}} for finite sequences I = (i_{1}, …, i_{t}) of indices from {1, …, n}, and similarly for other quantities like τ_{I} or δ_{I}.

We would like to build a ring whose right R-module structure is free with basis Y^{∗}, such that ry_{i} = y_{i}τ_{i}(r) + δ_{i}(r) for all i and all r ∈ R. To define a multiplication on this R-module and verify the ring axioms is tedious; instead, we build a ring of operators on this module, and afterwards carry over the structure.

Let V be the free right R-module with basis Y^{∗}, and let E = End_{ℤ}(V). We let maps in E act on the right of their arguments, so that we can identify R with the subring of right multiplication operators in E. Define z_{1}, …, z_{n} ∈ E so that

(yr)z_{i} = (yy_{i})τ_{i}(r) + yδ_{i}(r)

for all i, all r ∈ R, and all y ∈ Y^{∗}. Let S be the subring of E generated by R and z_{1}, …, z_{n}. For all i, all r, s ∈ R, and all y ∈ Y^{∗}, we have

(ys)(rz_{i}) = (ysr)z_{i} = (yy_{i})τ_{i}(sr) + yδ_{i}(sr)
= (yy_{i})τ_{i}(s)τ_{i}(r) + yδ_{i}(s)τ_{i}(r) + (ys)δ_{i}(r) = (ys)[z_{i}τ_{i}(r) + δ_{i}(r)].

Therefore

rz_{i} = z_{i}τ_{i}(r) + δ_{i}(r) (*)

for all i and all r ∈ R. In particular, (*) implies that S is generated as a right R-module by the monomials z_{I}. All τ_{i}(1) = 1 and δ_{i}(1) = 0, so (y)z_{i} = yy_{i} for all y ∈ Y^{∗}. Consequently, we get by induction that (1)z_{I} = y_{I} for all I. Any s ∈ S can be written as ∑_{I} z_{I}r_{I} with almost all r_{I} = 0, and (1)s = ∑_{I} y_{I}r_{I}. Since the y_{I} form a basis for V as a right R-module, the rule s ↦ (1)s thus gives a right R-module isomorphism from S onto V. We summarize our observations as follows:

Proposition 3.1. Given a ring R with right skew derivations (τ_{i}, δ_{i}) for i = 1, …, n, there is a ring S containing R as a subring such that:

(a) S_{R} is free with basis Y^{∗}, where Y^{∗} is the free monoid on an n-element set Y = {y_{1}, …, y_{n}}.

(b) The ring and module multiplications S × R → S coincide.

(c) The ring and monoid multiplications Y^{∗} × Y^{∗} → S coincide.

(d) ry_{i} = y_{i}τ_{i}(r) + δ_{i}(r) for all i and all r ∈ R.

Notation 3.2. We denote the ring S in Proposition 3.1 by R⟨Y; τ, δ⟩, where τ and δ are abbreviations for the n-tuples (τ_{1}, …, τ_{n}) and (δ_{1}, …, δ_{n}). To see that R⟨Y; τ, δ⟩ is unique in a suitable sense, we establish the following universal property:

Proposition 3.3. Let R be a ring and (τ_{1}, δ_{1}), …, (τ_{n}, δ_{n}) right skew derivations on R. Suppose ϕ : R → T is a ring homomorphism and t_{1}, …, t_{n} ∈ T are elements such that ϕ(r)t_{i} = t_{i}ϕτ_{i}(r) + ϕδ_{i}(r) for all i and all r ∈ R. Then there exists a unique ring homomorphism \overline{ϕ} : R⟨Y; τ, δ⟩ → T such that \overline{ϕ}|_{R} = ϕ and \overline{ϕ}(y_{i}) = t_{i} for all i.

Proof. It suffices to construct a ring with this universal property and show it is isomorphic to R⟨Y; τ, δ⟩. More precisely, let F = ℤ⟨z_{1}, …, z_{n}⟩ be the free ring on n letters, let S_{0} = F ∗_{ℤ} R be the ring coproduct of F and R, and let S_{1} be the factor ring of S_{0} by the ideal generated by (1 ∗ r)(z_{i} ∗ 1) − z_{i} ∗ τ_{i}(r) − 1 ∗ δ_{i}(r) for all i and all r ∈ R. Now let ψ : R → S_{1} be the map defined by ψ(r) = \overline{1 ∗ r} and set w_{i} = \overline{z_{i} ∗ 1} ∈ S_{1}.

Then for all i and all r ∈ R we have

ψ(r)w_{i} = w_{i}ψτ_{i}(r) + ψδ_{i}(r), (†)

and (S_{1}, ψ, w_{1}, …, w_{n}) is universal with respect to (†). In particular, there is a unique ring homomorphism θ : S_{1} → R⟨Y; τ, δ⟩ such that θψ is the inclusion map R → R⟨Y; τ, δ⟩ and θ(w_{i}) = y_{i} for all i. It is enough to show that θ is an isomorphism.

From (†), we see that S_{1} = ∑_{I} w_{I}ψ(R). Note that θ(∑_{I} w_{I}ψ(r_{I})) = ∑_{I} y_{I}r_{I} for all finite sums with r_{I} ∈ R. Since R⟨Y; τ, δ⟩_{R} is free with basis {y_{I}}, it follows that θ is an isomorphism.

Example 3.4. We give an example of the construction considered above, and show that it coincides with a ring constructed by Tyukavkin ([32], [33]).

Take R = k⟨⟨X⟩⟩, where X = {x_{1}, …, x_{n}}. Let τ_{1} = τ_{2} = ⋯ = τ_{n} = τ be the unique k-algebra homomorphism τ : R → R sending all x_{i} to 0. Write elements of R as infinite sums ∑_{w∈X^{∗}} λ_{w}w, where λ_{w} ∈ k and X^{∗} is the free monoid on X. Define k-linear maps δ_{i} : R → R by the rule

δ_{i}( ∑_{w∈X^{∗}} λ_{w}w) = ∑_{w∈X^{∗}} λ_{wx_{i}}w.

We want to check that the δ_{i} are right τ_{i}-derivations. Take two elements r = ∑_{w} λ_{w}w and r′ = ∑_{w} μ_{w}w in R, so that rr′ = ∑_{w}(∑_{uv=w} λ_{u}μ_{v})w. Then we have

δ_{i}(rr′) = ∑_{w}( ∑_{uv=wx_{i}} λ_{u}μ_{v})w = ∑_{w}(λ_{wx_{i}}μ_{1} + ∑_{uv=w} λ_{u}μ_{vx_{i}})w,

since (wx_{i})1 is the only factorization uv = wx_{i} where x_{i} is not a right factor of v. Also,

δ_{i}(r)τ(r′) + rδ_{i}(r′) = (∑_{w} λ_{wx_{i}}w)μ_{1} + (∑_{u} λ_{u}u)(∑_{v} μ_{vx_{i}}v)

= ∑_{w}(λ_{wx_{i}}μ_{1} + ∑_{uv=w} λ_{u}μ_{vx_{i}})w,

the same as above. This proves that δ_{i} is a right τ_{i}-derivation. It follows that there exists a k-algebra R⟨Y; τ, δ⟩. Note that τ_{i}(x_{j}) = 0 and δ_{i}(x_{j}) = δ_{ij} for all i, j. Hence, x_{i}y_{j} = δ_{ij} for all i, j. It follows that the algebra R⟨Y; τ, δ⟩ in this case coincides with the algebra constructed by Tyukavkin in [33, page 404].

The examples of Leavitt also arise from the R⟨Y; τ, δ⟩ construction, as we observe in the next section.

4. Leavitt’s algebras

This section is devoted to showing that some universal examples of non-IBN algebras lie in the class of purely infinite simple rings. Our interest in quoting them here is twofold: on one side, these examples are the algebraic precursors of Cuntz algebras; on the other side, Tyukavkin’s example quoted in Example 3.4 are, in some sense, completions of Leavitt’s examples.

For any field k, and for any two natural numbers m, n, Leavitt ([19]) introduced the k-algebras V_{m,n}, with a universal isomorphism i : nV_{m,n} ⟶ mV_{m,n}, and U_{m,n}, with a universal pair of morphisms i : nU_{m,n} ⟶ mU_{m,n} and j : mU_{m,n} ⟶ nU_{m,n} such that ji = 1_{nU_{m,n}}. Later, Cohn, Skornyakov and Bergman proved some fundamental properties of these algebras. The monoid V(R) of these examples was computed by Bergman in [4]. For the sake of completeness, we recall here Bergman’s result.

Theorem 4.1. ([4, Theorem 6.1]) For a field k and positive integers m, n, the rings V_{m,n} and U_{m,n} are hereditary. Moreover, V(V_{m,n}) = ⟨I | mI = nI⟩, and V(U_{m,n}) = ⟨I, P | mI = nI + P ⟩.

We will use the constructions in Section 3 to give some extra information on the algebras V_{1,n} and U_{1,n}. First of all, notice that U_{1,n} = k⟨X⟩⟨Y; τ, δ⟩, where X, Y, τ, δ are as in Example 3.4. This is clear once we observe that the row (y_{1}, …, y_{n}) and the column (x_{1}, …, x_{n})^{T} give a universal pair of morphisms i : nR_{n} ⟶ R_{n} and j : R_{n} ⟶ nR_{n} respectively such that ji = 1_{nR_{n}}, where we set R_{n} = k⟨X⟩⟨Y; τ, δ⟩. The algebra V_{1,n} is thus the algebra obtained by imposing the relation ij = 1_{R_{n}} in R_{n}, that is, V_{1,n} ≅ R_{n}/I_{n}, where I_{n} is the ideal of R_{n} generated by the idempotent e_{n} = 1 − ∑_{i=1}^{n} y_{i}x_{i}. Following Tyukavkin ([32], [33]) we will call the elements in X^{∗} monomials and the elements in Y^{∗} words. From our basic relations x_{i}y_{j} = δ_{ij} it follows that to each monomial x_{I} there corresponds a word y_{I^{∗}} such that x_{I}y_{I^{∗}} = 1, where I^{∗} = (i_{r}, …, i_{1}) for I = (i_{1}, …, i_{r}). A monoword is any element of the form y_{I}x_{J} for some indices I, J. Notice that the monowords y_{I}x_{J} form a k-basis of U_{1,n}.

Theorem 4.2. For every natural number n ≥ 2, V_{1,n} is a purely infinite simple ring, and K_{0}(V_{1,n}) ≅ ℤ/(n − 1)ℤ.

Proof. By Theorem 4.1, V(V_{1,n}) = ⟨I | nI = I⟩, from which it follows that K_{0}(V_{1,n}) ≅ ℤ/(n − 1)ℤ.

We identify k⟨X⟩⟨Y; τ, δ⟩ with U_{1,n}. Let I_{n} be the ideal of U_{1,n} generated by e_{n} := 1 − ∑_{i=1}^{n} y_{i}x_{i}. As observed before, we have V_{1,n} ≅ U_{1,n}/I_{n}. Every element α ∈ U_{1,n} can be written uniquely as α = ∑_{I,J} λ_{I,J}y_{I}x_{J} for scalars λ_{I,J} which are almost all zero. The support of α is the set of pairs (I, J) such that λ_{I,J} ≠ 0; denote by s(α) the cardinality of the support of α. Since x_{i}e_{n} = e_{n}y_{j} = 0 for all i, j, the ideal I_{n} is spanned (over k) by products of the form y_{I}e_{n}x_{J}. It follows that the support of any nonzero element of I_{n} must contain a pair (I, J) with both I and J nonempty. Consequently, I_{n} contains no nonzero elements of either k⟨X⟩ or k⟨Y ⟩.

We prove that for α ∈ U_{1,n} \ I_{n}, there exist β, γ ∈ U_{1,n} such that βαγ = 1. (In fact, β can be taken as a scalar times a monomial, and γ as a word.) If s(α) = 1, then α = λ_{I,J}y_{I}x_{J} with λ_{I,J} ≠ 0, and so

(λ_{I,J}^{−1}x_{I^{∗}})αy_{J^{∗}} = 1.

Now assume that s(α) = d > 1. There must be an index i such that αy_{i} ∉ I_{n}, since otherwise we would have α = α(e_{n} + ∑_{i=1}^{n} y_{i}x_{i}) ∈ I_{n}. Note that either α ∈ k⟨Y ⟩ or the total degree in X of αy_{i} is less than that of α. Hence, by induction on this degree, there is a word y_{K} such that αy_{K} is a nonzero polynomial in k⟨Y ⟩. Clearly αy_{K} ∉ I_{n}, and so there is a monomial x_{I} such that α′ = x_{I}αy_{K} is a polynomial in k⟨X⟩ with nonzero constant term. Clearly, we can choose a word y_{J} such that α′y_{J} ≠ 0 and s(α′y_{J}) < d. Since x_{J^{∗}}α′y_{J} is a polynomial in k⟨X⟩ with nonzero constant term, we see that α′y_{J} ∉ I_{n}. By induction, there exist β, γ ∈ U_{1,n} such that βx_{I}αy_{K}y_{J}γ = βα′y_{J}γ = 1. This establishes the claim.

By Theorem 1.6, since V_{1,n} is clearly not a division ring, it is simple and purely infinite.

Notice that we may identify each U_{1,n} with k⟨X_{n}⟩⟨Y_{n}; τ, δ⟩ where X_{n} = {x_{1}, …, x_{n}} and Y_{n} = {y_{1}, …, y_{n}} are subsets of fixed infinite sets {x_{1}, x_{2}, … } and {y_{1}, y_{2}, … }. In particular, U_{1,n} ⊂ U_{1,n+1} for all n, and we set U_{∞} equal to the union of the U_{1,n}.

Theorem 4.3. The ring U_{∞} is simple and purely infinite. Moreover, K_{0}(U_{∞}) ≅ ℤ.

Proof. Clearly, U_{∞} is not a division ring, because it contains at least one infinite idempotent.

Let α be a nonzero element of U_{∞}. Then α ∈ U_{1,n} for some n, and α = ∑_{I,J} λ_{I,J}y_{I}x_{J} where the λ_{I,J} ∈ k, the y_{I} are monomials in y_{1}, …, y_{n}, and the x_{J} are words in x_{1}, …, x_{n}. We choose an index I′ of minimal length with λ_{I′,J} ≠ 0 for some J, and then, we choose an index J′ of maximal length among the indices J such that (I′, J) is in the support of α. Then,

(λ_{I′,J′}^{−1}x_{I′}^{∗})α(y_{J′}^{∗}) = 1 + ∑_{I,J} λ′_{I,J}y_{I}x_{J} ,

and λ′_{∅,J} = 0 for all J ≠ ∅. Now, since x_{n+1}, y_{n+1} ∈ U_{∞} and x_{n+1}y_{I} = 0 for all nontrivial words y_{I} in U_{1,n}, we have (x_{n+1}λ_{I′,J′}^{−1}x_{I′}^{∗})α(y_{J′}^{∗}y_{n+1}) = 1. Thus U_{∞} is simple and purely infinite because of Theorem 1.6.

By Theorem 4.1, we have K_{0}(U_{1,n}) ≅ ℤ and a generator of K_{0}(U_{1,n}) is provided by the class [U_{1,n}]. It follows that the induced homomorphisms K_{0}(U_{1,n}) → K_{0}(U_{1,n+1}) are isomorphisms. Since the functor K_{0}(−) preserves direct limits, we get K_{0}(U_{∞}) ≅ \underrightarrow{lim} K_{0}(U_{1,n}) ≅ ℤ.

5. Algebras R⟨Y; τ, δ⟩ with other coefficients

Now we proceed to exploit the skew polynomial construction of Section 3, in order to get new examples of purely infinite simple rings. These will be, in some sense, “intermediate” algebras between Leavitt’s and Tyukavkin’s examples, and, as we will see, they all have the same K-theoretical behavior. We begin as in Example 3.4, except that we now index our variables starting at 0 rather than at 1. For a given natural number n, consider the algebra k⟨⟨X_{n}⟩⟩, where X_{n} = {x_{0}, x_{1}, …, x_{n}}, equipped with the skew derivations (τ_{i}, δ_{i}) defined in Example 3.4. Throughout this section, R_{n} will denote a local subalgebra of k⟨⟨X_{n}⟩⟩ containing k⟨X_{n}⟩ such that R_{n} is invariant under the δ_{i}. Of course, one possibility is R_{n} = k⟨⟨X_{n}⟩⟩; another choice of R_{n} will be important in Section 7. Since the augmentation ideal of k⟨⟨X_{n}⟩⟩ has codimension 1, its intersection with R_{n} must be the maximal ideal of R_{n}. Thus, all power series in R_{n} with nonzero constant term are invertible in R_{n}. Note that by [9, Proposition 2.9.18], R_{n} is a semifir.

By assumption, the (τ_{i}, δ_{i}) restrict to skew derivations on R_{n}. Thus, we may construct the skew polynomial algebra S_{n} = R_{n}⟨Y_{n}; τ, δ⟩, where Y_{n} = {y_{0}, y_{1}, …, y_{n}}, and our first goal is to determine the structure of this algebra. Except in Proposition 5.8, where we consider the sequence of algebras S_{1}, S_{2}, …, we shall throughout the remainder of the section keep n fixed and write R, S, X, Y for R_{n}, S_{n}, X_{n}, Y_{n}.

Lemma 5.1. (cf. [33]) For any nonzero element r ∈ R there are w ∈ Y^{∗} and r′ ∈ R such that rwr′ = 1. In fact, given any nonzero elements r_{1}, …, r_{m} ∈ R, there exists w ∈ Y^{∗} such that r_{j}w ∈ R for all j and some r_{i}w is invertible in R.

Proof. Let r_{1}, …, r_{m} be nonzero elements of R. For j = 1, …, m, let l_{j} denote the minimum length of monomials occurring in r_{j}. We may assume that l_{1} ≤ ⋯ ≤ l_{m}. Pick a monomial x_{I} of length l_{1} occurring in r_{1}, and set w = y_{I^{∗}}. Observe that for any monomial x_{J} of length at least l_{1}, either x_{J}w = 0 or x_{J}w is a monomial. Hence, r_{j}w ∈ R for all j, and r_{1}w has nonzero constant term. Consequently, r_{1}w is invertible in R, by our choice of R.

Proposition 5.2. (a) The algebra S is a prime ring, and its socle is the ideal SeS generated by the minimal idempotent e := 1 − ∑_{i=0}^{n} y_{i}x_{i}. Further, Soc(S) is a regular ideal of S.

(b) Suppose that ϕ : R → V is a k-algebra homomorphism and that there exist elements t_{0}, …, t_{n} ∈ V such that ϕ(x_{i})t_{j} = δ_{ij} for all i, j. Then ϕ extends uniquely to a k-algebra homomorphism \overline{ϕ} : S → V such that \overline{ϕ}(y_{j}) = t_{j} for all j.

Proof. (a) Note that ey_{j} = x_{j}e = 0 for all j. Consequently, eS = eR and eSe = ke.

Given a nonzero element α in S, either eα = α or there is some i such that x_{i}α ≠ 0. It follows by induction on the maximum length of words occurring in α that there is a monomial m ∈ X^{∗} such that emα ≠ 0. By the observation above, emα = er for some r ∈ R, and r is right invertible in S by Lemma 5.1. In particular, there is some γ ∈ S such that emαγ = e. It follows at once that S is a prime ring and that SeS is the unique minimal nonzero ideal of S. Then, because eSe = ke ≅ k we conclude that eS is a minimal right ideal of S. Therefore SeS = Soc(S).

Now SeS is a simple ring having a minimal one-sided ideal, and Litoff’s Theorem (see [13]) says that it is locally a matrix ring over a division ring (actually the division ring is k in our case). In particular Soc(S) = SeS is a regular ring.

(b) In view of Proposition 3.3, it suffices to show that

ϕ(r)t_{j} = t_{j}ϕτ(r) + ϕδ_{j}(r) (1)

for any r ∈ R and all j = 0, …, n. We can write r = α + r_{0}x_{0} + ⋯ + r_{n}x_{n} for some α ∈ k and r_{i} ∈ k⟨⟨X⟩⟩. Then α = τ(r) and r_{i} = δ_{i}(r); in particular, all the r_{i} ∈ R. Thus

ϕ(r)t_{j} = (α + ∑_{i=0}^{n} ϕ(r_{i})ϕ(x_{i}))t_{j} = αt_{j} + ϕ(r_{j}) = t_{j}ϕτ(r) + ϕδ_{j}(r),

proving (1).

The next lemma may be known, but we have not been able to locate any reference. Write M^{∗} = Hom_{R}(M, R) for R-modules M.

Lemma 5.3. Let R be a left semihereditary ring, and let T = R_{Σ} be a universal localization of R. Assume that for all finitely presented right R-modules M such that M^{∗} = 0, we have M ⊗_{R} T = 0. Then T is a regular ring and every finitely generated projective right T -module is induced from a finitely generated projective right R-module.

Proof. We first claim that any finitely presented right R-module M has a decomposition M = M_{1} ⊕ M_{2} with M_{1} projective and M_{2}^{∗} = 0. In case R is semihereditary on both sides, this follows from [22, Theorem 1.2(3)] (the hypothesis of an involution is not needed in the proof). Let

R^{m} ⟶^{f} R^{n} ⟶^{g} M ⟶ 0

be a resolution of M. Taking duals, we obtain an exact sequence

0 ⟶ M^{∗} ⟶^{g^{∗}} ^{n}R ⟶^{f^{∗}} ^{m}R.

Now f^{∗}(^{n}R) is a finitely generated submodule of ^{m}R, so it is projective because R is left semihereditary. Consequently, g^{∗}(M^{∗}) = (^{n}R)e for some idempotent matrix e ∈ M_{n}(R). It follows that

(1 − e)(R^{n}) = g^{−1}({x ∈ M | h(x) = 0 for all h ∈ M^{∗}}).

In particular, ker(g) ⊆ (1 − e)(R^{n}). Hence, M = M_{1} ⊕ M_{2} where M_{2} = g((1 − e)(R^{n})) satisfies M_{2}^{∗} = 0 and M_{1} ≅ e(R^{n}) is projective.

Now let N be a finitely presented right T -module. By [30, Corollary 4.5], there exists a finitely presented right R-module M such that M ⊗_{R} T ≅ N. By the above, we can write M = M_{1} ⊕ M_{2} with M_{1} projective and M_{2}^{∗} = 0. By hypothesis, we have M_{2} ⊗_{R} T = 0, and so N ≅ M_{1} ⊗_{R} T is an induced finitely generated projective right T -module.

Theorem 5.4. Let R, S, and e be as above, and set I = SeS. Let f : R → R^{n+1} be the homomorphism given by left multiplication by the column (x_{0}, …, x_{n})^{T}. Then the universal localization R_{f} is a purely infinite simple regular ring, and every finitely generated projective R_{f}-module is free. Moreover, R_{f} ≅ S/I, and S is regular.

Proof. We first prove that R_{f} ≅ S/I. For that, it is enough to show that the natural map R → S/I satisfies the universal property of the universal localization R → R_{f}. If ϕ : R → V is a k-algebra homomorphism such that f ⊗ 1_{V} is invertible, then there are elements t_{0}, …, t_{n} in V such that ϕ(x_{i})t_{j} = δ_{ij} and ∑_{i=0}^{n} t_{i}ϕ(x_{i}) = 1. By Proposition 5.2(b), there exists a unique k-algebra map \overline{ϕ} : S → V such that \overline{ϕ}|_{R} = ϕ and \overline{ϕ}(y_{i}) = t_{i} for all i. Clearly this map factors through S/I and so we get the desired map from S/I to V. This shows that we may identify S/I with R_{f}. It follows that the localization map R → R_{f} is injective (recall from Lemma 5.1 that nonzero elements of R are right invertible in S).

Write T = S/I = R_{f}, and identify R with its image in T. Let us check directly that T is purely infinite simple. (This also follows from the fact that T is regular with every finitely generated projective T -module being free, which we will prove later.) Let α be an element of S which is not in I. There must exist i such that x_{i}α ∉ I, since otherwise α = eα + ∑_{i=0}^{n} y_{i}x_{i}α ∈ I. Note that either α ∈ R or the degree in Y of x_{i}α is smaller than that of α. We conclude that there is a monomial m ∈ X^{∗} such that mα is a nonzero element of R. By Lemma 5.1, we get an element g ∈ S such that mαg = 1. This shows that T is a purely infinite simple ring.

Let M be a finitely presented right R-module such that M^{∗} = 0. We want to prove that M ⊗_{R} T = 0, in order to apply Lemma 5.3. Take a presentation

R^{s} → R^{t} → M → 0.

Since the functor (−) ⊗_{R} T is right exact, we get an exact sequence

T^{s} → T^{t} → M ⊗_{R} T → 0.

Write z_{i} for the image of y_{i} in T. The map R^{s} → R^{t} above is given by left multiplication by a t × s matrix A with coefficients in R, and we have to see that AT^{s} = T^{t}, that is, the columns of A generate T^{t} as a right T -module. We proceed by induction on t. If t = 1, then there exists a nonzero entry p in A, because M^{∗} = 0. By Lemma 5.1, the element p is right invertible in S, so in T, and thus the entries of A generate T as a right T -module. Now assume that t > 1. If some entry of A is invertible in R, then by a standard process we can find invertible matrices P and Q over R of appropriate sizes

such that P AQ = ( \begin{matrix}1 & 0 \\ 0 & A′ \end{matrix} ), where A′ is a matrix of size (t − 1) × (s − 1). In this case,

M can be generated by t − 1 elements, and induction applies.

In the general case, apply Lemma 5.1 to the entries a_{ij} of A, to obtain a word w ∈ Y^{∗} such that all a_{ij}w ∈ R with at least one a_{ij}w invertible. Consequently, there is a vector v ∈ T^{s} of the form (0, …, 0, \overline{w}, 0, …, 0)^{T} such that the column q = Av ∈ AT^{s} ∩ R^{s} has some entry which is invertible in R. Now consider the matrix C = ( q A ) with coefficients in R, of size t × (s + 1). The finitely presented module M′ := R^{t}/CR^{s+1} is a factor module of M, and consequently (M′)^{∗} = 0. Moreover, the matrix C has an invertible entry, and so we get as before by induction that M′ ⊗_{R} T = 0, or equivalently that T^{t} = CT^{s+1}. But q belongs to the T -submodule of T^{t} generated by the columns of A, and therefore the columns of A generate T^{t} as an T -module. We conclude that M ⊗_{R} T = 0, as desired.

By [9, Proposition 2.9.18], R is a semifir. It follows from Lemma 5.3 that T is a regular ring and that every finitely generated projective right T -module is free.

Finally, S/I is a regular ring, and by Proposition 5.2, I is a regular ideal of S. Thus it follows from [14, Lemma 1.3] that S is regular.

The regularity of S was proved by Tyukavkin ([32], [33]) in the case where R = k⟨⟨X⟩⟩. In order to compute K_{0} groups of both S and R_{f}, we will need the following technical lemma.

As in the proof of Theorem 5.4, we set T = S/I = R_{f}, and we write T_{n} in case n requires mention.

Lemma 5.5. If there is a right R-module map p : R → R^{s} which becomes invertible over T, then n divides s − 1.

Proof. Write p = (p_{1}, …, p_{s})^{T}, where each p_{i} ∈ R. We will construct, by induction on i, words w_{i} in Y^{∗} and invertible elements g_{i} in R such that the following statements hold: (P_{i}) There exists an invertible map p^{(i)} : T → T^{s} satisfying the following properties:

(1) p_{i+1}^{(i)}, …, p_{s}^{(i)} ∈ R.

(2) The inverse of p^{(i)} is the row (w_{1}g_{1}, …, w_{i}g_{i}, α_{i+1}, …, α_{s}) for some elements α_{i+1}, …, α_{s} ∈ T.

The statement is obvious when i = 0. Assume that 0 ≤ i < s, and that (P_{i}) holds. We will prove (P_{i+1}). Note that p_{j}^{(i)}w_{m} = 0 for j ≠ m. Without loss of generality, we can assume that the order of the series p_{i+1}^{(i)} is less than or equal to the order of p_{i+t}^{(i)} for all t ≥ 2. Choose a word w_{i+1} ∈ Y^{∗} with length equal to the order of p_{i+1}^{(i)} and such that p_{i+1}^{(i)}w_{i+1} is invertible in R. Let g_{i+1} ∈ R be the inverse of p_{i+1}^{(i)}w_{i+1} and note that

1 = p_{i+1}^{(i)}w_{i+1}g_{i+1} = p_{i+1}^{(i)}α_{i+1}.

It follows that u = α_{i+1}p_{i+1}^{(i)} + (1 − w_{i+1}g_{i+1}p_{i+1}^{(i)}) is invertible in T with inverse u^{−1} = w_{i+1}g_{i+1}p_{i+1}^{(i)} + (1 − α_{i+1}p_{i+1}^{(i)}). Therefore p^{(i+1)} := p^{(i)}u is invertible with inverse

u^{−1}(w_{1}g_{1}, …, w_{i}g_{i}, α_{i+1}, …, α_{s}).

Note that, for t > 1, we have

p_{i+t}^{(i)}u = p_{i+t}^{(i)}(1 − w_{i+1}g_{i+1}p_{i+1}^{(i)}).

Since the order of p_{i+t}^{(i)} is greater than or equal to the length of w_{i+1}, we conclude that p_{i+t}^{(i+1)} ∈ R, and condition (1) of (P_{i+1}) holds. On the other hand, for m ≤ i we have p_{i+1}^{(i)}w_{m} = 0 and so

u^{−1}w_{m}g_{m} = w_{m}g_{m}.

We also have

u^{−1}α_{i+1} = w_{i+1}g_{i+1}p_{i+1}^{(i)}α_{i+1} + (1 − α_{i+1}p_{i+1}^{(i)})α_{i+1} = w_{i+1}g_{i+1},

and so condition (2) of (P_{i+1}) is also satisfied. Therefore the induction works. Take q_{i} = g_{i}p_{i}^{(s)} ∈ T for i = 1, …, s. Then

∑_{i=1}^{s} w_{i}q_{i} = 1, (1)

q_{i}w_{i} ≠ 0 for all i, and q_{i}w_{j} = 0 for i ≠ j. We claim that these conditions imply s ≡ 1 (mod n). We proceed by induction on the maximum of the lengths of the w_{i}. This maximum is 0 if and only if s = 1 (and then q_{1} = 1). So assume that s > 1. In this case all w_{i} are different from 1. Fix ℓ ∈ {0, …, n}. Left multiplying (1) by x_{ℓ} and right multiplying it by y_{ℓ}, and letting A_{ℓ} = {i : x_{ℓ}w_{i} ≠ 0}, we have

∑_{i∈A_{ℓ}}(x_{ℓ}w_{i})(q_{i}y_{ℓ}) = 1.

Note that {1, …, s} is the disjoint union of the family {A_{ℓ} | ℓ = 0, …, n}. Observe also that for i, j ∈ A_{ℓ} we have (q_{i}y_{ℓ})(x_{ℓ}w_{j}) = q_{i}w_{j}. So this term is 0 if i ≠ j and nonzero if i = j. By induction, |A_{ℓ}| ≡ 1 (mod n). Therefore

s = ∑_{ℓ=0}^{n} |A_{ℓ}| ≡ n + 1 ≡ 1 (mod n),

as desired.

Theorem 5.6. Let R, T, and f : R → R^{n+1} be as above. Then K_{0}(T) is a cyclic group of order n, generated by [T].

Proof. Consider the homomorphism ι^{∗} : K_{0}(R) → K_{0}(T), where ι : R → T denotes the localization map. Since our ring R is a semifir, we have K_{0}(R) infinite cyclic, generated by [R]. Also, by Lemma 5.3 and the proof of Theorem 5.4, the map ι^{∗} is a group epimorphism. Thus, it is enough to show that the kernel of ι^{∗} is generated by n[R] to get the desired result.

Suppose that [R^{r}] − [R^{s}] lies in ker(ι^{∗}), for some nonnegative integers r, s, that is, [T^{r}] − [T^{s}] = 0 in K_{0}(T). Then T^{r+t} ≅ T^{s+t} for some t ≥ 0. Since [R^{r}] − [R^{s}] = [R^{r+t}] − [R^{s+t}], we may thus assume that T^{r} ≅ T^{s}. By [30, Corollary 4.4], there is an isomorphism T^{r+u} → T^{s+u}, for some u ≥ 0, which is induced from a map R^{r+u} → R^{s+u}. We can again reduce to the case that u = 0. Thus, it suffices to prove that if there exists a map g : R^{r} → R^{s} that becomes invertible over T, then n divides r − s. After taking the direct sum of g with an identity map R^{m} → R^{m} for a suitable m, we may assume that r = ℓn + 1 for some positive integer ℓ. Since the map f : R → R^{n+1} becomes an isomorphism over T, there exists a homomorphism h : R → R^{r} which becomes invertible over T, and the same holds for gh : R → R^{s}. By Lemma 5.5, n divides s − 1, and hence also r − s, as desired.

Corollary 5.7. Let R and S be as above. For ℓ, m ∈ ℕ, we have S^{ℓ} ≅ S^{m} if and only if ℓ = m. Moreover, K_{0}(S) is an infinite cyclic group, with generator [S].

Proof. Suppose that S^{ℓ} ≅ S^{m} for some ℓ ≥ m > 0. Consider p ∈ ℕ such that p ≥ n and p > ℓ − m. If we take R_{p} = k⟨⟨X_{p}⟩⟩, then S embeds in S_{p}, whence S_{p}^{ℓ} ≅ S_{p}^{m} and so T_{p}^{ℓ} ≅ T_{p}^{m}. Thus, by Theorem 5.6 (applied to T_{p}), we get that p divides ℓ − m, which implies that ℓ = m, as desired.

Now, by the above argument, we have that the class [S] ∈ K_{0}(S) generates an infinite cyclic subgroup. By Proposition 5.2, I is the socle of S, and I is generated by the minimal right ideal eS. Hence, K_{0}(I) is infinite cyclic, with generator [eS]. Since y_{0}x_{0}, …, y_{n}x_{n} are orthogonal idempotents equivalent to 1, we have eS ⊕ S^{n+1} ≅ S, and so [eS] = −n[S] in K_{0}(S). In particular, it follows that the natural map K_{0}(I) → K_{0}(S) is injective. Using that, we get the following commutative diagram, whose bottom row is exact because S is regular.

0 ⟶ ℤ ⟶^{−n} ℤ ⟶ ℤ/nℤ ⟶ 0
↓^{1↦[eS]}  ↓^{1↦[S]}  ↓^{[1]_{n}↦[T]}
0 ⟶ K_{0}(I) ⟶ K_{0}(S) ⟶ K_{0}(T) ⟶ 0

Moreover, the map ℤ/nℤ → K_{0}(T) is an isomorphism by Theorem 5.6. Thus, by the Five Lemma we conclude that K_{0}(S) is infinite cyclic with generator [S], as desired.

Now consider a sequence of algebras S_{1}, S_{2}, … of the type discussed in this section. Assume the sequence has been chosen so that R_{1} ⊂ R_{2} ⊂ …; we could, of course, take R_{n} = k⟨⟨X_{n}⟩⟩ for all n, but different choices will be needed in Section 8. We have natural inclusions S_{n} ⊂ S_{n+1} for all n, and we set S_{∞} = ⋃_{n=1}^{∞} S_{n}. Note that S_{∞} is a regular ring.

Proposition 5.8. Let S_{1}, S_{2}, …, S_{∞} be as above. The ring S_{∞} is simple, regular, and purely infinite, and K_{0}(S_{∞}) is an infinite cyclic group, with generator [S_{∞}].

Proof. Since each S_{n} is regular (Theorem 5.4), so is S_{∞}. Next, notice that S_{∞} is not a division ring, because it contains at least one infinite idempotent. Let α be a nonzero element of S_{∞}. Then α ∈ S_{n} for some n, and the proof of Proposition 5.2(a) shows that there exist s, t ∈ S_{n} such that sαt = 1 − ∑_{i=0}^{n} y_{i}x_{i}. Consequently, (x_{n+1}s)α(ty_{n+1}) = 1. Thus, S_{∞} is simple and purely infinite because of Theorem 1.6.

We finally compute K_{0}(S_{∞}). By Corollary 5.7, K_{0}(S_{n}) ≅ ℤ for all n, with [S_{n}] ↦ 1. Proceeding as in Theorem 4.3, we obtain K_{0}(S_{∞}) ≅ \underrightarrow{lim} K_{0}(S_{n}) = ℤ, with [S_{∞}] ↦ 1.

Theorem 5.6 and Proposition 5.8 provide us with examples of purely infinite simple regular rings whose K_{0} groups are cyclic of arbitrary order. Using these rings as basic building blocks, we can construct purely infinite simple regular rings whose K_{0}’s are arbitrary countable abelian groups – see Section 8.

6. The Rosenmann-Rosset and Schofield constructions

In this section we consider two constructions associated to the free algebra over a field k, and we will prove that they are isomorphic purely infinite simple regular k-algebras. In Section 7, we will relate these algebras to the ones constructed in Section 5. Throughout this section, R will denote a free k-algebra on the finite alphabet {x_{0}, …, x_{n}} with n > 0.

First, we briefly quote an example of Rosenmann and Rosset [27, Section 3]. For the general theory of rings of quotients, we refer the reader to [31]. Let F_{fc} be the Gabriel topology whose basic neighborhoods of 0 are the right ideals of finite codimension; see [27, Theorem 1.1]. Consider the F_{fc}-localization of R, denoted by R_{fc}. Then R_{fc} is a ring and we can associate to each right R-module a right R_{fc}-module M_{fc} by the rule M_{fc} = \underrightarrow{lim}_{I∈F_{fc}}Hom(I, M). (Note that since every right ideal in R is projective we have M_{fc} = M_{F_{fc}} by [27, Lemma 2.3].) Recall that a module M is torsion (with respect to F_{fc}) if for every x ∈ M, its annihilator, ann_{R}(x) is in F_{fc}. If M is torsion then M_{fc} = 0. Clearly, M is torsion if and only if all its cyclic submodules are finite dimensional.

We recall one of the main results in [27]:

Theorem 6.1. [27, Theorem 5.1] Let R and R_{fc} be as above. Then, for ℓ, m ∈ ℕ, we have R_{fc}^{ℓ} ≅ R_{fc}^{m} if and only if ℓ ≡ m (mod n).

Another construction was quoted by A. H. Schofield in a private communication [29]. We thank him for allowing us to present this construction here.

We denote by P the category of finitely generated projective right R-modules. Let Φ = Mor(P) denote the class of all homomorphisms between finitely generated projectives.

Proposition 6.2. [29] Let R be as above. Let Σ be the family of all \underline{mono}morphisms in Φ whose images have finite codimension.

(a) Σ is composition-closed: If f, g ∈ Σ and the composition fg is defined, then fg ∈ Σ.

(b) Every f ∈ Σ is a non-zero-divisor in Φ: If g ∈ Φ and fg is defined (respectively, gf is defined), then fg = 0 (respectively, gf = 0) implies g = 0.

(c) Σ satisfies the right Ore condition relative to Φ: If f ∈ Σ and g ∈ Φ with the same codomain, then there exist f′ ∈ Σ and g′ ∈ Φ such that gf′ = fg′ (with appropriate conditions on domains and codomains).

Proof. Part (a) is clear.

(b) Let f ∈ Σ and g ∈ Φ. Since f is injective it is clear that fg = 0 implies g = 0. Now assume that gf = 0. Let Q be the codomain of f. Then Q is a finitely generated free R-module, and since Im(f) ⊆ ker(g), we see that ker(g) has finite codimension. Hence, Im(g) is finite-dimensional and a free R-module, so it is zero.

(c) Now assume that f : X → Y is in Σ and that g : Z → Y is in Φ. Form the pullback

\begin{matrix}P & ⟶^{f′} & Z \\ ^{g′}↓ & & ↓^{g} \\ X & ⟶^{f} & Y \end{matrix}

Then gf′ = fg′. We will show that P ∈ P and f′ ∈ Σ, which will finish the proof. As is well known, since f is injective it follows that f′ is injective (e.g., [28, Exercise 2.47]). On the other hand f′(P) = g^{−1}(f(X)). Thus Z/f′(P) embeds in Y/f(X), which is finite dimensional, and so f′(P) has finite codimension in Z. Since f′ is injective, P is free of finite rank by [20, Theorem 4], and therefore f′ ∈ Σ.

Continue with R and Σ as above. Let C be the set of all pairs (f, s) with f ∈ Φ and s ∈ Σ such that f and s have the same domain. We define a relation on C. Let (f, s) and (f′, s′) be in C, with

Q ⟵^{f} P ⟶^{s} U  and  Q′ ⟵^{f′} P′ ⟶^{s′} U′.

Then (f, s) ∼ (f′, s′) if and only if Q = Q′ and U = U′ and there are maps h : U′′ → P and h′ : U′′ → P′ such that sh = s′h′ ∈ Σ and fh = f′h′. By [35, Section 10.3], ∼ is an equivalence relation, and we get a quotient category PΣ^{−1}. Now let Q be the ring consisting of all equivalence classes [(f, s)] with (f, s) ∈ C and f, s : P → R. Notice that Q = End_{PΣ^{−1}}(R). There is a canonical map ϕ : R → Q defined by ϕ(a) = [(a, 1)]. The full subcategory of Mod-Q consisting of all the induced finitely generated projective modules is equivalent to the category with the same objects as P and morphisms PΣ^{−1}. The map ϕ : R → Q gives the universal localization of R with respect to Σ. Note that since Σ is a non-zero-divisor class in Φ, the map ϕ is injective.

Since _{R}Q is the direct limit of Hom_{R}(eR^{m}, R)s^{−1}, where s : eR^{m} → R is in Σ and e ranges through all the idempotent matrices over R, and since Hom_{R}(eR^{m}, R) is projective as a left R-module, it follows that _{R}Q is flat.

Theorem 6.3. [29] Let R and Σ be as above. Then the universal localization Q = R_{Σ} is a purely infinite simple regular ring.

Proof. Let N be a finitely presented right R-module. By [20, Theorem 2], there is a finitely generated free submodule L of N of finite codimension. There is an exact sequence

0 → R^{ℓ} → R^{m} → N/L → 0,

and so the map R^{ℓ} → R^{m} must be in Σ, which gives (N/L) ⊗_{R} Q = 0. Now since _{R}Q is flat, we have an exact sequence

0 → L ⊗_{R} Q → N ⊗_{R} Q → (N/L) ⊗_{R} Q → 0.

Since (N/L)⊗_{R} Q = 0 we conclude that N ⊗_{R} Q is a free Q-module. Since every finitely presented Q-module is induced from a finitely presented R-module, we conclude that Q is a regular ring such that every finitely generated projective module is free. Thus Q is simple and purely infinite by Example 1.3(d).

Theorem 6.4. Let R, R_{fc} and Q be as above. Then, R_{fc} ≅ Q as k-algebras.

Proof. Let s ∈ Σ. Since finitely generated projective right R-modules are free, we can assume without loss of generality that s : R^{ℓ} → R^{m} for some ℓ, m ∈ ℕ, with N := coker(s) finite-dimensional. We have therefore a short exact sequence

0 ⟶ R^{ℓ} ⟶^{s} R^{m} ⟶ N ⟶ 0.

Proceeding as in [27, page 368] and taking into account that N is a torsion module with respect to F_{fc} and thus N_{fc} = 0, we get an exact sequence

0 ⟶ R_{fc}^{ℓ} ⟶^{s} R_{fc}^{m} ⟶ 0.

We conclude that every map in Σ becomes invertible over R_{fc}. Thus, using the universal property of Q with respect to R and Σ, we conclude that there exists a unique k-algebra homomorphism ρ : Q → R_{fc} which restricts to the identity map on R. Since Q is simple by Theorem 6.3, ρ is an injective homomorphism. Now, if I_{R} ≤ R_{R} is a right ideal of finite codimension, i : I → R is the natural inclusion map, and f : I → R is any right R-module homomorphism, then ρ([(f, i)]) = [f], whence ρ is an isomorphism, as desired.

Schofield also proved that K_{0}(Q) ≅ ℤ/nℤ. Because of Theorem 6.4 and Proposition 2.2, this implies the result in Theorem 6.1 ([27, Theorem 5.1]). We will give an alternate proof of this fact in Theorem 7.6, by using the techniques developed in Section 5.

7. Isomorphism of Q with T

In this section we will prove that the algebra Q (and so also the algebra R_{fc}) constructed in Section 6 is isomorphic to a particular instance of our construction in Section 5. For this, we need to introduce the algebra of rational series, denoted by krat⟨X⟩, associated to a finite alphabet X. This algebra plays an important role in other parts of mathematics, particularly formal language theory and the theory of codes, see [8].

We fix some notation for the rest of this section. Let X = X_{n} = {x_{0}, …, x_{n}} be a finite alphabet, with n > 0. The division closure of k⟨X⟩ in k⟨⟨X⟩⟩ is called the algebra of rational series and it is denoted by krat⟨X⟩. By definition, krat⟨X⟩ is the smallest subalgebra of k⟨⟨X⟩⟩ containing k⟨X⟩ and closed under inversion, in the sense that if an element of the subalgebra is invertible in k⟨⟨X⟩⟩, then it is already invertible in the subalgebra. The algebra krat⟨X⟩ is local, with maximal ideal consisting of the elements with constant term 0. By [9, Exercise 7.1.10], a square matrix over k⟨⟨X⟩⟩ (respectively krat⟨X⟩) is invertible if and only if its image under the augmentation map is invertible over k (see also [10, pp. 408-409]). It follows from this that krat⟨X⟩ coincides with the rational closure of k⟨X⟩ in k⟨⟨X⟩⟩, that is, krat⟨X⟩ is the smallest subalgebra of k⟨⟨X⟩⟩ containing k⟨X⟩ and such that every square matrix with coefficients in the subalgebra which is invertible over k⟨⟨X⟩⟩ is already invertible over the subalgebra. The right τ_{i}-derivations δ_{i} on k⟨⟨X⟩⟩ defined in Example 3.4 coincide with the left transductions associated to the monomials x_{i}. These left transductions are defined in [9, page 105] as follows. For a fixed monomial x_{i_{1}} ⋯ x_{i_{r}} of degree r define the left transduction for this monomial as the k-linear map a ↦ a^{∗} of k⟨X⟩ into itself which sends any monomial of the form bx_{i_{1}} ⋯ x_{i_{r}} to b and all other monomials to 0. This map extends in the obvious way to the power series algebra k⟨⟨X⟩⟩. By [9, page 135], the algebra of rational series krat⟨X⟩ is invariant under all transductions. We conclude that krat⟨X⟩ is a local subalgebra of k⟨⟨X⟩⟩ containing k⟨X⟩ and invariant under the δ_{i}. Therefore we can build the algebras S = S_{n} = krat⟨X⟩⟨Y; τ, δ⟩ and T = T_{n} = S/I = krat⟨X⟩_{f} based on krat⟨X⟩; see Section 5.

An important property of krat⟨X⟩ is that it is a universal localization of k⟨X⟩. Although this fact seems to be well known, we do not have an explicit reference, and so we sketch a proof as follows.

Proposition 7.1. Let Σ′ be the set of those square matrices over k⟨X⟩ which become invertible over k⟨⟨X⟩⟩. Then krat⟨X⟩ is the universal localization of k⟨X⟩ with respect to Σ′.

Proof. Note that Σ′ is the set of matrices of the form A_{0} + B, where A_{0} is an invertible matrix over k and all the entries of B have constant term 0. Since krat⟨X⟩ is the rational closure of k⟨X⟩ in k⟨⟨X⟩⟩, we have a unique algebra homomorphism ϕ : k⟨X⟩_{Σ′} → krat⟨X⟩ which is the identity on k⟨X⟩. It is easy to see from [9, Theorem 7.1.2] that ϕ is surjective. (This holds in general when we consider the rational closure of an inclusion of rings.)

By [9, Lemma 5.9.4], the inclusion map k⟨X⟩ → k⟨⟨X⟩⟩ is honest (i.e., it sends full matrices to full matrices), and hence so is the inclusion map k⟨X⟩ → krat⟨X⟩. Since Σ′ is a factor-closed, multiplicative set of matrices over k⟨X⟩, it now follows from [9, Proposition 7.5.7(ii)] that ϕ is injective. Therefore ϕ is an isomorphism.

By a result of Bergman and Dicks ([30, Theorem 4.9]), every universal localization of a hereditary ring is also hereditary. In particular, krat⟨X⟩ is a hereditary ring, and indeed it is a fir by [9, Theorem 7.10.7]. The algebra k⟨⟨X⟩⟩ is just a semifir, but not a fir; see [9, Proposition 2.9.18 and Proposition 5.10.9]. We summarize the results obtained in Section 5 for the algebras S and T based on the algebra of rational series krat⟨X⟩, along with the facts just mentioned.

The following notation will be helpful. If a_{0}, …, a_{n} are elements in a k-algebra V, let Σ′_{k}(a_{0}, …, a_{n}) denote the set of square matrices over V of the form A_{0} + B where A_{0} is an invertible matrix over k and the entries of B consist of noncommutative polynomials with zero constant term evaluated at (a_{0}, …, a_{n}). (Note that to check whether all such matrices are invertible over V, it suffices to consider those for which A_{0} is an identity matrix.) In particular, if V = k⟨X⟩ and the a_{i} = x_{i}, then Σ′_{k}(x_{0}, …, x_{n}) equals the set Σ′ in Proposition 7.1.

Theorem 7.2. Let Σ_{1} = Σ′_{k}(x_{0}, …, x_{n}) ∪ {(x_{0}, …, x_{n})^{T} }. Set S = krat⟨X⟩⟨Y; τ, δ⟩, let I be the ideal generated by the idempotent e = 1 − ∑_{i=0}^{n} y_{i}x_{i}, and write T = S/I. Then T is the universal localization of k⟨X⟩ with respect to Σ_{1}. In particular, T is a hereditary algebra.

To write the universal property of T in more elementary form, suppose that V is a k-algebra containing elements a_{0}, …, a_{n} and b_{0}, …, b_{n} such that

(1) a_{i}b_{j} = δ_{ij} for all i, j.

(2) b_{0}a_{0} + b_{1}a_{1} + ⋯ + b_{n}a_{n} = 1.

(3) All matrices in Σ′_{k}(a_{0}, …, a_{n}) are invertible over V.

Then there exists a unique k-algebra homomorphism ψ : T → V such that ψ(x_{i}) = a_{i} for all i and ψ(y_{j}) = b_{j} for all j.

Proof. That T = k⟨X⟩_{Σ_{1}} follows from Proposition 7.1 and Theorem 5.4. The fact that T is hereditary follows from [30, Theorem 4.9].

Now we bring the construction of Theorem 6.3 into play, but with a slight change of notation. Let Y = {y_{0}, …, y_{n}} and set R = k⟨Y ⟩. We may identify R with the ksubalgebra of S generated by Y. If r is any nonzero element of R and y_{J} is a monomial of maximum length occurring in r, then x_{J^{∗}}r is an element of k⟨X⟩ with nonzero constant term. Hence, x_{J^{∗}}r is invertible in S, and so r ∉ I. Therefore R ∩ I = 0, and thus we can identify R with its image in T under the quotient map.

Let Q be the universal localization of R with respect to the set Σ of monomorphisms between finitely generated projective right R-modules with finite dimensional cokernels. We want to prove that all maps in Σ are invertible over T.

We need the following fact:

Lemma 7.3. T is flat as a left R-module.

Proof. We first check that _{R}S is free. To see this, let {p_{α}} be a k-basis of krat⟨X⟩. Then the elements of S can be uniquely written as ∑ λ_{I,α}y_{I}p_{α}. We have

S = ⨁_{I} y_{I}krat⟨X⟩ = ⨁_{I,α} ky_{I}p_{α} = ⨁_{α}(⨁_{I} ky_{I})p_{α} = ⨁_{α} Rp_{α},

so _{R}S is free. Since S is regular, _{S}T is flat, hence _{R}T is flat.

Proposition 7.4. Every map in Σ is invertible over T, and so there is a unique kalgebra homomorphism Q → T which is the identity on R.

Proof. Notice that every finitely generated projective right R-module is free. Consider a short exact sequence

0 → R^{i} ⟶^{g} R^{j} ⟶ N ⟶ 0

with g ∈ Σ. Since _{R}T is flat, we have an exact sequence

0 → T^{i} ⟶^{g} T^{j} ⟶ N ⊗_{R} T ⟶ 0, (*)

and it is enough to see that N ⊗_{R} T = 0 to get the desired result.

Since g ∈ Σ we have dim_{k}(N) < ∞, and so J := ann_{R}(N) is an ideal of R of finite codimension. Since T is regular, the sequence (*) splits, so N ⊗_{R} T is a projective right T -module; in particular, it embeds in T_{T}. For γ ∈ N, (γ ⊗ 1)R has finite k-dimension since γJ = 0 and dim_{k}(R/J) < ∞. So it is enough to show that T does not contain nonzero finite-dimensional right R-submodules. We will prove this in the following form: If α ∈ S \ I, then αR contains an infinite family of elements which are linearly independent modulo I.

If α ∈ S \ I, then, as shown in the proof of Theorem 5.4, we can find m ∈ X^{∗} such that mα ∈ krat⟨X⟩ and mα ≠ 0. By Lemma 5.1, there is a word w ∈ Y^{∗} such that the product p = mαw is an invertible element of krat⟨X⟩. As noted earlier in the section, R ∩ I = {0}, whence 1, y_{1}, y_{1}^{2}, … are k-linearly independent modulo I. Thus, the sequence p^{−1}mαw, p^{−1}mαwy_{1}, p^{−1}mαwy_{1}^{2}, … is linearly independent modulo I. It follows immediately that αw, αwy_{1}, αwy_{1}^{2}, … is linearly independent modulo I, as desired.

Therefore all maps in Σ are indeed invertible over T, so we get a k-algebra homomorphism ψ : Q → T which is the identity on R.

We are now ready to prove the main result of this section.

Theorem 7.5. Let T be the universal localization of k⟨X⟩ with respect to the set Σ_{1} described in Theorem 7.2. Let Q be the universal localization of R = k⟨Y ⟩ with respect to the set Σ described in Section 6. Then there is a unique k-algebra isomorphism ψ : Q → T which is the identity on R.

Proof. By Proposition 7.4, there exists a unique k-algebra homomorphism ψ : Q → T which is the identity on R. Since Q is simple, the map ψ must be injective. Now ψ sends the row matrix (y_{0}, …, y_{n}) over R to the row matrix (y_{0}, …, y_{n}) over T, which is the inverse of the column (x_{0}, …, x_{n})^{T} over T. The row (y_{0}, …, y_{n}) is invertible over Q, so there exists a column (x′_{0}, …, x′_{n})^{T} over Q which is the inverse of (y_{0}, …, y_{n}), and obviously ψ(x′_{i}) = x_{i} for all i. Let A be an m × m matrix in Σ′_{k}(x′_{0}, …, x′_{n}). Then ψ(A) is invertible in M_{m}(T), and so, ψ being injective, A must be a non-zero-divisor in M_{m}(Q). Since Q is regular by Theorem 6.3, we get that A is invertible in M_{m}(Q). By the universal property of T = k⟨X⟩_{Σ_{1}}, there exists a unique k-algebra homomorphism φ : T → Q sending (x_{0}, …, x_{n})^{T} to (x′_{0}, …, x′_{n})^{T}. The row matrix (y_{0}, …, y_{n}), being the inverse of (x_{0}, …, x_{n})^{T} in T as well as the inverse of (x′_{0}, …, x′_{n})^{T} in Q, must be sent to itself by φ. Therefore we conclude that ψ and φ are mutually inverse isomorphisms.

As a consequence of our results in Section 5, we can derive an alternate proof of Schofield’s result that K_{0}(Q) ≅ ℤ/nℤ. As we observed at the end of Section 6, this implies in turn the Rosenmann-Rosset result [27, Theorem 5.1].

Theorem 7.6. Let Q be as above. Then K_{0}(Q) ≅ ℤ/nℤ, with [Q] ↦ [1]_{n}.

Proof. This follows immediately from Theorems 7.5 and 5.6.

8. Realizing groups as K_{0} of purely infinite simple regular rings

Rørdam has proved that all countable abelian groups appear as K_{0}’s of purely infinite simple C*-algebras [26, Theorem 8.1]. In this section we prove a similar result for purely infinite simple regular rings. Our construction follows the same pattern as Rørdam’s but requires much more care with the technical details, since we have to work with more complicated universal properties.

For a field K and an integer n ≥ 2, we will write S_{n,K} and T_{n,K} for the K-algebra versions of the algebras constructed in Section 7. Namely,

S_{n,K} = Krat⟨X_{n}⟩⟨Y_{n}; τ, δ⟩ ,

where X_{n} = {x_{0}, x_{1}, …, x_{n}} and Y_{n} = {y_{0}, y_{1}, …, y_{n}}, and T_{n,K} = S_{n,K}/I_{n,K}, where I_{n,K} is the ideal of S_{n,K} generated by the idempotent e_{n} := 1−∑_{p=0}^{n} y_{p}x_{p}. By Theorems 5.4 and 5.6, T_{n,K} is a purely infinite simple regular ring, and K_{0}(T_{n,K}) ≅ ℤ/nℤ with [T_{n,K}] ↦ [1]_{n}. We denote by T_{0,K} the inductive limit of the sequence (S_{n,K})_{n≥2} along the natural inclusions S_{n,K} ↪ S_{n+1,K}. Note that T_{0,K} contains the idempotents e_{0}, e_{1}, … and that

x_{i}e_{n} = \begin{cases}0 & (i ≤ n) \\ x_{i} & (i > n) , \end{cases} e_{n}y_{j} = \begin{cases}0 & (j ≤ n) \\ y_{j} & (j > n) . \end{cases}

By Proposition 5.8, T_{0,K} is simple, regular, and purely infinite. Moreover, K_{0}(T_{0,K}) ≅ ℤ with [T_{0,K}] ↦ 1.

We shall require the universal property for T_{n,K}, (n ≥ 2), given in Theorem 7.2. Analogously, T_{0,K} possesses the following universal property:

Lemma 8.1. Let V be a K-algebra containing elements a_{0}, a_{1}, … and b_{0}, b_{1}, … such that a_{i}b_{j} = δ_{i,j} for all i, j, and such that all the matrices in Σ′_{K}(a_{0}, a_{1}, …) are invertible over V. Then there exists a unique K-algebra homomorphism ψ : T_{0,K} → V such that ψ(x_{i}) = a_{i} for all i and ψ(y_{j}) = b_{j} for all j.

Proof. This follows from Lemmas 7.1 and 5.2(b).

We will take as canonical representatives of the nonzero cyclic groups the groups ℤ_{m}, the integers mod m, where m is either 0 or an integer larger than 1. For uniformity of notation, we write [a]_{0} = a for a ∈ ℤ. Every group homomorphism φ : ℤ_{n} → ℤ_{m} is given by multiplication by some integer ℓ, and we choose ℓ such that 1 ≤ ℓ ≤ m in case m ≥ 2. Recall that for any ring R and ℓ > 0, there is a canonical group isomorphism K_{0}(M_{ℓ}(R)) ≅ K_{0}(R) with [M_{ℓ}(R)] ↦ ℓ[R]. Hence, we identify

(K_{0}(M_{ℓ}(T_{m,K})), [M_{ℓ}(T_{m,K})]) = (ℤ_{m}, [ℓ]_{m})

for all ℓ > 0 and m ∈ {0} ∪ {2, 3, … }.

It will be convenient to set e_{−ℓ} = e_{ℓ} = 1 − y_{0}x_{0} − y_{1}x_{1} − ⋯ − y_{ℓ}x_{ℓ} for all ℓ ≥ 0. If ℓ ≤ 0, then by M_{ℓ}(T_{0,K}) we will understand the corner ring e_{ℓ}T_{0,K}e_{ℓ}. In this case K_{0}(M_{ℓ}(T_{0,K})) ≅ K_{0}(T_{0,K}) with [M_{ℓ}(T_{0,K})] ↦ [e_{ℓ}T_{0,K}]. Note that y_{0}x_{0}, …, y_{−ℓ}x_{−ℓ} are pairwise orthogonal idempotents equivalent to 1, and also orthogonal to e_{ℓ}. Since e_{ℓ} +y_{0}x_{0} +y_{1}x_{1} +⋯+y_{−ℓ}x_{−ℓ} = 1, we have [e_{ℓ}T_{0,K}]+(−ℓ+1)[T_{0,K}] = [T_{0,K}] in K_{0}(T_{0,K}), whence [e_{ℓ}T_{0,K}] = ℓ[T_{0,K}]. Hence, we can make the identification

(K_{0}(M_{ℓ}(T_{0,K})), [M_{ℓ}(T_{0,K})]) = (ℤ_{0}, [ℓ]_{0}),

in parallel with the previous identifications.

The following lemma and corollary give the key for our construction.

Lemma 8.2. Let φ : ℤ_{n} → ℤ_{m} be a group homomorphism given by multiplication by ℓ, where we adopt the above conventions. Let K be a field and t an indeterminate. Then there exists a K-algebra homomorphism ψ : T_{n,K} → M_{ℓ}(T_{m,K(t)}) such that K_{0}(ψ) = φ.

Proof. With the canonical identifications we have made for the K_{0} groups, it is automatic that K_{0}(ψ) = φ will hold for any K-algebra homomorphism ψ : T_{n,K} → M_{ℓ}(T_{m,K(t)}). Thus, only the existence of such homomorphisms needs to be established. We will consider several cases, in the first two of which ℓ > 0.

Case 1. Assume that n, m ≥ 2.

We can write ℓn = hm for some positive integer h. There are elements x′_{0}, …, x′_{hm} and y′_{0}, …, y′_{hm} in T_{m,K(t)} implementing an isomorphism T_{m,K(t)} ≅ T_{m,K(t)}^{hm+1} and with each x′_{i} being a nontrivial product of the standard elements x_{i} of T_{m,K(t)}. For example, take x′_{0}, …, x′_{hm} and y′_{0}, …, y′_{hm} to be the sequences

x_{0}^{h} ,x_{1}x_{0}^{h−1} , …, x_{m}x_{0}^{h−1} , x_{1}x_{0}^{h−2} , …, x_{m}x_{0}^{h−2} , …, x_{1} , …, x_{m} and
y_{0}^{h} ,y_{0}^{h−1}y_{1} , …, y_{0}^{h−1}y_{m} , y_{0}^{h−2}y_{1} , …, y_{0}^{h−2}y_{m} , …, y_{1} , …, y_{m} ,

respectively. Define matrices A_{i}, B_{j} ∈ M_{ℓ}(T_{m,K(t)}) for i, j = 0, 1, …, n as follows:

A_{0} = diag(t, …, t, x′_{0}) , (A_{i})_{α,β} = δ_{β,ℓ}x′_{(i−1)ℓ+α} (1 ≤ i ≤ n) ,
B_{0} = diag(t^{−1}, …, t^{−1}, y′_{0}) , (B_{j})_{α,β} = δ_{α,ℓ}y′_{(i−1)ℓ+β} (1 ≤ j ≤ n) .

It is easy to check that A_{i}B_{j} = δ_{i,j}I for all i, j and ∑_{i=0}^{n} B_{i}A_{i} = I.

Observe that if p(Z_{0}, …, Z_{n}) is a noncommutative polynomial over K with zero constant term, then p(A_{0}, …, A_{n}) is a matrix in M_{ℓ}(T_{m,K(t)}) whose entries come from either tK[t] or K[t]⟨X_{m}⟩X_{m} (where the latter notation refers to the left ideal of K[t]⟨X_{m}⟩ generated by X_{m}). Consequently, any r × r matrix in Σ′_{K}(A_{0}, …, A_{n}), when viewed as a block form of an ℓr × ℓr matrix over T_{m,K(t)}, consists of a sum C_{0} + C_{1} + C_{2} where C_{0} is an invertible matrix over K, all entries of C_{1} lie in tK[t], and all entries of C_{2} lie in K[t]⟨X_{m}⟩X_{m}. Now C_{0} + C_{1} is an invertible matrix over K(t) (note that we need the variable t to achieve this statement), whence C_{0} + C_{1} + C_{2} lies in Σ′_{K(t)}(x_{0}, …, x_{m}). It follows that every matrix in Σ′_{K}(A_{0}, …, A_{n}) is invertible over M_{ℓ}(T_{m,K(t)}). By the universal property of the algebras T_{n,K} (Theorem 7.2), there is a unique K-algebra homomorphism T_{n,K} → M_{ℓ}(T_{m,K(t)}) sending (x_{i}) to (A_{i}) and (y_{j}) to (B_{j}).

Case 2. Assume that n = 0 and ℓ > 0 (with m arbitrary).

This case is easier than the previous one. Just take an infinite sequence of diagonal matrices A_{i}, B_{j} in M_{ℓ}(T_{m,K}), where the A_{i}’s consist of nontrivial products of the x’s in the diagonal, the B_{j}’s consist of nontrivial products of the y’s in the diagonal, and they satisfy the rules A_{i}B_{j} = δ_{ij}. For example, take

A_{i} = diag(x_{0}x_{1}^{i}, …, x_{0}x_{1}^{i})  and  B_{j} = diag(y_{1}^{j}y_{0}, …, y_{1}^{j}y_{0}).

for i, j = 0, 1, . . .. Then we can identify Σ′_{K}(A_{0}, A_{1}, …) with a subset of Σ′_{K}(x_{0}, x_{1}), and so every matrix in Σ′_{K}(A_{0}, A_{1}, …) is invertible over M_{ℓ}(T_{m,K}). By the universal property of T_{0,K} (Lemma 8.1), there is a K-algebra homomorphism ψ : T_{0,K} → M_{ℓ}(T_{m,K}). (Here the use of the new variable t is not necessary.)

Case 3. Now assume that n = 0 and ℓ ≤ 0. Necessarily, m = 0.

Recall that in this case our convention is to set M_{ℓ}(T_{0,K}) = e_{ℓ}T_{0,K}e_{ℓ}. Define the following elements A_{i}, B_{j} in M_{ℓ}(T_{0,K}) for i, j ≥ 0:

A_{i} = e_{ℓ}x_{−ℓ+1+i}  and  B_{j} = y_{−ℓ+1+j}e_{ℓ}.

Then one has A_{i}B_{j} = δ_{ij}e_{ℓ} for all i, j. If C is a matrix in Σ′_{K}(A_{0}, A_{1}, …), then C = ẽ_{ℓ}Γ = ẽ_{ℓ}Γẽ_{ℓ} for some matrix Γ in Σ′_{K}(x_{−ℓ+1}, x_{−ℓ+2}, …), where ẽ_{ℓ} = diag(e_{ℓ}, …, e_{ℓ}). Note that Γ is invertible over Krat⟨x_{−ℓ+1}, x_{−ℓ+2}, … ⟩, with ẽ_{ℓ}Γ^{−1} = ẽ_{ℓ}Γ^{−1}ẽ_{ℓ}, and so C is invertible over M_{ℓ}(T_{0,K}) with inverse ẽ_{ℓ}Γ^{−1}. Thus in this case too we get our desired homomorphism T_{0,K} → M_{ℓ}(T_{0,K}).

Case 4. Finally, suppose that m = 0 and n ≥ 2. Then necessarily ℓ = 0.

Set E = diag(e_{n}, 1, 1, …, 1) in M_{n+1}(T_{0,K(t)}), and observe that E ∼ e_{0}. Hence, M_{0}(T_{0,K(t)}) is isomorphic to the K(t)-algebra M = EM_{n+1}(T_{0,K(t)})E, and so it suffices to produce a K-algebra homomorphism T_{n,K} → M. Let us index rows and columns of matrices in M_{n+1}(T_{0,K(t)}) by 0, 1, …, n.

Now consider matrices A_{i}, B_{j} in M for i, j = 0, 1, …, n where

A_{0} = diag(te_{n}, x_{0}, …, x_{0}) , (A_{i})_{α,β} = δ_{β,i} \begin{cases}te_{n} & (α = 0) \\ x_{α} & (α ≥ 1) \end{cases} (1 ≤ i ≤ n) ,

B_{0} = diag(t^{−1}e_{n}, y_{0}, …, y_{0}) , (B_{j})_{α,β} = δ_{α,j} \begin{cases}t^{−1}e_{n} & (β = 0) \\ y_{β} & (β ≥ 1) \end{cases} (1 ≤ j ≤ n) .

Then we have A_{i}B_{j} = δ_{i,j}E for all i, j and ∑_{i=0}^{n} B_{i}A_{i} = E. If p(Z_{0}, …, Z_{n}) is a noncommutative polynomial over K with zero constant term, then p(A_{0}, …, A_{n}) = EC = ECE for some matrix C ∈ M_{n+1}(T_{0,K(t)}) whose entries come from either tK[t]⟨X_{n}⟩ or K⟨X_{n}⟩X_{n}, and where C_{i0} = 0 for i = 1, …, n. Consequently, any matrix D in Σ′_{K}(A_{0}, …, A_{n}) can be written as D = ẼΔ = ẼΔẼ for some Δ in Σ′_{K(t)}(x_{0}, …, x_{n}), where Ẽ = diag(E, …, E). Now Δ is invertible over K(t)rat⟨X_{n}⟩, and we compute that ẼΔ^{−1} = ẼΔ^{−1}Ẽ. Thus D is invertible over M, with inverse ẼΔ^{−1}. This shows that every matrix in Σ′_{K}(A_{0}, …, A_{n}) is invertible over M. Therefore by the universal property of T_{n,K}, there is a K-algebra homomorphism T_{n,K} → M, and we are done.

Corollary 8.3. Let K be a field and t an indeterminate. Let R be a K-algebra Morita equivalent to some T_{n,K} and S a K(t)-algebra Morita equivalent to some T_{m,K(t)}, where n, m ∈ {0} ∪ {2, 3, … }. Let φ : K_{0}(R) → K_{0}(S) be a group homomorphism such that φ([R]) = [S]. Then there exists a K-algebra homomorphism ψ : R → S such that K_{0}(ψ) = φ.

Proof. There exists a finitely generated projective right T_{n,K}-module A such that R ≅ End(A), and [A] = ℓ[T_{n,K}] for some ℓ ∈ ℤ, where we may assume 1 ≤ ℓ ≤ n in case n ≥ 2. If ℓ > 0, then A ≅ T_{n,K}^{ℓ} by Proposition 2.2, whence R ≅ M_{ℓ}(T_{n,K}). If ℓ ≤ 0, then n = 0 and [A] = [e_{ℓ}T_{0,K}], in which case R ≅ e_{ℓ}T_{0,K}e_{ℓ} = M_{ℓ}(T_{0,K}). Hence, there is no loss of generality in assuming that R = M_{ℓ}(T_{n,K}). Likewise, we may assume that S = M_{ℓ′}(T_{m,K(t)}) for some ℓ′ ∈ ℤ, where 1 ≤ ℓ′ ≤ m in case m ≥ 2.

As above, we can make the identifications

(K_{0}(T_{n,K}), [T_{n,K}]) = (ℤ_{n}, [1]_{n})
(K_{0}(R), [R]) = (K_{0}(T_{n,K}), ℓ[T_{n,K}]) = (ℤ_{n}, [ℓ]_{n})
(K_{0}(T_{m,K(t)}), [T_{m,K(t)}]) = (ℤ_{m}, [1]_{m})
(K_{0}(S), [S]) = (K_{0}(T_{m,K(t)}), ℓ′[T_{m,K(t)}]) = (ℤ_{m}, [ℓ′]_{m}).

Then φ is identified with a homomorphism ℤ_{n} → ℤ_{m} given by multiplication by an integer h, where we can assume that 1 ≤ h ≤ m in case m ≥ 2, and [hℓ]_{m} = [ℓ′]_{m}. Finally, identify

(K_{0}(M_{h}(T_{m,K(t)})), [M_{h}(T_{m,K(t)})]) = (K_{0}(T_{m,K(t)}), h[T_{m,K(t)}]) = (ℤ_{m}, [h]_{m}).

By Lemma 8.2, there exists a K-algebra homomorphism ψ_{0} : T_{n,K} → M_{h}(T_{m,K(t)}) such that K_{0}(ψ_{0}) = φ. If ℓ > 0, then ψ_{0} induces a K-algebra homomorphism ψ_{1} : R → M_{ℓ}(M_{h}(T_{m,K(t)})) such that K_{0}(ψ_{1}) = φ. There is a nonzero finitely generated projective right S-module B such that [B] = [h]_{m} in K_{0}(S) and End_{S}(B) ≅ M_{h}(T_{m,K(t)}). Since ℓ[B] = [ℓh]_{m} = [ℓ′]_{m} = [S], it follows from Proposition 2.2 that ℓB ≅ S_{S}, whence M_{ℓ}(M_{h}(T_{m,K(t)})) ≅ S. On composing the latter isomorphism with ψ_{1}, we obtain a K-algebra homomorphism ψ : R → S such that K_{0}(ψ) = φ.

Now suppose that ℓ ≤ 0, so that n = 0 and R = e_{ℓ}T_{0,K}e_{ℓ}. In this case, ψ_{0} restricts to a K-algebra homomorphism ψ_{1} : R → fM_{h}(T_{m,K(t)})f, where f = ψ_{0}(e_{ℓ}). Since [e_{ℓ}T_{0,K}] = ℓ[T_{0,K}] in K_{0}(T_{0,K}), we have

[fM_{h}(T_{m,K(t)})] = ℓ[M_{h}(T_{m,K(t)})] = ℓ[h]_{m} = [ℓ′]_{m}

in K_{0}(M_{h}(T_{m,K(t)})). There is a nonzero finitely generated projective right M_{h}(T_{m,K(t)})- module C such that [C] = [ℓ′]_{m} in K_{0}(M_{h}(T_{m,K(t)})) and End(C) ≅ S. By Proposition 2.2, fM_{h}(T_{m,K(t)}) ≅ C, whence fM_{h}(T_{m,K(t)})f ≅ S. As in the previous paragraph, on composing the latter isomorphism with ψ_{1}, we obtain the desired K-algebra homomorphism ψ : R → S.

We are now ready to prove our realization result.

Theorem 8.4. Let G be a countable abelian group, u ∈ G, and k any field. Then there exists a purely infinite simple regular k-algebra R such that K_{0}(R) ≅ G with [R] ↦ u.

Proof. We can write (G, u) as the inductive limit of a sequence

(G_{0}, u_{0}) ⟶^{φ_{0}} (G_{1}, u_{1}) ⟶^{φ_{1}} ⋯,

where each G_{n} is a nonzero finitely generated abelian group, u_{n} ∈ G_{n}, and φ_{n} : G_{n} → G_{n+1} is a group homomorphism such that φ_{n}(u_{n}) = u_{n+1}. For each n, we may assume that

(G_{n}, u_{n}) = (G_{n,1}, u_{n,1}) × ⋯ × (G_{n,r(n)}, u_{n,r(n)})

where each G_{n,i} is nonzero and cyclic.

Let t_{1}, t_{2}, … be independent indeterminates over k, and set K_{n} = k(t_{1}, …, t_{n}) for n = 0, 1, … (thus K_{0} = k). For each n, let M_{n} denote the class of K_{n}-algebras Morita equivalent to ones of the form T_{∗,K_{n}}. Choose R_{n,1}, …, R_{n,r(n)} in M_{n} together with identifications (K_{0}(R_{n,i}), [R_{n,i}]) = (G_{n,i}, u_{n,i}) for all i. Set R_{n} = R_{n,1} × ⋯ × R_{n,r(n)}, so that (K_{0}(R_{n}), [R_{n}]) = (G_{n}, u_{n}). We shall construct K_{n}-algebra homomorphisms ψ_{n} : R_{n} → R_{n+1} such that K_{0}(ψ_{n}) = φ_{n} and each of the component maps ψ_{n,j} : R_{n} → R_{n+1,j} is an embedding. Then the inductive limit of the sequence

R_{0} ⟶^{ψ_{0}} R_{1} ⟶^{ψ_{1}} ⋯

will be a regular k-algebra with (K_{0}(R), [R]) ≅ (G, u). Because the ψ_{n,j} are embeddings, we see that for any nonzero element a ∈ R_{n}, there exist elements x, y ∈ R_{n+1} such that xψ_{n}(a)y = 1. Therefore R will be a purely infinite simple ring.

It only remains to build the homomorphisms ψ_{n}. Fix n, write φ_{n} as a matrix of group homomorphisms

η_{i,j} : G_{n,i} → G_{n+1,j} ,

and note that η_{1,j}(u_{n,1})+⋯+η_{r(n),j}(u_{n,r(n)}) = u_{n+1,j} for all j. Hence, there exist finitely generated projective right R_{n+1,j}-modules P_{1}, …, P_{r(n)} such that [P_{i}] = η_{i,j}(u_{n,i}) for all i and P_{1}⊕⋯⊕P_{r(n)} ≅ R_{n+1,j}. Since R_{n+1,j} is purely infinite simple, there exists a nonzero finitely generated projective right R_{n+1,j}-module P such that [P] is the identity element of the group V(R_{n+1,j})^{∗} (see Proposition 2.2). We can replace each P_{i} by P ⊕ P_{i}, and so we may assume that all P_{i} ≠ 0. Consequently, there exist nonzero pairwise orthogonal idempotents f_{1,j}, …, f_{r(n),j} in R_{n+1,j} such that f_{1,j} + ⋯ + f_{r(n),j} = 1 and [f_{i,j}R_{n+1,j}] = η_{i,j}(u_{n,i}) for all i.

Each corner f_{i,j}R_{n+1,j}f_{i,j} belongs to M_{n+1}, and we can make the identification

(K_{0}(f_{i,j}R_{n+1,j}f_{i,j}), [f_{i,j}R_{n+1,j}f_{i,j}]) = (G_{n+1,j}, η_{i,j}(u_{n,i})).

By Corollary 8.3, there exist K_{n}-algebra homomorphisms θ_{i,j} : R_{n,i} → f_{i,j}R_{n+1,j}f_{i,j} such that K_{0}(θ_{i,j}) = η_{i,j}. Since the R_{n,i} are simple algebras, the θ_{i,j} are embeddings. Consequently, the rule ψ_{n,j}(a) = θ_{1,j}(a)+⋯+θ_{r(n),j}(a) defines a K_{n}-algebra embedding ψ_{n,j} : R_{n} → R_{n+1,j} such that K_{0}(ψ_{n,j}) = (η_{1,j}, …, η_{r(n),j}). These ψ_{n,j}, finally, are the components for the desired K_{n}-algebra homomorphism ψ_{n} : R_{n} → R_{n+1}.

Remark 8.5. Note that we get a countable k-algebra R in Theorem 8.4 in case we start with a countable field k.

Acknowledgements

Part of this work was done during a visit of the first author to the Department of Mathematics of the University of California at Santa Barbara, and visits of the second and third authors to the Centre de Recerca Matemàtica, Institut d’Estudis Catalans in Barcelona. The three authors are very grateful to the host centers for their warm hospitality. We also thank Mikael Rørdam for interesting discussions on the topic of Section 8.

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

E-mail address: para@mat.uab.es

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

E-mail address: goodearl@math.ucsb.edu

Departamento de Matemáticas, Universidad de Cádiz, Aptdo. 40, 11510 Puerto Real (Cádiz), Spain

E-mail address: enrique.pardo@uca.es