FRACTIONAL SKEW MONOID RINGS

P. ARA, M.A. GONZÁLEZ-BARROSO, K.R. GOODEARL, AND E. PARDO

Dedicated to the memory of Dmitry Tyukavkin

Abstract. Given an action α of a monoid T on a ring A by ring endomorphisms, and an Ore subset S of T, a general construction of a fractional skew monoid ring S^{op} ∗_{α} A ∗_{α} T is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case S is a subsemigroup of a group G such that G = S^{−1}S, we obtain a G-graded ring S^{op} ∗_{α} A ∗_{α} S with the property that, for each s ∈ S, the s-component contains a left invertible element and the s^{−1}-component contains a right invertible element. In the most basic case, where G = ℤ and S = T = ℤ^{+}, the construction is fully determined by a single ring endomorphism α of A. If α is an isomorphism onto a proper corner pAp, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by A[t_{+}, t_{−}; α]. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type (1, n), can be presented in the form A[t_{+}, t_{−}; α]. Finally, mild and reasonably natural conditions are obtained under which S^{op} ∗_{α} A ∗_{α} S is a purely infinite simple ring.

Introduction

Let α: G → Aut(A), g ↦ α_{g}, be an action of a group G on a unital ring A. A useful construction attached to these data is the skew group ring A ∗_{α} G, see [18] and [20]. This is the ring of formal expressions ∑_{g∈G} a_{g}g, where a_{g} ∈ A and almost all the coefficients a_{g} are 0. Addition is defined componentwise and multiplication is defined according to the rule (ag)(bh) = (aα_{g}(b))(gh). The skew group ring A ∗_{α} G can also be defined as the unital ring R such that there are a unital ring homomorphism ϕ: A → R and a unital monoid homomorphism i: G → R from G to the multiplicative structure of R, universal with respect to the property that i(g)ϕ(a) = ϕ(α_{g}(a))i(g) for all a ∈ A and all g ∈ G. In his pioneering paper [19], Paschke gave a construction of a C^{∗}-algebraic crossed product A ⋊_{α} ℕ associated to a not necessarily unital C^{∗}-algebra endomorphism α on a C^{∗}-algebra A. Paschke’s C^{∗}- algebraic construction has been generalized to other semigroups, see [13], [14], [15] and [16]. Moreover, Rørdam [22] used Paschke’s construction together with the Pimsner-Voiculescu exact sequence associated to an automorphism [6, Theorem 10.2.1] to realize any pair of countable abelian groups (G_{0}, G_{1}) as (K_{0}(B), K_{1}(B)) for a certain purely infinite, simple, nuclear separable C^{∗}-algebra B.

1991 Mathematics Subject Classification. Primary 16A50, 16D70. Key words and phrases. Skew monoid ring, purely infinite simple ring, Leavitt algebra. The first author was partially supported by the DGI and European Regional Development Fund, jointly,

through Project BFM2002-01390, the second by an FPU fellowship of the Junta de Andalucía, the third by an NSF grant, and the fourth by DGESIC grant BFM2001–3141. Also, the first and fourth authors are partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya, and the second and fourth by PAI III grant FQM-298 of the Junta de Andalucía.

In this paper, we develop a systematic purely algebraic theory of fractional skew monoid rings with respect to monoid actions on rings by not necessarily unital ring endomorphisms, in which an Ore submonoid is inverted. (Recall that a monoid is a semigroup with a neutral element.) More precisely, we assume the following data are given (see (1.1) for the detailed definitions of the properties):

(1) A monoid T acting on a unital ring A by endomorphisms;

(2) A submonoid S of T satisfying the left denominator conditions, and such that S is

left saturated in T. Then a fractional skew monoid ring S^{op} ∗_{α} A ∗_{α} T is constructed, with suitable maps from A, S^{op} and T to S^{op} ∗_{α} A ∗_{α} T, which satisfy a universal property analogous to the one for the skew group ring described above, see Definition 1.2. It is not difficult to show that such a ring exists by using a construction with generators and relations, but it is rather non-obvious to determine the algebraic structure of S^{op}∗_{α}A∗_{α}T. The ring S^{op}∗_{α}A∗_{α}T is best understood by means of its S^{−1}T -graded structure, obtained in Proposition 1.6. The structure is completely pinned down in (1.12) in the case where T acts by injective endomorphisms.

The general construction of S^{op} ∗_{α} A ∗_{α} T is given in Section 1. In the other sections, we specialize the construction to the case of a submonoid S of a group G such that G = S^{−1}S (taking T = S), and to an action α of S on A by corner isomorphisms, meaning that α_{s} is an isomorphism from A onto the corner ring α_{s}(1)Aα_{s}(1) for all s ∈ S. Several examples of interest are considered in Section 2 in the case where S = T = ℤ^{+}. In particular, the Leavitt algebras V_{1,n}(k) and U_{1,n}(k), already considered by Leavitt, Skornyakov, Cohn, Bergman and others, are seen here to be particular cases of our construction.

For S = T = ℤ^{+}, the construction is determined by a single corner isomorphism α, and the elements of the fractional skew monoid ring R = ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} can all be written as ‘polynomials’ of the form

r = a_{n}t_{+}^{n} + … + a_{1}t_{+} + a_{0} + t_{−}a_{−1} + … t_{−}^{m}a_{−m},

with coefficients a_{i} ∈ A. Because of this similarity of R with a skew-Laurent polynomial ring, we shall use the notation R = A[t_{+}, t_{−}; α]. Using this construction and the Bass-Heller-Swan-Farrell-Hsiang-Siebenmann Theorem, the K_{1} group of these algebras is computed in [2].

A general source of interesting examples is provided in Section 3. Namely, assume that G is a group acting on a ring A by automorphisms, and that there are a submonoid S of G such that G = S^{−1}S and a non-trivial idempotent e in A such that α_{s}(e) ∈ eAe for all s ∈ S. Then the corner ring e(A ∗_{α} G)e of the skew group ring A ∗_{α} G is isomorphic as a G-graded ring to a fractional skew monoid ring S^{op} ∗_{α′} (eAe) ∗_{α′} S (Proposition 3.3). Under the standing assumption that S acts by corner isomorphisms, we prove that all S^{op} ∗_{α} A ∗_{α} S can be exhibited in the form e(A ∗_{α} G)e (Proposition 3.8).

Sections 4 and 5 deal with actions on simple rings. Using a suitable definition of outer action of a monoid S on a ring A, we prove in Theorem 4.1 that S^{op} ∗_{α} A ∗_{α} S is a simple ring for any outer action α of S on a simple ring A. This is a generalization of a well-known sufficient condition for simplicity of skew group rings, see [18, Theorem 2.3]. Section 5 shows that, under mild conditions on A and on the outer action α of S on A, the fractional skew monoid ring S^{op} ∗_{α} A ∗_{α} S is a purely infinite simple ring (Theorem 5.3). In particular, this holds whenever A is either a simple ultramatricial algebra over some field or a purely infinite simple ring. The class of purely infinite simple rings has been recently studied by the first, third and fourth authors in [3], and constitute an important and large class of relatively wellbehaved simple rings. They can be thought of as the nice rings in the wild universe of the directly infinite simple rings; see specially [3, Corollary 2.2 and Theorem 2.3] for the good behaviour of K-theory of purely infinite simple rings. A further nice property of them has been recently established by the first author in [1]: Every purely infinite simple ring satisfies the exchange property.

All rings and modules in this paper will be assumed to be unital unless explicitly noted. (The main exception is the ring S^{−1}A constructed in Section 3.) However, many of the subrings we deal with will have units different from the unit of the larger ring; specifically, we will deal with many corners pAp in a ring A, where p is an idempotent. Note that any ring endomorphism ε of A, even if not unital when considered as a map A → A, \underline{is} unital when viewed as a ring homomorphism A → ε(1)Aε(1).

We will use the standard order structure on the set of idempotents in a ring A; that is, for idempotents e and f in A, we have e ≤ f if and only if e = ef = fe. Two idempotents e and f are said to be equivalent, written e ∼ f, if there are elements x, y ∈ A such that e = xy and f = yx. This is equivalent to saying that the right A-modules eA and fA are isomorphic. We write e ≲ f in case e ∼ f′ for some idempotent f′ ≤ f. We say that an idempotent e is infinite if there are nonzero orthogonal idempotents e′ and g such that e = e′ + g and e ∼ e′. If no such decomposition exists, e is called a finite idempotent.

1. The general construction

We present the construction of a fractional skew monoid ring in full generality in this section, and establish the precise graded structure of this ring. The basic data consist of a ring A, a monoid T acting on A by ring endomorphisms, and a left denominator set S ⊆ T; the fractional skew monoid ring we construct is graded by S^{−1}T, and its identity component is the quotient of A modulo the union of the kernels of the endomorphisms by which S acts.

1.1. We begin by fixing the basic data needed for our construction; these data and conventions will remain in force throughout the paper. Let A be a (unital) ring, and Endr(A) the monoid of non-unital (i.e., not necessarily unital) ring endomorphisms of A.

Let T be a monoid and α : T → Endr(A) a monoid homomorphism, written t ↦ α_{t}. In general, we will write T multiplicatively, with its identity element denoted 1, but in some applications it will be convenient to switch to additive notation for T. For t ∈ T, set p_{t} = α_{t}(1), an idempotent in A. Then α_{t} can be viewed as a unital ring homomorphism from A to the corner p_{t}Ap_{t}. For s, t ∈ T, we have p_{st} = α_{st}(1) = α_{s}α_{t}(1) = α_{s}(p_{t}).

Let S ⊆ T be a submonoid satisfying the left denominator conditions, i.e., the left Ore condition and the monoid version of left reversibility: whenever t, u ∈ T with ts = us for some s ∈ S, there exists s′ ∈ S such that s′t = s′u. Then there exists a monoid of fractions, S^{−1}T, with the usual properties (e.g., see [7, §1.10] or [8, §0.8]).

We shall also assume that S is left saturated in T: whenever s ∈ S and t ∈ T such that ts ∈ S, we must have t ∈ S. This assumption means that equality in S^{−1}T can be described as follows: if s_{1}^{−1}t_{1} = s_{2}^{−1}t_{2} for some s_{i} ∈ S and t_{i} ∈ T, there exist u_{1}, u_{2} ∈ S such that u_{1}s_{1} = u_{2}s_{2} and u_{1}t_{1} = u_{2}t_{2}. (The usual denominator conditions only yield the latter equations for, say, some u_{1} ∈ S and u_{2} ∈ T. But then u_{2}s_{2} = u_{1}s_{1} ∈ S, and left saturation implies u_{2} ∈ S.)

Definition 1.2. The label S^{op} ∗_{α} A ∗_{α} T stands for a (unital) ring R equipped with a (unital) ring homomorphism ϕ : A → R and monoid homomorphisms s ↦ s_{−} from S^{op} → R and t ↦ t_{+} from T → R, universal with respect to the following relations:

(1) t_{+}ϕ(a) = ϕα_{t}(a)t_{+} for all a ∈ A and t ∈ T;

(2) ϕ(a)s_{−} = s_{−}ϕα_{s}(a) for all a ∈ A and s ∈ S;

(3) s_{−}s_{+} = 1 for all s ∈ S;

(4) s_{+}s_{−} = ϕ(p_{s}) for all s ∈ S.

Note that condition (2) follows from the others. Given a ∈ A and s ∈ S, we have s_{+}ϕ(a) = ϕα_{s}(a)s_{+} by (1), and on multiplying each term of this equation on the left and on the right by s_{−}, we obtain ϕ(a)s_{−} = s_{−}ϕα_{s}(a)ϕ(p_{s}) = s_{−}ϕ(α_{s}(a)p_{s}) from (3) and (4), whence (2) follows because α_{s}(a)p_{s} = α_{s}(a).

1.3. At this point, we sketch the existence of the ring R = S^{op}∗_{α}A∗_{α}T. The existence of a ring satisfying the universal property of Definition 1.2 follows from a construction with generators and relations, which does not use at all any property of S; in fact, S can be an arbitrary subset of T. Take B = A ∗ ℤ⟨t_{+}, s_{−} | t ∈ T, s ∈ S⟩ to be the free product of A and the free ring on the disjoint union T ⊔ S, and let i_{1} : A → B and i_{2} : ℤ⟨t_{+}, s_{−} | t ∈ T, s ∈ S⟩ → B be the canonical maps. Let J be the two-sided ideal of B generated by

(a) i_{2}(t_{+})i_{1}(a) − i_{1}(α_{t}(a))i_{2}(t_{+}) for all a ∈ A and t ∈ T;

(b) i_{2}((tt′)_{+}) − i_{2}(t_{+})i_{2}(t′_{+}) for all t, t′ ∈ T;
(c) i_{2}(s_{−})i_{2}(s_{+}) − i_{1}(1) for all s ∈ S;
(d) i_{2}(s_{+})i_{2}(s_{−}) − i_{1}(p_{s}) for all s ∈ S.

Then R = B/J is the ring we are looking for, and ϕ is the composite map π ∘ i_{1}, where π: B → B/J is the canonical projection. (Here we identify the elements s_{−}, for s ∈ S, with their images πi_{2}(s_{−}), and similarly for the elements t_{+}.) Note that the relations (ss′)_{−} = s′_{−}s_{−}, for all s, s′ ∈ S such that ss′ ∈ S, hold automatically from (a)–(d) above. Also, we have already observed that condition (2) in 1.2 follows from conditions (1),(3) and (4), and so it follows from (a)–(d) too.

Rather than introduce a notation for the product in S^{op}, we view the map (−)_{−} as a monoid anti-homomorphism S → R, so that (su)_{−} = u_{−}s_{−} for s, u ∈ S.

The construction above will also be applied when A is an algebra over a field k and the ring endomorphisms α_{t} for t ∈ T are k-linear. In this case, it is easily checked that ϕ maps k = k·1 into the center of B (use relations (1),(2) above and part (c) of the following lemma to see that ϕ(k) commutes with each s_{−} and t_{+}), so that B becomes a k-algebra and ϕ becomes a k-algebra homomorphism. The universal property of B then holds also in the category of k-algebras.

The following lemma and subsequent results pin down the structure of R = S^{op} ∗_{α} A ∗_{α} T. This structure simplifies considerably when the maps α_{s} are injective – see (1.12).

Lemma 1.4. Let a, b ∈ A, s, u ∈ S, and t, v ∈ T.

(a) s_{+}ϕ(a)s_{−} = ϕα_{s}(a).

(b) s_{−}ϕα_{s}(a)s_{+} = ϕ(a).

(c) s_{−} = s_{−}ϕ(p_{s}) and t_{+} = ϕ(p_{t})t_{+}.

(d) s_{−}ϕ(a)t_{+} = s_{−}ϕ(p_{s}ap_{t})t_{+}.

(e) s_{−}ϕ(a)t_{+} = (us)_{−}ϕα_{u}(a)(ut)_{+}.

(f) There exist x ∈ S and y ∈ T such that xt = yu. For any such x, y,

[s_{−}ϕ(a)t_{+}][u_{−}ϕ(b)v_{+}] = (xs)_{−}ϕ(α_{x}(ap_{t})α_{y}(b))(yv)_{+}.

In particular, t_{+}u_{−} = x_{−}p_{xt}y_{+}.

Proof. (a) s_{+}ϕ(a)s_{−} = ϕα_{s}(a)s_{+}s_{−} = ϕα_{s}(a)ϕ(p_{s}) = ϕ(α_{s}(a)p_{s}) = ϕα_{s}(a).

(b) This follows from (a) because s_{−}s_{+} = 1.

(c) s_{−} = ϕ(1)s_{−} = s_{−}ϕα_{s}(1) = s_{−}ϕ(p_{s}). Similarly, t_{+} = t_{+}ϕ(1) = ϕα_{t}(1)t_{+} = ϕ(p_{t})t_{+}.

(d) This is clear from (c).

(e) From (b), we have ϕ(a) = u_{−}ϕα_{u}(a)u_{+}, and the desired equation follows because s_{−}u_{−} = (us)_{−}.

(f) Note that (xt)_{+}(yu)_{−} = (xt)_{+}(xt)_{−} = ϕ(p_{xt}) = ϕα_{x}(p_{t}). Using (e), we get

[s_{−}ϕ(a)t_{+}][u_{−}ϕ(b)v_{+}] = [(xs)_{−}ϕα_{x}(a)(xt)_{+}][(yu)_{−}ϕα_{y}(b)(yv)_{+}]
= (xs)_{−}ϕα_{x}(a)ϕα_{x}(p_{t})ϕα_{y}(b)(yv)_{+}
= (xs)_{−}ϕ(α_{x}(ap_{t})α_{y}(b))(yv)_{+}. □

Corollary 1.5. R = ∑_{s∈S, t∈T} s_{−}ϕ(A)t_{+} = ∑_{s∈S, t∈T} s_{−}ϕ(p_{s}Ap_{t})t_{+}.

Proof. The second equality is clear from Lemma 1.4(d). Let R′ denote the sum in question. Clearly R′ is closed under addition, and it is closed under multiplication by Lemma 1.4(f). Also, 1_{R} = 1_{−}ϕ(1_{A})1_{+} ∈ R′. Thus, R′ is a unital subring of R.

Since the images of ϕ, s ↦ s_{−}, and t ↦ t_{+} are contained in R′, we can view these as maps into R′. The universal property for R then implies that there is a unique unital ring homomorphism ψ : R → R′ such that ψϕ = ϕ while ψ(s_{−}) = s_{−} for s ∈ S and ψ(t_{+}) = t_{+} for t ∈ T. Consequently, ψ acts as the identity on R′, whence ψ(R) = R′. Moreover, if we view ψ as a ring homomorphism R → R, we have ψϕ = id_{R}ϕ while ψ(s_{−}) = id_{R}(s_{−}) for s ∈ S and ψ(t_{+}) = id_{R}(t_{+}) for t ∈ T. Now the universal property for R implies that ψ = id_{R}, and therefore R = ψ(R) = R′. □

We next exhibit the graded ring structure of R. As the reader will note, this result can also be obtained from the proof of Proposition 1.10 below, and so Propositions 1.6 and 1.10 could have been combined. However, we think that separating the two results is helpful in orienting the reader.

Proposition 1.6. The ring R has an S^{−1}T -grading R = ⨁_{x∈S^{−1}T} R_{x} where each R_{x} = ⋃_{s^{−1}t=x} s_{−}ϕ(A)t_{+}.

Proof. We can view R as a left A-module via ϕ, and the relations in R imply that each s_{−}ϕ(A)t_{+} is a left A-submodule. If s_{1}, s_{2} ∈ S and t_{1}, t_{2} ∈ T such that s_{1}^{−1}t_{1} = s_{2}^{−1}t_{2}, there exist u_{1}, u_{2} ∈ S such that u_{1}s_{1} = u_{2}s_{2} and u_{1}t_{1} = u_{2}t_{2}, whence Lemma 1.4(e) implies that (s_{i})_{−}ϕ(A)(t_{i})_{+} ⊆ (u_{1}s_{1})_{−}ϕ(A)(u_{1}t_{1})_{+} for i = 1, 2. Thus, each R_{x} is a directed union of left A-submodules of R, and so is a left A-submodule itself.

It is clear from Corollary 1.5 that R = ∑_{x∈S^{−1}T} R_{x}, and from Lemma 1.4(f) that R_{x}R_{y} ⊆ R_{xy} for all x, y ∈ S^{−1}T. Hence, it only remains to show that the sum of the R_{x} is a direct sum.

Let R′ denote the external direct sum of the R_{x}, and set E′ = End_{ℤ}(R′). There is a unital ring homomorphism λ : A → E′ such that each λ(a) is the left A-module multiplication by a ∈ A.

Given s ∈ S, observe that s_{−}R_{x} ⊆ R_{s^{−1}x} for all x ∈ S^{−1}T. Hence, there exists μ_{s} ∈ E′ such that μ_{s}(b)_{y} = s_{−}b_{sy} for all b ∈ R′ and y ∈ S^{−1}T. Since ϕ(a)s_{−} = s_{−}ϕα_{s}(a) for a ∈ A, we see that λ(a)μ_{s} = μ_{s}λα_{s}(a) for a ∈ A. Observe also that s ↦ μ_{s} is a monoid homomorphism S^{op} → E′.

Given t ∈ T, it follows from Lemma 1.4(f) that t_{+}R_{x} ⊆ R_{tx} for all x ∈ S^{−1}T. Hence, there exists ν_{t} ∈ E′ such that ν_{t}(b)_{y} = ∑_{tx=y} t_{+}b_{x} for b ∈ R′ and y ∈ S^{−1}T. Since t_{+}ϕ(a) = ϕα_{t}(a)t_{+} for a ∈ A, we see that ν_{t}λ(a) = λα_{t}(a)ν_{t} for a ∈ A. Observe also that t ↦ ν_{t} is a monoid homomorphism T → E′.

Since s_{−}s_{+} = 1 and s_{+}s_{−} = ϕ(p_{s}) for s ∈ S, we see that μ_{s}ν_{s} = id_{R′} = 1_{E′} and ν_{s}μ_{s} = λ(p_{s}) for s ∈ S. Now by the universal property of R, there exists a unital ring homomorphism ψ : R → E′ such that ψϕ = λ while ψ(s_{−}) = μ_{s} for s ∈ S and ψ(t_{+}) = ν_{t} for t ∈ T.

Note that 1_{R} = 1_{−}ϕ(1)1_{+} ∈ R_{1}, so there exists e ∈ R′ such that e_{1} = 1 while e_{z} = 0 for all z ≠ 1. Given s ∈ S, a ∈ A, and t ∈ T, we observe that

[ψ(s_{−}ϕ(a)t_{+})(e)]_{s^{−1}t} = [μ_{s}λ(a)ν_{t}(e)]_{s^{−1}t} = s_{−}ϕ(a)t_{+}

and all other components of ψ(s_{−}ϕ(a)t_{+})(e) are zero. Hence, for x ∈ S^{−1}T and b ∈ R_{x}, we have [ψ(b)(e)]_{x} = b while [ψ(b)(e)]_{y} = 0 for all y ≠ x. Consequently, if b_{1} + ⋯ + b_{n} = 0 for some b_{i} ∈ R_{x_{i}} where the x_{i} are distinct elements of S^{−1}T, then b_{i} = [ψ(b_{1} + ⋯ + b_{n})(e)]_{x_{i}} = 0 for all i. Therefore ∑_{x∈S^{−1}T} R_{x} = ⨁_{x∈S^{−1}T} R_{x}, as desired. □

To completely pin down the elements of R, we need to know the relations holding in each homogeneous component R_{x}. In particular, if p_{s}ap_{t} ∈ ker(ϕ), then s_{−}ϕ(a)t_{+} = 0 by Lemma 1.4(d), and we would like to show that s_{−}ϕ(a)t_{+} = 0 only when p_{s}ap_{t} ∈ ker(ϕ). For this purpose, we set up another representation of R on a left A-module.

Lemma 1.7. Let u, s ∈ S and t ∈ T.

(a) The map ∗ : A × p_{s}Ap_{t} → p_{s}Ap_{t} given by the rule a ∗ b := α_{s}(a)b turns the abelian group p_{s}Ap_{t} into a left A-module.

(b) The restriction of α_{u} to p_{s}Ap_{t} is a left A-module homomorphism p_{s}Ap_{t} → p_{us}Ap_{ut}.

Proof. Part (a) is clear because α_{s} is a unital ring homomorphism from A to p_{s}Ap_{s}, while part (b) follows because α_{us} = α_{u}α_{s}. □

Each homogeneous component R_{x} of R turns out to be a direct limit of the rectangular corners p_{s}Ap_{t} over pairs (s, t) such that s^{−1}t = x. However, there is no natural partial order on the set of these pairs – the limit has to be taken over a small category.

Definition 1.8. For x ∈ S^{−1}T, let D_{x} be the small category in which the objects are all pairs (s, t) ∈ S × T such that s^{−1}t = x, the morphisms from an object (s, t) to an object (s′, t′) are those elements u ∈ S such that us = s′ and ut = t′, and composition of morphisms is given by the multiplication in S. The Ore and saturation conditions on S imply that D_{x} is directed: given any objects (s_{1}, t_{1}) and (s_{2}, t_{2}) in D_{x}, there exist an object (s, t) and morphisms u_{i} : (s_{i}, t_{i}) → (s, t) in D_{x} for i = 1, 2. Consequently, colimits based on D_{x} are directed colimits.

Taking account of Lemma 1.7, there is a functor F_{x} : D_{x} → A-Mod such that F_{x}(s, t) = p_{s}Ap_{t} for all objects (s, t) in D_{x} and F_{x}(u) = α_{u}|_{p_{s}Ap_{t}} for all morphisms u : (s, t) → (us, ut) in D_{x}. Let M_{x} denote the colimit of F_{x}, with natural maps η_{s,t} : p_{s}Ap_{t} → M_{x} for objects (s, t) in D_{x}. Since M_{x} is a directed colimit, it is the union of its submodules η_{s,t}(p_{s}Ap_{t}) for (s, t) ∈ D_{x}. Note that if b_{i} ∈ p_{s_{i}}Ap_{t_{i}} for i = 1, 2, where (s_{i}, t_{i}) ∈ D_{x}, then η_{s_{1},t_{1}}(b_{1}) = η_{s_{2},t_{2}}(b_{2}) if and only if there exist u_{1}, u_{2} ∈ S such that u_{1}s_{1} = u_{2}s_{2} and u_{1}t_{1} = u_{2}t_{2} while also α_{u_{1}}(b_{1}) = α_{u_{2}}(b_{2}).

Lemma 1.9. Let s ∈ S, t ∈ T, and x ∈ S^{−1}T.

(a) There exists an additive map σ_{s} : M_{x} → M_{s^{−1}x} such that σ_{s}η_{u,v}(b) = η_{us,v}(p_{us}b) for u^{−1}v = x and b ∈ p_{u}Ap_{v}.

(b) aσ_{s}(m) = σ_{s}(α_{s}(a)m) for a ∈ A and m ∈ M_{x}.

(c) There exists an additive map τ_{t} : M_{x} → M_{tx} such that τ_{t}η_{u,v}(b) = η_{w,zv}α_{z}(b) for u^{−1}v = x, b ∈ p_{u}Ap_{v}, and w ∈ S, z ∈ T such that wt = zu.

(d) τ_{t}(am) = α_{t}(a)τ_{t}(m) for a ∈ A and m ∈ M_{x}.

Proof. (a) For each (u, v) ∈ D_{x}, we have (us, v) ∈ D_{s^{−1}x}, and there is an additive map p_{u}Ap_{v} → M_{s^{−1}x} given by b ↦ η_{us,v}(p_{us}b). Moreover, if w ∈ S then η_{wus,wv}(p_{wus}α_{w}(b)) = η_{wus,wv}α_{w}(p_{us}b) = η_{us,v}(p_{us}b). Thus, our maps to M_{s^{−1}x} are compatible with the functor F_{x}, and so there exists a unique additive map σ_{s} as described.

(b) If m = η_{u,v}(b) for u, v, b as in (a), then

aσ_{s}(m) = aη_{us,v}(p_{us}b) = η_{us,v}(a ∗ (p_{us}b)) = η_{us,v}(α_{us}(a)p_{us}b) = η_{us,v}(p_{us}α_{us}(a)b)
= η_{us,v}(p_{us}(α_{s}(a) ∗ b)) = σ_{s}η_{u,v}(α_{s}(a) ∗ b) = σ_{s}(α_{s}(a)m).

(c) Fix (u, v) ∈ D_{x}, choose w ∈ S, z ∈ T such that wt = zu, and note that tx = w^{−1}zv. Since α_{z}(p_{u}Ap_{v}) ⊆ p_{zu}Ap_{zv} ⊆ p_{w}Ap_{zv}, the composition of η_{w,zv} with the restriction of α_{z} to p_{u}Ap_{v} gives an additive map p_{u}Ap_{v} → M_{tx}. Suppose also w_{1} ∈ S and z_{1} ∈ T such that w_{1}t = z_{1}u. Then w_{1}^{−1}z_{1} = tu^{−1} = w^{−1}z, so there exist r_{1}, r ∈ S such that r_{1}w_{1} = rw and r_{1}z_{1} = rz. Since also r_{1}z_{1}v = rzv and α_{r_{1}}α_{z_{1}} = α_{r}α_{z}, it follows that η_{w_{1},z_{1}v}α_{z_{1}} = η_{w,zv}α_{z} on p_{u}Ap_{v}. Thus, we obtain a well-defined additive map f_{u,v} : p_{u}Ap_{v} → M_{tx} which agrees with η_{w,zv}α_{z} for any w ∈ S and z ∈ T with wt = zu.

Now consider a morphism r : (u, v) → (ru, rv) in D_{x}. There exist w ∈ S and z ∈ T such that wt = z(ru), so that f_{ru,rv} is given by η_{w,zrv}α_{z}. Since wt = (zr)u, we also have that f_{u,v} is given by η_{w,zrv}α_{zr}, and so f_{u,v} equals the composition of f_{ru,rv} with the restriction of α_{r} to p_{u}Ap_{v}. Thus, the maps f_{.,.} are compatible with F_{x}, and so there exists a unique additive map τ_{t} as described.

(d) If m = η_{u,v}(b) with u, v, b, w, z as in (c), then

τ_{t}(am) = τ_{t}η_{u,v}(a ∗ b) = τ_{t}η_{u,v}(α_{u}(a)b) = η_{w,zv}α_{z}(α_{u}(a)b) = η_{w,zv}(α_{w}α_{t}(a)α_{z}(b))
= η_{w,zv}(α_{t}(a) ∗ α_{z}(b)) = α_{t}(a)η_{w,zv}α_{z}(b) = α_{t}(a)τ_{t}(m). □

Proposition 1.10. For each x ∈ S^{−1}T, there is a left A-module isomorphism θ_{x} : M_{x} → R_{x} such that θ_{x}η_{u,v}(b) = u_{−}ϕ(b)v_{+} for u^{−1}v = x and b ∈ p_{u}Ap_{v}.

Proof. In view of Lemma 1.4(e), for each x ∈ S^{−1}T there is a unique additive map θ_{x} : M_{x} → R_{x} as described. If m = η_{u,v}(b) with u, v, b as above, then for a ∈ A we have

θ_{x}(am) = θ_{x}η_{u,v}(a ∗ b) = θ_{x}η_{u,v}(α_{u}(a)b) = u_{−}ϕα_{u}(a)ϕ(b)v_{+} = ϕ(a)u_{−}ϕ(b)v_{+} = aθ_{x}(m).

Thus, θ_{x} is a left A-module homomorphism. It is surjective by definition of R_{x}, and so it only remains to show that ker(θ_{x}) = 0.

Form the left A-module M := ⨁_{x∈S^{−1}T} M_{x}, set E = End_{ℤ}(M), and for each a ∈ A let λ(a) ∈ E be the map given by left multiplication by a. Then we have a unital ring homomorphism λ : A → E.

For all x ∈ S^{−1}T, use the same notations σ_{s} and τ_{t} for the additive maps M_{x} → M_{s^{−1}x} and M_{x} → M_{tx} described in Lemma 1.9, and also for the corresponding homogeneous maps on M. Thus, for s ∈ S and t ∈ T we have additive maps σ_{s}, τ_{t} ∈ E such that σ_{s}(m)_{y} = σ_{s}(m_{sy}) and τ_{t}(m)_{y} = ∑_{tx=y} τ_{t}(m_{x}) for m ∈ M and y ∈ S^{−1}T. Lemma 1.9 also shows that λ(a)σ_{s} = σ_{s}λα_{s}(a) and τ_{t}λ(a) = λα_{t}(a)τ_{t} for a ∈ A.

It is easily checked that s ↦ σ_{s} and t ↦ τ_{t} are monoid homomorphisms S^{op} → E and T → E. Now consider m = η_{u,v}(b) ∈ M_{x} for x, u, v, b as in Lemma 1.9. There exist w ∈ S and z ∈ T such that ws = zu, and

σ_{s}τ_{s}(m) = σ_{s}η_{w,zv}α_{z}(b) = η_{ws,zv}(p_{ws}α_{z}(b)) = η_{zu,zv}(p_{zu}α_{z}(b))
= η_{zu,zv}α_{z}(p_{u}b) = η_{u,v}(b) = m.

It follows that σ_{s}τ_{s} = 1_{E} in E. Next, note that u ∈ S and 1 ∈ T with u · s = 1 · us. Hence,

τ_{s}σ_{s}(m) = τ_{s}η_{us,v}(p_{us}b) = η_{u,v}α_{1}(p_{us}b) = η_{u,v}(p_{s} ∗ b) = p_{s}m.

It follows that τ_{s}σ_{s} = λ(p_{s}) in E.

By the universal property of R, there is a unital ring homomorphism ψ : R → E such that ψϕ = λ while ψ(s_{−}) = σ_{s} for s ∈ S and ψ(t_{+}) = τ_{t} for t ∈ T.

Define e ∈ M so that e_{1} = η_{1,1}(1) while e_{z} = 0 for all z ≠ 1. We claim that [(ψθ_{x}(m))(e)]_{x} = m for x ∈ S^{−1}T and m ∈ M_{x}. Write m = η_{u,v}(b) where u^{−1}v = x and b ∈ p_{u}Ap_{v}. Then ψθ_{x}(m) = ψ(u_{−}ϕ(b)v_{+}) = σ_{u}λ(b)τ_{v} and so

[(ψθ_{x}(m))(e)]_{x} = σ_{u}λ(b)τ_{v}η_{1,1}(1) = σ_{u}λ(b)η_{1,v}α_{v}(1) = σ_{u}η_{1,v}(b ∗ p_{v})
= σ_{u}η_{1,v}(b) = η_{u,v}(p_{u}b) = η_{u,v}(b) = m,

as claimed.

The claim immediately implies that ker(θ_{x}) = 0 for all x ∈ S^{−1}T, as desired. □

Corollary 1.11. (a) Let s ∈ S, t ∈ T, and a ∈ A. Then s_{−}ϕ(a)t_{+} = 0 if and only if p_{s}ap_{t} ∈ ker(α_{s′}) for some s′ ∈ S. In particular, ker(ϕ) = ⋃_{s′∈S} ker(α_{s′}).

(b) The ideal I = ker(ϕ) satisfies α_{s}^{−1}(I) = I for all s ∈ S and α_{t}(I) ⊆ I for all t ∈ T.

(c) α induces a monoid homomorphism α′ : T → End_{ℤ}(A/I), and α′_{s} is injective for all s ∈ S.

(d) S^{op} ∗_{α} A ∗_{α} T = S^{op} ∗_{α′} (A/I) ∗_{α′} T.

Proof. (a) By Lemma 1.4(d), s_{−}ϕ(a)t_{+} = s_{−}ϕ(b)t_{+} where b = p_{s}ap_{t}. Then Proposition 1.10 yields θ_{x}η_{s,t}(b) = s_{−}ϕ(a)t_{+} where x = s^{−1}t. Since θ_{x} is an isomorphism, s_{−}ϕ(a)t_{+} = 0 if and only if η_{s,t}(b) = 0, which happens if and only if α_{s′}(b) = 0 for some s′ ∈ S. This verifies the first statement in (a). The second follows on taking s = t = 1.

(b) If t ∈ T and s ∈ S, there exist s′ ∈ S and t′ ∈ T such that s′t = t′s. Then α_{s′}α_{t}(ker(α_{s})) = 0, and so α_{t}(ker(α_{s})) ⊆ ker(α_{s′}) ⊆ I. This shows that α_{t}(I) ⊆ I for all t ∈ T.

Now if s ∈ S, the previous paragraph implies that I ⊆ α_{s}^{−1}(I). If a ∈ α_{s}^{−1}(I), then α_{s}(a) ∈ ker(α_{s′}) for some s′ ∈ S, whence a ∈ ker(α_{s′s}) ⊆ I. Therefore α_{s}^{−1}(I) = I.

(c)(d) These are clear from (a) and (b). □

1.12. As Corollary 1.11 shows, we can always reduce to the case where α_{s} is injective for all s ∈ S. In that case, ϕ is injective by Corollary 1.11(a), and so we can identify A with the unital subring ϕ(A) ⊆ R. All of the relations in R simplify in this case:

(1) t_{+}a = α_{t}(a)t_{+} for all a ∈ A and t ∈ T;

(2) as_{−} = s_{−}α_{s}(a) for all a ∈ A and s ∈ S;

(3) s_{−}s_{+} = 1 for all s ∈ S;

(4) s_{+}s_{−} = p_{s} for all s ∈ S;

(5) R has an S^{−1}T -grading R = ⨁_{x∈S^{−1}T} R_{x} where each R_{x} = ⋃_{s^{−1}t=x} s_{−}At_{+};

(6) s_{−}at_{+} = s_{−}p_{s}ap_{t}t_{+} for s ∈ S, t ∈ T, and a ∈ A, and s_{−}at_{+} = 0 if and only if p_{s}ap_{t} = 0;

(7) Let x = s_{1}^{−1}t_{1} = s_{2}^{−1}t_{2} ∈ S^{−1}T for some s_{1}, s_{2} ∈ S, t_{1}, t_{2} ∈ T, and let a_{1}, a_{2} ∈ A. Then (s_{1})_{−}a_{1}(t_{1})_{+} = (s_{2})_{−}a_{2}(t_{2})_{+} if and only if there exist u_{1}, u_{2} ∈ S such that u_{1}s_{1} = u_{2}s_{2} and u_{1}t_{1} = u_{2}t_{2} while also α_{u_{1}}(p_{s_{1}}a_{1}p_{t_{1}}) = α_{u_{2}}(p_{s_{2}}a_{2}p_{t_{2}}).

2. The case S = T = ℤ^{+}; Examples

2.1. For the remainder of the paper, we take advantage of Corollary 1.11 and assume that α_{s} is injective for all s ∈ S. Thus, the relations in R = S^{op} ∗_{α} A ∗_{α} T take the simplified form given in (1.12). Moreover, we assume that the maps α_{s} are corner isomorphisms, that is, each α_{s} is an isomorphism of A onto p_{s}Ap_{s}. Finally, we assume that S = T is a submonoid of a group G which is its group of left fractions, that is, G = S^{−1}S. These conventions are to remain in effect for the rest of the paper.

2.2. A particularly nice setting is the case when G is a left totally ordered group with positive cone G^{+} = S (thus G = S^{−1} ∪ S and S^{−1} ∩ S = {1}). In this case, the elements of R can be expressed in a simpler way, namely in the form ∑_{s∈S} s_{−}a_{s} + ∑_{t∈S} a_{t}t_{+}. To achieve this, we need to be able to simplify individual terms s_{−}at_{+}, for s, t ∈ S and a ∈ A. If s ≤ t, then s^{−1}t ≥ 1, whence u := s^{−1}t ∈ S. Then s_{−}at_{+} = s_{−}a(su)_{+} = s_{−}p_{s}ap_{s}s_{+}u_{+}. Because of our current convention that α_{s} : A → p_{s}Ap_{s} is an isomorphism, p_{s}ap_{s} = α_{s}(b) for some b ∈ A, and therefore s_{−}at_{+} = s_{−}α_{s}(b)s_{+}u_{+} = bs_{−}s_{+}u_{+} = bu_{+}. On the other hand, if s ≥ t, then v := t^{−1}s ∈ S and s_{−}at_{+} = v_{−}c where c = α_{t}^{−1}(p_{t}ap_{t}).

2.3. We now specialize to the case where S is the additive monoid ℤ^{+}, so that G = ℤ. Here the monoid homomorphism α : S → Endr(A) is determined by α_{1}, and so we change notation, writing α and p for α_{1} and p_{1}. Thus, α is now an isomorphism A → pAp, and the monoid homomorphism S → Endr(A) is given by the rule n ↦ α^{n}. Let t denote the generator 1 ∈ ℤ^{+} = S. Since the maps s ↦ s_{±} are monoid homomorphisms into the multiplicative structure of R, we have n_{±} = (t_{±})^{n} =: t_{±}^{n} for n ∈ ℤ^{+}, and

at_{−}^{n} = t_{−}^{n}α^{n}(a)  and  t_{+}^{n}a = α^{n}(a)t_{+}^{n}

for all a ∈ A and n ∈ ℤ^{+}.

In view of (2.2), the elements r ∈ R = ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} can all be written as ‘polynomials’ of the form

r = a_{n}t_{+}^{n} + … + a_{1}t_{+} + a_{0} + t_{−}a_{−1} + … t_{−}^{m}a_{−m},

with coefficients a_{i} ∈ A. Because of this similarity of R with a skew-Laurent polynomial ring, we shall use the notation R = A[t_{+}, t_{−}; α]. Proposition 1.6 shows that R is a ℤ-graded ring R = ⨁_{i∈ℤ} R_{i}, and from the discussion above we see that R_{i} = At_{+}^{i} for i > 0 and R_{i} = t_{−}^{−i}A for i < 0, while A_{0} = A.

Our construction of ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} is an exact algebraic analog of the construction of the crossed product of a C*-algebra by an endomorphism introduced by Paschke [19]. In fact, if A is a C*-algebra and the corner isomorphism α is a *-homomorphism, then Paschke’s C*-crossed product, which he denotes A ⋊_{α} ℕ, is just the completion of ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} in a suitable norm.

Note again that any ring R = A[t_{+}, t_{−}; α] is ℤ-graded, with A = R_{0}. Moreover, t_{+} is a left invertible element of R_{1} with a particular left inverse t_{−} ∈ R_{−1}, and α can be recovered from the rule α(a) = t_{+}at_{−}. These observations allow us to recognize rings of the form A[t_{+}, t_{−}; α] among ℤ-graded rings, as follows.

Lemma 2.4. Let D = ⨁_{i∈ℤ} D_{i} be a ℤ-graded ring containing elements t_{+} ∈ D_{1} and t_{−} ∈ D_{−1} such that t_{−}t_{+} = 1. Then there is a corner isomorphism α : D_{0} → t_{+}t_{−}D_{0}t_{+}t_{−} given by the rule α(d) = t_{+}dt_{−}, and D = D_{0}[t_{+}, t_{−}; α].

Proof. It is clear that t_{+}t_{−} is an idempotent in D_{0}, and that the given rule defines an isomorphism α : D_{0} → t_{+}t_{−}D_{0}t_{+}t_{−}. Hence, there exists a fractional skew monoid ring D̃ = D_{0}[t̃_{+}, t̃_{−}; α]. Since t_{+}d = α(d)t_{+} and dt_{−} = t_{−}α(d) for all d ∈ D, the identity map on D_{0} extends uniquely to a ring homomorphism ϕ : D̃ → D such that ϕ(t̃_{±}) = t_{±}. It remains to show that ϕ is an isomorphism. Note that since t_{+}^{i} ∈ D_{i} and t_{−}^{i} ∈ D_{−i} for all i ∈ ℕ, the map ϕ is a homomorphism of graded rings. Thus, we need only show that ϕ maps each homogeneous component D̃_{i} isomorphically onto D_{i}. This is already given when i = 0.

Now let i > 0. If x ∈ D̃_{i}, then x = dt̃_{+}^{i} for some d ∈ D_{0}, and ϕ(x) = dt_{+}^{i}. If ϕ(x) = 0, then dα^{i}(1) = dt_{+}^{i}t_{−}^{i} = 0 in D_{0}, whence x = dα^{i}(1)t̃_{+}^{i} = 0 in D̃. Thus, the restriction of ϕ to D̃_{i} is injective. Further, if y ∈ D_{i}, then yt_{−}^{i} ∈ D_{0} and ϕ((yt_{−}^{i})t̃_{+}^{i}) = yt_{−}^{i}t_{+}^{i} = y. Therefore ϕ maps D̃_{i} isomorphically onto D_{i}. A symmetric argument shows that this also holds for i < 0, completing the proof. □

Example 2.5. An algebraic version of the Cuntz-Krieger algebras. We give an algebraic version of the C*-algebras O_{A} introduced in [10] (now called “Cuntz-Krieger algebras” in the literature), and show that they may be expressed in the form B[t_{+}, t_{−}; α] for ultramatricial algebras B and proper corner isomorphisms α. The latter statement is parallel to the corresponding C*-algebra result: O_{A} = B ⋊_{α} ℕ for a suitable approximately finite dimensional C*-algebra B (essentially in [10]; discussed explicitly in [22, Example 2.5]).

Let k be an arbitrary field and A = (a_{ij}) an n×n matrix over k, with a_{ij} ∈ {0, 1} for all i, j. To avoid degenerate and trivial cases, we assume that no row or column of A is identically zero, and that A is not a permutation matrix. We define the algebraic Cuntz-Krieger algebra associated to A to be the k-algebra C = CK_{A}(k) with generators x_{1}, y_{1}, …, x_{n}, y_{n} and relations

(1) x_{i}y_{i}x_{i} = x_{i} and y_{i}x_{i}y_{i} = y_{i} for all i;

(2) x_{i}y_{j} = 0 for all i ≠ j;

(3) x_{i}y_{i} = ∑_{j=1}^{n} a_{ij}y_{j}x_{j} for all i;

(4) ∑_{j=1}^{n} y_{j}x_{j} = 1.

Note that all the x_{i}y_{i} and y_{j}x_{j} are idempotents, and that the y_{j}x_{j} are pairwise orthogonal. The free algebra k⟨X_{1}, Y_{1}, …, X_{n}, Y_{n}⟩ can be given a ℤ-grading in which the X_{i} have degree −1 while the Y_{i} have degree 1, and the relators X_{i}Y_{i}X_{i} − X_{i} etc. corresponding to (1)–(4) are all homogeneous. Hence, C inherits a ℤ-grading such that each x_{i} ∈ C_{−1} and each y_{i} ∈ C_{1}.

Now set N = {1, …, n}. Given μ = (μ_{1}, …, μ_{ℓ}) ∈ N^{ℓ} for some ℓ, we set x_{μ} = x_{μ_{1}}x_{μ_{2}} ⋯ x_{μ_{ℓ}} and y_{μ} = y_{μ_{1}}y_{μ_{2}} ⋯ y_{μ_{ℓ}}. The case ℓ = 0 is allowed, with the conventions that N^{0} = {∅} and x_{∅} = y_{∅} = 1. The subalgebra B = C_{0} of C is the k-linear span of the set

{y_{μ}x_{ν} | μ, ν ∈ N^{ℓ}, ℓ ∈ ℤ^{+}}.

As in [10, Proposition 2.3 and following discussion], B is an ultramatricial k-algebra, and K_{0}(B) is isomorphic (as an ordered group) to the direct limit of the sequence

ℤ^{n} ⟶^{A} ℤ^{n} ⟶^{A} ℤ^{n} ⟶^{A} ⋯,

with the class [B] ∈ K_{0}(B) corresponding to the image of the order-unit (1, 1, …, 1)^{tr} in the first ℤ^{n}. (See [11, Chapter 15] for a development of ultramatricial algebras and their classification via K_{0}.)

For i = 1, …, n, let e_{i} denote the sum of those y_{j}x_{j} for which y_{j}x_{j} ≤ x_{i}y_{i} but y_{j}x_{j} ≰ x_{m}y_{m} for any m < i. These e_{i} are pairwise orthogonal idempotents in B, with each e_{i} ≤ x_{i}y_{i}. Since the matrix A has no identically zero columns, each y_{j}x_{j} lies below some x_{i}y_{i}, and so each y_{j}x_{j} lies below some e_{i}. In fact, y_{j}x_{j} ≤ e_{i} where i is the least index such that a_{ij} = 1. From relation (4), it follows that ∑_{i=1}^{n} e_{i} = 1. Next, note that the elements y_{i}e_{i}x_{i} are pairwise orthogonal idempotents in B (because e_{i}x_{i}y_{i} = e_{i} for all i), whence the sum p := y_{1}e_{1}x_{1} + ⋯ + y_{n}e_{n}x_{n} is an idempotent in B. Moreover, x_{i}p = e_{i}x_{i} and py_{i} = y_{i}e_{i} for all i. We claim that p ≠ 1.

If p = 1, then each x_{i} = e_{i}x_{i}, whence each x_{i}y_{i} = e_{i}. Then the x_{i}y_{i} are pairwise orthogonal. In view of the relations (3), it follows that each column of A has only one nonzero entry. Since A has no identically zero rows, it must be a permutation matrix, contradicting our assumptions. Therefore p ≠ 1, as claimed.

Now set t_{−} = e_{1}x_{1} + ⋯ + e_{n}x_{n} ∈ C_{−1} and t_{+} = y_{1}e_{1} + ⋯ + y_{n}e_{n} ∈ C_{1}. Then t_{+}t_{−} = p, and

t_{−}t_{+} = ∑_{i=1}^{n} e_{i}x_{i}y_{i}e_{i} = ∑_{i,j=1}^{n} a_{ij}e_{i}y_{j}x_{j}e_{i} = ∑_{j=1}^{n} y_{j}x_{j} = 1,

because each y_{j}x_{j} ≤ e_{i} for precisely one i, and a_{ij} = 1 for that i. Hence, there is a proper corner isomorphism α : B → pBp given by the rule α(b) = t_{+}bt_{−}, and we conclude from Lemma 2.4 that

C = CK_{A}(k) = B[t_{+}, t_{−}; α]. □

In case the matrix A in Example 2.5 has all of its entries equal to 1, the relations for the algebra CK_{A}(k) reduce to

(1) x_{i}y_{j} = δ_{i,j} for all i, j;

(2) ∑_{j=1}^{n} y_{j}x_{j} = 1. Thus in this case, CK_{A}(k) is the Leavitt algebra V_{1,n}(k) first studied in [17]. (The notation V_{1,n} was introduced in [5].) There is a related Leavitt algebra U_{1,n}(k) which, as we now show, can also be presented as a fractional skew monoid ring.

Example 2.6. Let k be a field and n ∈ ℕ. The algebra U = U_{1,n}(k) is the k-algebra with generators x_{1}, y_{1}, …, x_{n}, y_{n} and relations x_{i}y_{j} = δ_{i,j} for all i, j. (Thus, V_{1,n}(k) is the factor algebra of U_{1,n}(k) modulo the ideal generated by 1−∑_{j=1}^{n} y_{j}x_{j}.) The elements y_{1}x_{1}, …, y_{n}x_{n} are pairwise orthogonal idempotents in U. As in Example 2.5, there is a ℤ-grading on U such that each x_{i} ∈ U_{−1} and each y_{i} ∈ U_{1}.

Set N = {1, …, n} and define x_{μ}, y_{μ} ∈ U for μ ∈ N^{ℓ} as in Example 2.5. In U, the set

{y_{μ}x_{ν} | μ ∈ N^{ℓ}, ν ∈ N^{m}, ℓ, m ∈ ℤ^{+}}

forms a k-basis. We again set B = U_{0}, which is the k-linear span of the set

{y_{μ}x_{ν} | μ, ν ∈ N^{ℓ}, ℓ ∈ ℤ^{+}},

and as before, B is ultramatricial. It is isomorphic to a direct limit of the algebras

M_{n^{i}}(k) × M_{n^{i−1}}(k) × ⋯ × M_{n}(k) × k,

the ordered group K_{0}(B) is isomorphic to the direct limit of a sequence ℤ → ℤ^{2} → ℤ^{3} → ⋯ where each transition map ℤ^{i} → ℤ^{i+1} is given by an (i + 1) × i matrix of the form

(n 0 0 ⋯ 0 0 ; 1 0 0 ⋯ 0 0 ; 0 1 0 ⋯ 0 0 ; ⋮ ; 0 0 0 ⋯ 1 0 ; 0 0 0 ⋯ 0 1) ,

and the class [B] ∈ K_{0}(B) corresponds to the image of 1 ∈ ℤ.

Set p = y_{1}x_{1} ∈ B, a proper idempotent. Then set t_{−} = x_{1} ∈ U_{−1} and t_{+} = y_{1} ∈ U_{1}, so that t_{+}t_{−} = p and t_{−}t_{+} = 1. Hence, the rule b ↦ t_{+}bt_{−} gives a proper corner isomorphism α : B → pBp, and Lemma 2.4 shows that

U = U_{1,n}(k) = B[t_{+}, t_{−}; α]. □

Example 2.7. Let k be a field, and note that there are natural inclusions

U_{1,1}(k) ⊂ U_{1,2}(k) ⊂ U_{1,3}(k) ⊂ ⋯

among the algebras U_{1,n}(k). Set U_{∞}(k) = ⋃_{n=1}^{∞} U_{1,n}(k), which is a simple algebra (e.g., [3, Theorem 4.3]). We may also view U_{∞}(k) as the k-algebra with an infinite sequence of generators x_{1}, y_{1}, x_{2}, y_{2}, … and relations x_{i}y_{j} = δ_{i,j} for all i, j. This algebra is ℤ-graded as before, with the x_{i} having degree −1 and the y_{i} degree 1. Set B = U_{∞}(k)_{0}, which is the k-linear span of the set

{y_{μ}x_{ν} | μ, ν ∈ {1, …, n}^{ℓ}, n ∈ ℕ, ℓ ∈ ℤ^{+}}.

In the present case, B is an ultramatricial k-algebra isomorphic to a direct limit of the algebras

M_{n^{n}}(k) × M_{n^{n−1}}(k) × ⋯ × M_{n}(k) × k.

Here K_{0}(B) is isomorphic to the direct limit of a sequence ℤ^{2} → ℤ^{3} → ℤ^{4} → ⋯ with transition maps

(n n n^{2} n^{3} ⋯ n^{n−2} n^{n−1} ; 1 1 n n^{2} ⋯ n^{n−3} n^{n−2} ; 0 1 1 n ⋯ n^{n−4} n^{n−3} ; ⋮ ; 0 0 0 0 ⋯ 1  1 ; 0 0 0 0 ⋯ 0  1) ,

and [B] corresponds to ( \begin{matrix}1 \\ 1 \end{matrix} ) ∈ ℤ^{2}. If we define p, t_{±}, α exactly as in Example 2.6, we conclude from Lemma 2.4 that

U_{∞}(k) = B[t_{+}, t_{−}; α]. □

3. Fractional skew monoid rings versus corners of skew group rings

Paschke [19] and Rørdam [22, Section 2] have shown that a C*-algebra crossed product by an endomorphism corresponds naturally to a corner in a crossed product by an automorphism. In other words, the C*-algebra versions of fractional skew monoid rings ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} are isomorphic to corners e(B ∗_{α′} ℤ)e in certain skew group rings. This leads us to ask whether, in general, our rings S^{op} ∗_{α} A ∗_{α} S should appear as corner rings e(B ∗ G)e, where B ∗ G is some skew group ring over the group G = S^{−1}S. This is indeed the case, as we prove in Proposition 3.8. We prepare the way by studying corner rings of the form e(A ∗ G)e (for G = S^{−1}S as above), and showing that they fall into the class of fractional skew monoid rings under appropriate conditions on the action.

3.1. Let A be a unital ring, G a group, and α : G → Aut(A) an action. Assume that S is a submonoid of G with G = S^{−1}S, and let R = A ∗_{α} G. Suppose that there exists a nontrivial idempotent e ∈ A such that α_{s}(e) ≤ e for all s ∈ S.

Lemma 3.2. Under the above assumptions, the following hold:

(a) The action α restricts to an action α′ : S → Endr(eAe) by corner isomorphisms.

(b) There are natural monoid morphisms S^{op} → eRe, given by s ↦ es^{−1}, and S → eRe, given by t ↦ te, satisfying the conditions (1)–(4) in Definition 1.2 with respect to α′ and the inclusion map ϕ : eAe → eRe.

Proof. (a) This is clear from the hypothesis on e.

(b) Notice that, since e ≤ α_{s}^{−1}(e) for all s ∈ S, we have es^{−1} = es^{−1}α_{s}(e) ∈ eRe and (es^{−1})(et^{−1}) = e(ts)^{−1} for s, t ∈ S. Similarly, se ∈ eRe and (se)(te) = (st)e. So, the defined maps are monoid morphisms. It is straightforward to check conditions (1)–(4) in Definition 1.2. □

Because of Lemma 3.2, we have the data to construct a fractional skew monoid ring of the form S^{op} ∗_{α′} (eAe) ∗_{α′} S. Since the maps α′_{s} = α_{s}|_{eAe} are injective for all s ∈ S, the ring homomorphism eAe → S^{op} ∗_{α′} (eAe) ∗_{α′} S going with the construction of S^{op} ∗_{α′} (eAe) ∗_{α′} S is injective by Corollary 1.11. Hence, we identify eAe with its image in S^{op} ∗_{α′} (eAe) ∗_{α′} S, as in (1.12).

Proposition 3.3. Under the assumptions of (3.1), the rings S^{op}∗_{α′} (eAe)∗_{α′} S and e(A∗_{α}G)e are isomorphic as G-graded rings.

Proof. By the universal property of S^{op} ∗_{α′} (eAe) ∗_{α′} S, there exists a unique ring homomorphism ψ : S^{op} ∗_{α′} (eAe) ∗_{α′} S → e(A ∗_{α} G)e such that ψ(s_{−}at_{+}) = (es^{−1})a(te) for all s, t ∈ S and a ∈ eAe. Clearly, ψ is G-graded. To see that ψ is onto, consider e(ag)e ∈ e(A ∗ G)e where a ∈ A and g ∈ G, and write g = s^{−1}t for some s, t ∈ S. Then we have

e(ag)e = eas^{−1}te = (es^{−1})(α_{s}(ea)α_{t}(e))(te) ∈ ψ(S^{op} ∗_{α′} (eAe) ∗_{α′} S),

which proves that ψ is onto. It only remains to check that ψ is one-to one.

Since ψ is G-graded, we only have to check that ψ(s_{−}at_{+}) = 0 implies a = 0, when s, t ∈ S and a ∈ p_{s}(eAe)p_{t}. Note that p_{s} = α′_{s}(1_{eAe}) = α_{s}(e), and likewise p_{t} = α_{t}(e), so that a = α_{s}(e)aα_{t}(e). Now

0 = (es^{−1})a(te) = eα_{s}^{−1}(aα_{t}(e))(s^{−1}t) = α_{s}^{−1}(α_{s}(e)aα_{t}(e))(s^{−1}t) = α_{s}^{−1}(a)(s^{−1}t),

whence α_{s}^{−1}(a) = 0 and a = 0, as desired. □

The following procedure gives a generic way to obtain a situation as in (3.1).

Example 3.4. Let α: G → Aut(A) be an action of an abelian group G on a unital ring A, and let e be an idempotent in A. Set S := {s ∈ G | α_{s}(e) ≤ e}. Then S is a submonoid of G and G′ := S^{−1}S is a subgroup of G acting on A via α. Moreover, e(A∗_{α} G′)e ≅ S^{op} ∗_{α′} (eAe)∗_{α′} S, where α′: S → Endr(eAe) is the induced action of S on eAe by corner isomorphisms.

Proof. It is clear that S is a submonoid of G, and we can apply Proposition 3.3 to get the result. □

Now we go in the reverse direction, looking for a representation of a fractional skew monoid ring S^{op}∗_{α}A∗_{α}S as a corner ring of a skew group ring. Our original approach utilized a direct limit construction based on ideas of Rørdam [22]; that approach required S to be abelian. In the meantime, we learned of the work of Picavet [21], whose construction we can make use of without needing S to be abelian.

3.5. Let A be a unital ring, G a group and S a submonoid of G such that G = S^{−1}S. Thus, S satisfies the left Ore condition, and left reversibility holds trivially because S has cancellation. Let α : S → Endr(A) be an action of S on A by corner isomorphisms, and for s ∈ S let p_{s} denote the idempotent α_{s}(1). We construct a ring S^{−1}A as in [21], but with some changes of notation to fit our situation. As written, the development in [21] would require S to act on A by unital ring endomorphisms. However, almost all the results we shall quote do not make use of this assumption, the exception being the question of an identity – in our situation, S^{−1}A can be a non-unital ring.

First, define a relation ∼ on S × A as follows:

(s_{1}, a_{1}) ∼ (s_{2}, a_{2}) if and only if there exist t_{1}, t_{2} ∈ S such that t_{1}s_{1} = t_{2}s_{2} and

α_{t_{1}}(a_{1}) = α_{t_{2}}(a_{2}).

This is an equivalence relation [21, Lemma 2.1], and we write [s, a] for the equivalence class of a pair (s, a). Let S^{−1}A = (S × A)/∼ be the set of these equivalence classes. The left Ore condition guarantees “common denominators” in S^{−1}A: Given any x_{1}, x_{2} ∈ S^{−1}A, there exist s ∈ S and a_{1}, a_{2} ∈ A such that each x_{i} = [s, a_{i}]. By [21, Lemma 2.2 ff.], there is a well-defined associative multiplication on S^{−1}A as follows:

Given any [s_{1}, a_{1}], [s_{2}, a_{2}] ∈ S^{−1}A, choose t_{1}, t_{2} ∈ S such that t_{1}s_{1} = t_{2}s_{2}, and set

[s_{1}, a_{1}] · [s_{2}, a_{2}] = [t_{1}s_{1}, α_{t_{1}}(a_{1})α_{t_{2}}(a_{2})].

(This multiplication rule is simpler than the Ore-Asano rule for multiplication of noncommutative fractions, because the classes [s, a] model elements that would have the form α_{s}^{−1}(a) if α_{s} extended to an automorphism of an overring of A.) It is routine to build a well-defined, commutative, associative addition on S^{−1}A by the corresponding rule:

Given any [s_{1}, a_{1}], [s_{2}, a_{2}] ∈ S^{−1}A, choose t_{1}, t_{2} ∈ S such that t_{1}s_{1} = t_{2}s_{2}, and set

[s_{1}, a_{1}] + [s_{2}, a_{2}] = [t_{1}s_{1}, α_{t_{1}}(a_{1}) + α_{t_{2}}(a_{2})].

The distributive law is also routine, and so S^{−1}A becomes a (possibly non-unital) ring. In fact, for [s, a] ∈ S^{−1}A we have [1, 1] · [s, a] = [s, p_{s}a] and [s, a] · [1, 1] = [a, ap_{s}].

Next, we extend α to an action of S on S^{−1}A. Since this is done without proof in [21, Theorem 2.4 ff.], we sketch the details.

Lemma 3.6. The action of α on A extends to an action α : S → Aut(S^{−1}A) as follows: Given any s ∈ S and [t, a] ∈ S^{−1}A, choose s′, t′ ∈ S such that s′s = t′t, and set

α_{s}([t, a]) = [s′, α_{t′}(a)].

Proof. First, let s ∈ S and [t_{1}, a_{1}] = [t_{2}, a_{2}] in S^{−1}A. Let s_{1}, u_{1}, s_{2}, u_{2} ∈ S such that s_{1}s = u_{1}t_{1} and s_{2}s = u_{2}t_{2}; we must show that [s_{1}, α_{u_{1}}(a_{1})] = [s_{2}, α_{u_{2}}(a_{2})]. There exist r_{1}, r_{2} ∈ S such that r_{1}s_{1} = r_{2}s_{2}, and each [s_{i}, α_{u_{i}}(a_{i})] = [r_{i}s_{i}, α_{r_{i}u_{i}}(a_{i})]. Hence, we may assume that s_{1} = s_{2}. Note that now u_{1}t_{1} = u_{2}t_{2}.

Since [t_{1}, a_{1}] = [t_{2}, a_{2}], there exist v_{1}, v_{2} ∈ S such that v_{1}t_{1} = v_{2}t_{2} and α_{v_{1}}(a_{1}) = α_{v_{2}}(a_{2}). Further, there are p, q ∈ S with pv_{1} = qu_{1}. Then pv_{2}t_{2} = pv_{1}t_{1} = qu_{1}t_{1} = qu_{2}t_{2}, and so pv_{2} = qu_{2}. After replacing s_{1}, u_{1}, s_{2}, u_{2} by qs_{1}, qu_{1}, qs_{2}, qu_{2}, we may assume that pv_{i} = u_{i} for i = 1, 2. Consequently, α_{u_{1}}(a_{1}) = α_{u_{2}}(a_{2}), whence [s_{1}, α_{u_{1}}(a_{1})] = [s_{2}, α_{u_{2}}(a_{2})]. Therefore α_{s}([t, a]) is well-defined.

Consider s ∈ S and [t, a_{1}], [t, a_{2}] ∈ S^{−1}A. Choose s′, t′ ∈ S such that s′s = t′t; then

α_{s}([t, a_{1}] · [t, a_{2}]) = α_{s}([t, a_{1}a_{2}]) = [s′, α_{t′}(a_{1}a_{2})]
= [s′, α_{t′}(a_{1})] · [s′, α_{t′}(a_{2})] = α_{s}([t, a_{1}]) · α_{s}([t, a_{2}]),

and similarly for addition. This shows that α_{s} is a ring endomorphism of S^{−1}A. If α_{s}([t, a_{1}]) = α_{s}([t, a_{2}]), there exist u_{1}, u_{2} ∈ S such that u_{1}s′ = u_{2}s′ and α_{u_{1}t′}(a_{1}) = α_{u_{2}t′}(a_{2}). Since then u_{1} = u_{2}, it follows that [t, a_{1}] = [t, a_{2}]. Thus, α_{s} is injective. Moreover, for any [t, a] ∈ S^{−1}A we see that α_{s}([ts, a]) = [t, a]. Therefore α_{s} ∈ Aut(S^{−1}A).

It is clear that α_{1} is the identity map. Finally, consider s_{1}, s_{2} ∈ S and [t, a] ∈ S^{−1}A. There exist s′_{2}, t_{2} ∈ S such that s′_{2}s_{2} = t_{2}t, so that α_{s_{2}}([t, a]) = [s′_{2}, α_{t_{2}}(a)]. There exist s′_{1}, t_{1} ∈ S such that s′_{1}s_{1} = t_{1}s′_{2}, so that α_{s_{1}}([s′_{2}, α_{t_{2}}(a)]) = [s′_{1}, α_{t_{1}t_{2}}(a)]. But s′_{1}s_{1}s_{2} = t_{1}t_{2}t, and so α_{s_{1}s_{2}}([t, a]) = [s′_{1}, α_{t_{1}t_{2}}(a)] = α_{s_{1}}(α_{s_{2}}([t, a])). Therefore the map α : S → Aut(S^{−1}A) is a monoid homomorphism. □

There is a shortcut that can be taken for part of the above work. The given action of S induces on A the structure of a left module over the monoid ring ℤS. Moreover, S is a left denominator set in ℤS, and the Ore localization S^{−1}(ℤS) is just the group ring ℤG. By standard localization theory, there exists a module of fractions S^{−1}A, which is a left ℤGmodule. Thus, one obtains the construction of S^{−1}A as an additive group and the action of S on S^{−1}A by ℤ-module automorphisms.

Lemma 3.7. The rule a ↦ [1, a] defines an S-equivariant ring embedding ϕ : A → S^{−1}A with image [1, 1] · S^{−1}A · [1, 1].

Proof. It is clear that ϕ is a ring homomorphism and that it is S-equivariant, i.e., ϕ(α_{s}(a)) = α_{s}(ϕ(a)) for s ∈ S and a ∈ A. If a ∈ ker(ϕ), then [1, a] = [1, 0], and so α_{s}(a) = 0 for some s ∈ S. Since α_{s} is injective, a = 0. Thus, ϕ is an embedding.

Set e = [1, 1] = ϕ(1), and note that ϕ(a) = eϕ(a)e for a ∈ A. Recall that e[s, a]e = [s, p_{s}ap_{s}] for any s ∈ S and a ∈ A. Since α_{s}(A) = p_{s}Ap_{s}, there exists b ∈ A with α_{s}(b) = p_{s}ap_{s}, whence e[s, a]e = [s, α_{s}(b)] = [1, b]. Therefore the image of ϕ equals e(S^{−1}A)e. □

Proposition 3.8. Let G be a group and S a submonoid of G such that G = S^{−1}S. Let α : S → Endr(A) be an action of S on A by corner isomorphisms. Then there exist a unital ring B, an action α̂ : G → Aut(B), and an idempotent e in B such that α̂_{s}(e) ≤ e for all s ∈ S and S^{op} ∗_{α} A ∗_{α} S ≅ e(B ∗α̂ G)e (as G-graded rings).

Proof. Construct S^{−1}A as above, set e = [1, 1], and identify A with the corner e(S^{−1}A)e via Lemma 3.7. Let B be the unitization of S^{−1}A; then also A = eBe. In view of Lemma 3.6, α extends to an action G → Aut(S^{−1}A), and thus to an action α̂ : G → Aut(B). It is clear that α̂_{s}(e) ≤ e for s ∈ S, and we conclude from Proposition 3.3 that e(B ∗α̂ G)e ≅ S^{op} ∗α̂ (eBe) ∗α̂ S = S^{op} ∗_{α} A ∗_{α} S as G-graded rings. □

4. Simplicity

We continue the general assumptions of (1.1) and (2.1), and seek conditions on A, S, and α under which R = S^{op} ∗_{α} A ∗_{α} S is a simple ring. In the case of a group action (i.e., S = G and α : G → Aut(A)), sufficient conditions for simplicity are well known [18, Theorem 2.3]: If A is simple and the action α is outer, then the skew group ring A ∗_{α} G is simple. It turns out that a suitable modification of the notion of an outer action also leads to simplicity in our more general situation.

We shall say that a pair (α_{s}, α_{t}), where s, t ∈ S, is inner provided there exist elements u ∈ p_{s}Ap_{t} and v ∈ p_{t}Ap_{s} such that uv = p_{s}, vu = p_{t} and α_{s}(x) = uα_{t}(x)v for all x ∈ A. Note that then α_{s}α_{t}^{−1}(x) = uxv for every x ∈ p_{t}Ap_{t}, and α_{t}α_{s}^{−1}(x) = vxu for all x ∈ p_{s}Ap_{s}. Let us say that α is outer in case (α_{s}, α_{t}) is not inner for any distinct s, t ∈ S.

We will use the following standard terminology. The support of an element r = ∑_{x} r_{x} in R = ⨁_{x∈G} R_{x} is the set Supp(r) = {x ∈ G | r_{x} ≠ 0}. The length of r is the number of elements in the support of r, and is denoted len(r).

Theorem 4.1. If A is simple and α is outer, then R = S^{op} ∗_{α} A ∗_{α} S is simple.

Proof. Suppose that R is not simple. Let I be a proper nonzero ideal of R, and let ρ ∈ I be a nonzero element with minimal length, say length n. Write ρ = ∑_{i=1}^{n}(s_{i})_{−}a_{i}(t_{i})_{+} where the s_{i}^{−1}t_{i} are distinct elements of S^{−1}S and each a_{i} is a nonzero element of p_{s_{i}}Ap_{t_{i}}. Observe that (s_{1})_{+}ρ(t_{1})_{−} = a_{1} + ∑_{i=2}^{n} ρ_{i} where each ρ_{i} lies in the s_{1}s_{i}^{−1}t_{i}t_{1}^{−1}-component of R. Hence, (s_{1})_{+}ρ(t_{1})_{−} = a_{1} + ∑_{i=2}^{n}(u_{i})_{−}b_{i}(v_{i})_{+} where the u_{i}^{−1}v_{i} are distinct elements of S^{−1}S, different from 1, and each b_{i} ∈ p_{u_{i}}Ap_{v_{i}}. Moreover, a_{1} ≠ 0 implies (s_{1})_{+}ρ(t_{1})_{−} ≠ 0, and so (s_{1})_{+}ρ(t_{1})_{−} has length n by minimality. Thus, after replacing ρ by (s_{1})_{+}ρ(t_{1})_{−}, we may assume that s_{1} = t_{1} = 1.

Since A is simple, ∑_{j=1}^{m} c_{j}a_{1}d_{j} = 1 for some c_{j}, d_{j} ∈ A. Then we can replace ρ by

∑_{j=1}^{m} c_{j}ρd_{j} = 1 + ∑_{i=2}^{n}(s_{i})_{−}(∑_{j=1}^{m} α_{s_{i}}(c_{j})a_{i}α_{t_{i}}(d_{j}))(t_{i})_{+},

and so we may now assume that a_{1} = 1. Of course ρ ≠ 1 because I ≠ R, whence n ≥ 2. Set s = s_{2}, t = t_{2}, and a = a_{2} ∈ p_{s}Ap_{t}, so that

ρ = 1 + s_{−}at_{+} + ∑_{i=3}^{n}(s_{i})_{−}a_{i}(t_{i})_{+}.

For any x ∈ A, we have xρ − ρx ∈ I and

xρ − ρx = s_{−}(α_{s}(x)a − aα_{t}(x))t_{+} + ∑_{i=3}^{n}(s_{i})_{−}♢_{i}(t_{i})_{+}

for some elements ♢_{i} ∈ p_{s_{i}}Ap_{t_{i}} that we need not specify. Thus xρ − ρx has length less than n, and so xρ − ρx = 0 by the minimality of n. Therefore

α_{s}(x)a = aα_{t}(x)

for all x ∈ A. In particular, p_{s}Aa = p_{s}Ap_{s}a = α_{s}(A)a = aα_{t}(A) = aAp_{t}.

Since A is simple, Ap_{t}A = AaA = A, and so

aAp_{s} = aAp_{t}Ap_{s} = p_{s}AaAp_{s} = p_{s}Ap_{s},

whence there is some b ∈ p_{t}Ap_{s} such that ab = p_{s}. Similarly, there is some c ∈ p_{t}Ap_{s} such that ca = p_{t}. But c = cp_{s} = cab = p_{t}b = b, so that ba = p_{t}. Now

aα_{t}(x)b = α_{s}(x)ab = α_{s}(x)p_{s} = α_{s}(x)

for all x ∈ A, and so we conclude that the pair (α_{s}, α_{t}) is inner. Since α is assumed to be outer, we must have s = t. But then s_{2}^{−1}t_{2} = s^{−1}t = 1 = s_{1}^{−1}t_{1}, contradicting the distinctness of the s_{i}^{−1}t_{i}. Therefore R is simple. □

Corollary 4.2. If A is simple and p_{s} ≁ p_{t} for all distinct s, t ∈ S, then R is simple. □

Corollary 4.3. If A is a directly finite simple ring, p ∈ A is a proper idempotent (i.e., p ≠ 1), and α : A → pAp is a corner isomorphism, then ℤ^{+} ∗_{α} A ∗_{α} ℤ^{+} is simple.

Proof. The idempotents corresponding to the monoid homomorphism ℤ^{+} → Endr(A) in this case are the α^{i}(1) for i ∈ ℤ^{+}. Since α(1) = p ≠ 1, we have 1 > α(1) > α^{2}(1) > ⋯, and it follows from the direct finiteness of A that α^{i}(1) ≁ α^{j}(1) for all distinct i, j ∈ ℤ^{+}. □

5. Purely infinite simplicity

We recall from [3] that a simple ring T is said to be purely infinite if every nonzero right ideal of T contains an infinite idempotent. This concept is left-right symmetric, as the following characterization shows: T is purely infinite if and only if (1) T is not a division ring; (2) for every nonzero element a ∈ T, there exist elements x, y ∈ T such that xay = 1 [3, Theorem 1.6]. For instance, the Leavitt algebras V_{1,n}(k) and U_{∞}(k) are purely infinite simple rings [3, Theorems 4.2, 4.3]. As we have seen above (Examples 2.5 and 2.7), the V_{1,n}(k) and U_{∞}(k) can be presented in the form ℤ^{+} ∗_{α} B ∗_{α} ℤ^{+}. This suggests that fractional skew monoid rings might be purely infinite simple in some generality. Our goal in this section is to establish sufficient conditions for a fractional skew monoid ring R = S^{op} ∗_{α} A∗_{α} S to be a purely infinite simple ring, under the general assumptions of (1.1) and (2.1).

The following concept will be needed. A ring T is said to be strictly unperforated provided the finitely generated projective right (or left) T -modules enjoy the following property: If mA ≺ mB for some m ∈ ℕ, then A ≺ B. (Here mA denotes the direct sum of m copies of A, and the notation X ≺ Y means that X is isomorphic to a \underline{proper} direct summand of Y. Similarly, e ≺ f, for idempotents e, f ∈ T, means that e ∼ e′ < f for some idempotent e′ in T.) Stated in terms of idempotents in matrix rings over T, strict unperforation is the condition (m·p ≺ m·q ⟹ p ≺ q), where m·p denotes the orthogonal sum of m copies of an idempotent p. For instance, ultramatricial algebras are strictly unperforated [11, Theorem 15.24(a)]. Also, any purely infinite simple ring T is strictly unperforated, because A ≺ B for all nonzero finitely generated projective T -modules A and B [3, Proposition 1.5].

Lemma 5.1. , Assume that A is simple and strictly unperforated, and that there exists u ∈ S such that p_{u} ≠ 1. For any nonzero idempotent e ∈ A, there exists v = u^{j} ∈ S for some j ∈ ℕ such that p_{v} ≲ e.

Proof. Set p_{i} = p_{u^{i}} = α_{u}^{i}(1) for i ≥ 0. Since A is simple, there exists m ∈ ℕ such that 1 ≺ m·e and 1 ≲ m·(1 − p_{1}). Note that

(m + 1)·p_{1} ≲ m·p_{1} ⊕ 1 ≲ m·p_{1} ⊕ m·(1 − p_{1}) ∼ m·1.

Applying the isomorphisms α_{u}^{i} : A → p_{i}Ap_{i}, we obtain that (m + 1)·p_{i+1} ≲ m·p_{i} for all i. It follows by induction that (m + 1)^{i}·p_{i} ≲ m^{i}·1 for all i.

Now choose j ∈ ℕ such that m^{j+1} < (m + 1)^{j}, and observe that

m^{j+1}·p_{j} ≺ (m + 1)^{j}·p_{j} ≲ m^{j}·1 ≺ m^{j+1}·e,

whence m^{j+1}·p_{j} ≺ m^{j+1}·e. Therefore p_{j} ≺ e, because A is strictly unperforated. □

The following lemma is a variation on results such as [11, Proposition 3.3].

Lemma 5.2. If T is a simple ring containing an idempotent p ≠ 0, 1, then T is generated (as a ring) by its idempotents.

Proof. Let T′ be the subring of T generated by the idempotents. Since p + pt(1 − p) is idempotent for any t ∈ T, we see that pT (1 − p) ⊆ T′, and likewise (1 − p)T p ⊆ T′. The simplicity of T implies that T (1 − p)T = T, whence pT p = [pT (1 − p)][(1 − p)T p] ⊆ T′, and similarly (1 − p)T (1 − p) ⊆ T′. Therefore T′ = T. □

Theorem 5.3. Assume that A is a simple, strictly unperforated ring, in which every nonzero right (left) ideal contains a nonzero idempotent. Assume also that α is outer, and that there exists u ∈ S with p_{u} ≠ 1. Then R = S^{op} ∗_{α} A ∗_{α} S is a purely infinite simple ring.

Proof. The hypothesis that p_{u} ≠ 1 will allow us later to apply Lemma 5.1. Moreover, it implies that R is not a division ring.

Let ρ be an arbitrary nonzero element of R. Choose ρ′, ρ′′ ∈ R such that ρ′ρρ′′ is nonzero and has minimal length for such nonzero products, say length n. Since it suffices to find x, y ∈ R such that xρ′ρρ′′y = 1, we may replace ρ by ρ′ρρ′′. Thus, without loss of generality, all nonzero products σρσ′ in R have length at least n. Now write ρ = ∑_{i=1}^{n}(s_{i})_{−}a_{i}(t_{i})_{+} where the s_{i}^{−1}t_{i} are distinct elements of S^{−1}S and each a_{i} is a nonzero element of p_{s_{i}}Ap_{t_{i}}. As in the proof of Theorem 4.1, after replacing ρ by (s_{1})_{+}ρ(t_{1})_{−} we may assume that s_{1} = t_{1} = 1, so that ρ = a_{1} + ∑_{i=2}^{n}(s_{i})_{−}a_{i}(t_{i})_{+}.

By our hypothesis on idempotents, there exists a′_{1} ∈ A such that a_{1}a′_{1} is a nonzero idempotent. By Lemma 5.1, there exist x, y ∈ A such that xa_{1}a′_{1}y = p_{v} for some v ∈ S. Note that v_{−}xa_{1}a′_{1}yv_{+} = 1. Hence, after replacing ρ by v_{−}xρa′_{1}yv_{+}, we may assume that a_{1} = 1. We are thus done in case n = 1.

Suppose that n ≥ 2, and set s = s_{2}, t = t_{2}, and a = a_{2} ∈ p_{s}Ap_{t}. Thus,

ρ = 1 + s_{−}at_{+} + ∑_{i=3}^{n}(s_{i})_{−}a_{i}(t_{i})_{+}

at this point. For any idempotent e ∈ A, we have

eρ(1 − e) = s_{−}α_{s}(e)a(p_{t} − α_{t}(e))t_{+} + ∑_{i=3}^{n}(s_{i})_{−}♢_{i}(t_{i})_{+}.

Since eρ(1 − e) has length less than n, it must be zero, whence α_{s}(e)a(p_{t} − α_{t}(e)) = 0. Thus, α_{s}(e)a = α_{s}(e)aα_{t}(e). A symmetric argument involving (1 − e)ρe shows that aα_{t}(e) = α_{s}(e)aα_{t}(e), and so α_{s}(e)a = aα_{t}(e).

By Lemma 5.2, A is generated by its idempotents. Hence, it follows from the equations α_{s}(e)a = aα_{t}(e) that α_{s}(x)a = aα_{t}(x) for all x ∈ A. As in the proof of Theorem 4.1, this implies that the pair (α_{s}, α_{t}) is inner, yielding s = t and s_{2}^{−1}t_{2} = s_{1}^{−1}t_{1}, which contradicts our assumptions. Therefore n = 1, and the proof is complete. □

It is perhaps not so surprising that the purely infinite simple property carries over from A to R under suitable conditions. More interesting is that R can be purely infinite simple even when A is directly finite. We single out an important case of this phenomenon in the following corollary.

Corollary 5.4. Suppose that A is either a purely infinite simple ring or a simple ultramatricial algebra over some field. Assume also that α is outer, and that there exists u ∈ S with p_{u} ≠ 1. Then R = S^{op} ∗_{α} A ∗_{α} S is a purely infinite simple ring. □

Acknowledgments

Parts of this work were done during visits of the first author to the Department of Mathematics of the University of California at Santa Barbara and to the Departamento de Matemáticas de la Universidad de Cádiz, and during visits of the last three authors to the Departament de Matemàtiques de l’Universitat Autònoma de Barcelona and to the Centre de Recerca Matemàtica (UAB). The authors would like to thank these host centers for their warm hospitality.

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

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

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

  E-mail address: mariangeles.gonzalezbarroso@alum.uca.es

Department of Mathematics, University of California Santa Barbara CA 93106, U.S.A.

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

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

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