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Commutativity and ideals in category crossed products/Kommutatiivsus ja ideaalid kategooriaga gradueeritud ristkorrutistes.

1. INTRODUCTION

Let R be a ring. By this we always mean that R is an additive group equipped with a multiplication which is associative and unital. The identity element of R is denoted [1.sub.R] and the set of ring endomorphisms of R is denoted End(R). We always assume that ring homomorphisms respect the multiplicative identities. The centre of R is denoted Z(R) and by the commutant of a subset of R we mean the collection of elements in R that commute with all the elements in the subset.

Suppose that [R.sub.1] is a subring of R, i.e. there is an injective ring homomorphism [R.sub.1] [right arrow] R. Recall that if [R.sub.1] is commutative, then it is called a maximal commutative subring of R if it coincides with its commutant in R. A lot of work has been devoted to investigating the connection between on the one hand maximal commutativity of [R.sub.1] in R and on the other hand nonemptiness of intersections of [R.sub.1] with nonzero two-sided ideals of R (see [2,3,5,6,9-11,16]). Recently (see [18-22]) such a connection was established for the commutant [R.sub.1] of the coefficient ring of crossed products R (see Theorem 1 below). Recall that crossed products are defined by first specifying a crossed system, i.e. a quadruple {A, G, [sigma], [alpha]} where A is a ring, G is a group (written multiplicatively and with identity element e) and [sigma] : G [right arrow] End (A) and [alpha] : G x G [right arrow] A are maps satisfying the following four conditions:

[[sigma].sub.e] = [id.sub.A], (1)

[alpha](s, e) = [alpha](e, s) = [1.sub.A] (2)

[alpha](s, t )[alpha](st, r) = [[sigma].sub.s] ([alpha](t, r))[alpha](s, tr), (3)

[[sigma].sub.s] ([[sigma].sub.t] (a))[alpha](s, t) = [alpha](s, t)[[sigma].sub.st](a) (4)

for all s, t, r [member of] G and all a [member of] A. The crossed product, denoted A [[??].sup.[sigma].sub.[alpha]] G, associated to this quadruple, is the collection of formal sums [[summation].sub.s[member of]G] [a.sub.s][u.sub.s], where [a.sub.s] [member of] A, s [member of] G, are chosen so that all but finitely many of them are zero. By abuse of notation we write [u.sub.s] instead of [1.sub.A][u.sub.s] for all s [member of] G. The addition on A [[??].sup.[sigma].sub.[alpha]] G is defined pointwise

[summation over (s[member of]G)] [a.sub.s][u.sub.s] + [summation over (s[member of]G)] [b.sub.s][u.sub.s] = [summation over (s[member of]G)] ([a.sub.s] + [b.sub.s])[u.sub.s] (5)

and the multiplication on A [[??].sup.[sigma].sub.[alpha]] G is defined by the bilinear extension of the relation

([a.sub.s][u.sub.s])([b.sub.t][u.sub.t]) = [a.sub.s][[sigma].sub.s]([b.sub.t])[alpha](s, t)[u.sub.st] (6)

for all s, t [member of] G and all [a.sub.s], [b.sub.t] [member of] A. By (1) and (2) [u.sub.e] is a multiplicative identity of A [[??].sup.[sigma].sub.[alpha]] G and by (3) the multiplication on A [[??].sup.[sigma].sub.[alpha]] G is associative. There is also an A-bimodule structure on A [[??].sup.[sigma].sub.[alpha]] G defined by the linear extension of the relations a([bu.sub.s]) = (ab)[u.sub.s] and ([au.sub.s])b = (a[[sigma].sub.s](b))[u.sub.s] for all a, b [member of] A and all s, t [member of] G, which, by (4), makes A [[??].sup.[sigma].sub.[alpha]] G an A-algebra. In the article [18], the first author and Silvestrov show the following result.

Theorem 1. If A [[??].sup.[sigma].sub.[alpha]] G is a crossed product with A commutative, all [[sigma].sub.s], s [member of] G, are ring automorphisms and all [alpha](s, [s.sup.-1]), s [member of] G, are units in A, then every intersection of a nonzero two-sided ideal of A [[??].sup.[sigma].sub.[alpha]] G with the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is nonzero.

In [18] the first author and Silvestrov determine the centre of crossed products and in particular when crossed products are commutative; they also give a description of the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G. Theorem 1 has been generalized somewhat by relaxing the conditions on [sigma] and [alpha] (see [20,21]) and by considering general strongly group graded rings (see [22]). For more details concerning group graded rings in general and crossed product algebras in particular, see e.g. [1,7,17].

Many natural examples of rings, such as rings of matrices, crossed product algebras defined by separable extensions and category rings, are not in any natural way graded by groups, but instead by categories (see [12-14] and Remark 1). The purpose of this article is to define a category graded generalization of crossed products and to analyse commutativity questions similar to the ones discussed above for such algebras. In particular, we wish to generalize Theorem 1 from groups to groupoids (see Theorem 2 in Section 4). To be more precise, suppose that G is a category. The family of objects of G is denoted ob(G); we will often identify an object in G with its associated identity morphism. The family of morphisms in G is denoted mor(G); by abuse of notation, we will often write s [member of] G when we mean s [member of] mor(G). The domain and codomain of a morphism s in G are denoted d(s) and c(s) respectively. We let [G.sup.(2)] denote the collection of composable pairs of morphisms in G, i.e. all (s, t) in mor(G) x mor(G) satisfying d(s) = c(t). Analogously, we let [G.sup.(3)] denote the collection of all composable triples of morphisms in G, i.e. all (s, t, r) in mor(G) x mor(G) x mor(G) satisfying (s, t) [member of] [G.sup.(2)] and (t, r) [member of] [G.sup.(2)]. Throughout the article G is assumed to be small, i.e. with the property that mor(G) is a set. A category is called a groupoid (1) if all its morphisms are invertible. By a crossed system we mean a quadruple {A, G, [sigma], [alpha]} where A is the direct sum of rings [A.sub.e], e [member of] ob(G), [[sigma].sub.s] : [A.sub.d(s)] [right arrow] [A.sub.s(s)], for s [member of] G, are ring homomorphisms and [alpha] is a map from [G.sup.(2)] to the disjoint union of the sets [A.sub.e], for e [member of] ob(G), with [alpha](s, t) [member of] [A.sub.c(s)], for (s, t) [member of] [G.sup.(2)], satisfying the following five conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[alpha](s,t)[alpha](st, r) = [[sigma].sub.s]([alpha](t, r))[alpha](s, tr), (10)

[[sigma].sub.s]([[sigma].sub.t](a))[alpha](s, t) = [alpha](s, t)[[sigma].sub.st] (a) (11)

for all e [member of] ob(G), all (s, t, r) [member of] [G.sup.(3)] and all a [member of] [A.sub.d(t)]. Let A [[??].sup.[sigma].sub.[alpha]] G denote the collection of formal sums [[summation].sub.s[member of]G] [a.sub.s][u.sub.s], where [a.sub.s] [member of] [A.sub.c(s)], s [member of] G, are chosen so that all but finitely many of them are zero. Define addition on A [[??].sup.[sigma].sub.[alpha]] G by (5) and define multiplication on A [[??].sup.[sigma].sub.[alpha]] G by (6) if (s, t) [member of] [G.sup.(2)] and ([a.sub.s][u.sub.s])([b.sub.t][u.sub.t]) = 0 otherwise where [a.sub.s] [member of] [A.sub.c(s)] and [b.sub.t] [member of] [A.sub.c(t)]. By (7), (8), and (9) it follows that A [[??].sup.[sigma].sub.[alpha]] G has a multiplicative identity if and only if ob(G) is finite; in that case the multiplicative identity is [[summation].sub.e[member of]ob(G)] [u.sub.e]. By (10) the multiplication on A [[??].sup.[sigma].sub.[alpha]] G is associative. Define a left A-module structure on A [[??].sup.[sigma].sub.[alpha]] G by the bilinear extension of the rule [a.sub.e]([b.sub.s][u.sub.s]) = ([a.sub.e][b.sub.s])[u.sub.s] if e = c(s) and [a.sub.e]([b.sub.s][u.sub.s]) = 0 otherwise for all [a.sub.e] [member of] [A.sub.e], [b.sub.s] [member of] [A.sub.c(s)], e [member of] ob(G), s [member of] G. Analogously, define a right A-module structure on A [[??].sup.[sigma].sub.[alpha]] G by the bilinear extension of the rule ([b.sub.s][u.sub.s])[c.sub.f] = ([b.sub.s][[sigma].sub.s]([c.sub.f]))[u.sub.s] if f = d(s) and ([b.sub.s][u.sub.s])[c.sub.f] = 0 otherwise for all [b.sub.s] [member of] [A.sub.c(s)], [c.sub.f] [member of] [A.sub.f], f [member of] ob(G), s [member of] G. By (11) this A-bimodule structure makes A [[??].sup.[sigma].sub.[alpha]] G an A-algebra. We will often identify A with [[direct sum].sub.e[member of]ob(G)] [A.sub.e][u.sub.e]; this ring will be referred to as the coefficient ring of A [[??].sup.[sigma].sub.[alpha]] G. It is clear that A [[??].sup.[sigma].sub.[alpha]] G is a category graded ring in the sense defined in [13] and it is strongly graded if and only if each [alpha](s, t), (s, t) [member of] [G.sup.(2)], has a left inverse in [A.sub.c(s)]. We call A [[??].sup.[sigma].sub.[alpha]] G the category crossed product algebra associated to the crossed system {A, G, [sigma], [alpha]}.

In Section 2, we determine the centre of category crossed products. In particular, we determine when category crossed products are commutative. In Section 3, we describe the commutant of the coefficient ring in category crossed products. In Section 4, we investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptiness of intersections of the coefficient ring by nonzero two-sided ideals. At the end of each section, we indicate how our results generalize earlier results for other algebraic structures such as group crossed products and matrix rings (see Remarks 1-6 and Remark 8).

2. THE CENTRE

For the rest of the article, unless otherwise stated, we suppose that A [[??].sup.[sigma].sub.[alpha]] G is a category crossed product. We say that a is symmetric if [alpha](s, t) = [alpha](t, s) for all s, t [member of] G with d(s) = c(s) = d(t) = c(t). We say that A [[??].sup.[sigma].sub.[alpha]] G is a monoid (groupoid, group) crossed product if G is a monoid (groupoid, group). We say that A [[??].sup.[sigma].sub.[alpha]] G is a twisted category (monoid, groupoid, group) algebra if each [[sigma].sub.s], s [member of] G, with d(s) = c(s) equals the identity map on [A.sub.d(s)] = [A.sub.c(s)]; in that case the category (monoid, groupoid, group) crossed product is denoted A [[??].sub.[alpha]] G. We say that A [[??].sup.[sigma].sub.[alpha]] G is a skew category (monoid, groupoid, group) algebra if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for (s, t) [member of] [G.sup.(2)]; in that case the category (monoid, groupoid, group) crossed product is denoted A [[??].sup.[sigma]] G. If G is a monoid, then we let [A.sup.G] denote the set of elements in A fixed by all [[sigma].sub.s], s [member of] G. We say that G is cancellable if any equality of the form [s.sub.1][t.sub.1] = [s.sub.2][t.sub.2], when ([s.sub.i], [t.sub.i]) [member of] [G.sup.(2)], for i = 1, 2, implies that [s.sub.1] = [s.sub.2] (or [t.sub.1] = [t.sub.2]) whenever [t.sub.1] = [t.sub.2] (or [s.sub.1] = [s.sub.2]). For e, f [member of] ob(G) we let [G.sub.f], e] denote the collection of s [member of] G with c(s) = f and d(s) = e; we let [G.sub.e] denote the monoid [G.sub.e,e]. We let the restriction of [alpha](or [sigma]) to [G.sup.2.sub.e] (or [G.sub.e]) be denoted by [[alpha].sub.e] (or [[sigma].sub.e]). With this notation all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for e [member of] ob(G), are monoid crossed products.

Proposition 1. The centre of a monoid crossed product A [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G satisfying the following two conditions: (i) [a.sub.s][[sigma].sub.s](a) = [aa.sub.s] for s [member of] G and a [member of] A; (ii) for all t, r [member of] G the following equality holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let e denote the identity element of G. Take x : = [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in the centre of A [[??].sup.[sigma].sub.[alpha]] G. Condition (i) follows from the fact that [xau.sub.e] = [au.sub.e]x for all a [member of] A. Condition (ii) follows from the fact that [xu.sub.t] = [u.sub.t]x for all t [member of] G. Conversely, it is clear that conditions (i) and (ii) are sufficient for x to be in the centre of A [[??].sup.[sigma].sub.[alpha]] G. []

Corollary 1. The centre of a twisted monoid ring A [[??].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sub.[alpha]] G satisfying the following two conditions: (i) [a.sub.s] [member of] Z(A), for s [member of] G; (ii) for all t, r [member of] G, the following equality holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. This follows immediately from Proposition 1. []

Corollary 2. If G is an abelian cancellable monoid, [alpha] is symmetric and has the property that none of the [alpha](s, t),for (s, t) [member of] [G.sup.(2)], is a zero-divisor, then the centre of A [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G][a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G satisfying the following two conditions: (i) [a.sub.s][[sigma].sub.s](a) = [aa.sub.s], for s [member of] G and a [member of] A; (ii) [a.sub.s] [member of] [A.sup.G], for s [member of] G. In particular, if A [[??].sup.[sigma]] G is a skew monoid ring where G is abelian and cancellable, then the same description of the centre is valid.

Proof. Take x := [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G. Suppose that x belongs to the centre of A [[??].sup.[sigma].sub.[alpha]] G. Condition (i) follows from the first part of Proposition 1. Now we show condition (ii). Take s, t [member of] G and let r = st. Since G is commutative and cancellable, we get, by the second part of Proposition 1, that [a.sub.s][alpha](s, t) = [[sigma].sub.t] ([a.sub.s])[alpha](t, s). Since [alpha] is symmetric and [alpha](s, t) is not a zero-divisor, this implies that [a.sub.s] = [[sigma].sub.t]([a.sub.s]). Since s and t were arbitrarily chosen from G, this implies that [a.sub.s] [member of] [A.sup.G], for s [member of] G. On the other hand, by Proposition 1, it is clear that (i) and (ii) are sufficient conditions for x to be in the centre of A [[??].sup.[sigma].sub.[alpha]] G. The second part of the claim is obvious.

Now we show that the centre of a category crossed product is a particular subring of the direct sum of the centres of the corresponding monoid crossed products.

Proposition 2. The centre of a category crossed product A [[??].sup.[sigma].sub.[alpha]] G equals the collection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all e, f [member of] ob(G) with e [not equal to] f, and all r, g [member of] [G.sub.f,e].

Proof. Take x := [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in the centre of A [[??].sup.[sigma].sub.[alpha]] G. By the equalities [u.sub.e]x = [xu.sub.e], for e [member of] ob(G), it follows that [a.sub.s] = 0 for all s [member of] G with d(s) [not equal to] c(s). Therefore we can write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The last part of the claim follows from the fact that the equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds for all e, f [member of] ob(G), all e [not equal to] f, and all r [member of] [G.sub.f,e].

Proposition 3. Suppose that A [[??].sup.[sigma].sub.[alpha]] G is a category crossed product and consider the following six conditions: (0) all [alpha](s, t), for (s, t) [member of] [G.sup.(2)], are nonzero; (i) A [[??].sup.[sigma].sub.[alpha]] G is commutative; (ii) G is the disjoint union of the monoids [G.sub.e], for e [member of] ob(G), and they are all abelian; (iii) each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for e [member of] ob(G), is a twisted monoid algebra; (iv) A is commutative; (v) [alpha] is symmetric. Then (a) conditions (0) and (i) imply conditions (ii)-(v); (b) conditions (ii)-(v) imply condition (i).

Proof. (a) Suppose that conditions (0) and (i) hold. By Proposition 2, we get that G is the direct sum of [G.sub.e], for e [member of] ob(G), and that each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for e [member of] ob(G), is commutative. The latter and Proposition 1(i) imply that (iii) holds. Corollary 1 now implies that (iv) holds. For the rest of the proof we can suppose that G is a monoid. Take s, t [member of] G. By the commutativity of A [[??].sup.[sigma].sub.[alpha]] G we get that [alpha](s, t) [u.sub.st] = [u.sub.s][u.sub.t] = [u.sub.t][u.sub.s] = [alpha](t, s)[u.sub.ts] for all s, t [member of] G. Since [alpha] is nonzero this implies that st = ts and that [alpha](s, t) = [alpha](t, s) for all s, t [member of] G. Therefore, G is abelian and (v) holds.

Conversely, by Corollary 1 and Corollary 2 we get that conditions (ii)-(iv) are sufficient for commutativity of A [[??].sup.[sigma].sub.[alpha]] G.

Remark 1. Proposition 2, Corollary 1, Corollary 2, and Proposition 3 generalize Proposition 3 and Corollaries 1-4 in [18] from groups to categories.

Remark 2. Let A [??] G be a category algebra where all the rings [A.sub.e], for e [member of] ob(G), coincide with a fixed ring D. Then A [??] G is the usual category algebra DG of G over D. Let H denote the disjoint union of the monoids [G.sub.e], for e [member of] ob(G). By Proposition 1 and Proposition 2 the centre of DG is the collection of [[summation].sub.s[member of]H] [a.sub.s][u.sub.s], for [a.sub.s] [member of] Z(D), and s [member of] H, in the induced category algebra Z(D)H satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all r, t [member of] G. Note that if G is a groupoid, then the last condition simplifies to [a.sub.rt.sup.-1] = [a.sub.t.] - [1.sub.r] for all r, t [member of] G with c(r) = c(t) and d(r) = d(t). This result specializes to two well-known cases. First of all, if G is a group, then we retrieve the usual description of the centre of a group ring (see e.g. [23]). Secondly, if G is the groupoid with the n first positive integers as objects and as arrows all pairs (i, j), for 1 [less than or equal to] i, j [less than or equal to] n, equipped with the partial binary operation defined by letting (i, j)(k, l) be defined and equal to (i, l) precisely when j = k, then DG is the ring of square matrices over D of size n and we retrieve the result that Z([M.sub.n](D)) equals the Z(D)[1.sub.n] where [1.sub.n] is the unit n x n matrix.

Remark 3. Let L/K be a finite separable (not necessarily normal) field extension. Let N denote a normal closure of L/K and let Gal(N/K) denote the Galois group of N/K. Furthermore, let L = [L.sub.1], [L.sub.2], ..., [L.sub.n] denote the different conjugate fields of L under the action of Gal (N/K) and put F = [[direct sum].sup.n.sub.i=1] [L.sub.i]. If 1 [less than or equal to] i, j [less than or equal to] n, then let [G.sub.ij] denote the set of field isomorphisms from [L.sub.j] to [L.sub.i]. If s [member of] [G.sub.ij], then we indicate this by writing d(s) = j and c(s) = i. If we let G be the union of the [G.sub.ij], for 1 [less than or equal to] i, j [less than or equal to] n, then G is a groupoid. For each s [member of] G, let [[sigma].sub.s] = s. Suppose that [alpha] is a map [G.sup.(2)] [right arrow] [[??].sup.n.sub.i=1] [L.sub.i] with [alpha](s, t) [member of] [L.sub.c(s)], for (s,t) [member of] [G.sup.(2)] satisfying (2), (3) and (4) for all (s, t, r) [member of] [G.sup.(3)] and all a [member of] [L.sub.d(t)]. The category crossed product F [[??].sup.[sigma].sub.[alpha]] G extends the construction usually defined by Galois field extensions L/K. By Proposition 2, the centre of F [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.e[member of]ob(G)] [a.sub.e][u.sub.e] with [a.sub.e] = s([a.sub.f]) for all e, f [member of] ob(G) and all s [member of] G with c(s) = e and d(s) = f. Therefore the centre is a field isomorphic to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we retrieve the first part of Theorem 4 in [12].

3. THE COMMUTANT OF THE COEFFICIENT RING

Proposition 4. The commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G satisfying [a.sub.s] = 0, for s [member of] G, with d(s) [not equal to] c(s), and [a.sub.s][[sigma].sub.s](a) = [aa.sub.s], for s [member of] G with d(s) = c(s) and a [member of] [A.sub.d(s)].

Proof. The first claim follows from the fact that the equality ([[summation].sub.s[member of]G] [a.sub.s][u.sub.s])[u.sub.e] = [u.sub.e]([[summation].sub.s[member of]G] [a.sub.s][u.sub.s]) holds for all e [member of] ob(G). The second claim follows from the fact that the equality ([[summation].sub.s[member of]G] [a.sub.s][u.sub.s])[au.sub.e] = [au.sub.e]([[summation].sub.s[member of]G] [a.sub.s][u.sub.s]) holds for all e [member of] ob(G) and all a [member of] [A.sub.e].

Recall that the annihilator of an element r in a commutative ring R is the collection, denoted ann(r), of elements s in R with the property that rs = 0.

Corollary 3. Suppose that A is commutative. Then the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G satisfying [a.sub.s] = 0, for s [member of] G with d(s) [not equal to] c(s), and [[sigma].sub.s](a) - a [member of] ann([a.sub.s]), for s [member of] G with d(s) = c(s) and a [member of] [A.sub.d(s)]. In particular, A is maximal commutative in A [[??].sup.[sigma].sub.[alpha]] G if and only if for all choices of e [member of] ob(G), s [member of] [G.sub.e] \ {e}, as [member of] [A.sub.e], there is a nonzero a [member of] [A.sub.e] with the property that [[sigma].sub.s](a) - a [not member of] ann([a.sub.s]).

Proof. This follows immediately from Proposition 4. []

Corollary 4. Suppose that each [A.sub.e], e [member of] ob(G), is an integral domain. Then the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G satisfying [a.sub.s] = 0 whenever [[sigma].sub.s] is not an identity map. In particular, A is maximal commutative in A [[??].sup.[sigma].sub.[alpha]] G if and only if for all nonidentity s [member of] G, the map [[sigma].sub.s] is not an identity map.

Proof. This follows immediately from Corollary 3.

Proposition 5. If A is commutative, G a disjoint union of abelian monoids and [alpha] is symmetric, then the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is the unique maximal commutative subalgebra of A [[??].sup.[sigma].sub.[alpha]] G containing A.

Proof. We need to show that the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G is commutative. By the first part of Proposition 4, we can assume that G is an abelian monoid. If we take [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] and [[summation].sub.t[member of]G] [b.sub.t][u.sub.t] G in the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G, then, by the second part of Proposition 4 and the fact that [alpha] is symmetric, we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 4. Proposition 4, Corollary 3, Corollary 4, and Proposition 5 together generalize Theorem 1, Corollaries 5-10, and Proposition 4 in [18] from groups to categories.

Remark 5. Let A [??] G be a category algebra where all the rings [A.sub.e], e [member of] ob(G), coincide with a fixed integral domain D. Then A [??] G is the usual category algebra DG of G over D. By Corollary 4, the commutant of D in DG is DG itself. In particular, A is maximal commutative in DG if and only if G is the disjoint union of [absolute value of ob(G)] copies of the trivial group.

Remark 6. Let L/K be a finite separable (not necessarily normal) field extension. We use the same notation as in Remark 3. By Corollary 4, the commutant of F in F [[??].sup.[sigma].sub.[alpha]] G is the collection of [[summation].sup.n.sub.i=1] [[summation].sub.s[member of][G.sub.ii]] [a.sub.s][u.sub.s] satisfying [a.sub.s] = 0 whenever [[sigma].sub.s] is not an identity map. In particular, F is maximal commutative in F [[??].sup.[sigma].sub.[alpha]] G if all groups [G.sub.ii], i = 1, ..., n, are nontrivial; this of course happens in the case when L/K is a Galois field extension.

4. COMMUTATIVITY AND IDEALS

In this section, we investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptiness of intersections of the coefficient ring by nonzero two-sided ideals. For the rest of the article, we assume that ob(G) is finite. Recall (from Section 1) that this is equivalent to the fact that A [[??].sup.[sigma].sub.[alpha]] G has a multiplicative identity; in that case the multiplicative identity is [[summation].sub.e[member of]ob(G)] [u.sub.e].

Theorem 2. If A [[??].sup.[sigma].sub.[alpha]] G is a groupoid crossed product such that for every s [member of] G, [alpha](s, [s.sup.-1]) is not a zero-divisor in [A.sub.c(s)], then every intersection of a nonzero two-sided ideal of A [[??].sup.[sigma].sub.[alpha]] G with the commutant of Z(A) in A [[??].sup.[sigma].sub.[alpha]] G is nonzero.

Proof. We show the contrapositive statement. Let C denote the commutant of Z(A) in A [[??].sup.[sigma].sub.[alpha]] G and suppose that I is a two-sided ideal of A [[??].sup.[sigma].sub.[alpha]] G with the property that I [intersection] C = {0}. We wish to show that I = {0}. Take x [member of] I. If x [member of] C, then by the assumption x = 0. Therefore we now assume that x = [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] [member of] I, [a.sub.s] [member of] [A.sub.c(s)], s [member of] G, and that x is chosen so that x [not member of] C with the set S := {s [member of] G | [a.sub.s] [not equal to] 0} of least possible cardinality N. Seeking a contradiction, suppose that N is positive. First note that there is e [member of] ob(G) with [u.sub.e]x [member of] I \ C. In fact, if [u.sub.e]x [member of] C for all e [member of] ob(G), then x = 1x = [[summation].sub.e[member of]ob(G)] [u.sub.e]x [member of] C, which is a contradiction. By minimality of N we can assume that c(s) = e, s [member of] S, for some fixed e [member of] ob(G). Take t [member of] S and consider the element x' := [xu.sub.t-1] [member of] I. Since [alpha](t, [t.sup.-1]) is not a zero-divisor we get that x' [not equal to] 0. Therefore, since I [intersection] C = {0}, we get that x' [member of] I \ C. Take a = [[summation].sub.f[member of]ob(G)] [b.sub.f][u.sub.f] [member of] Z(A) and note that Z(A) = [[direct sum].sub.f[member of]ob(G)] Z([A.sub.f]). Then I [contain as member] x" := ax' - x'a = [[summation].sub.s[member of]S] ([b.sub.c(s)][a.sub.s] - [a.sub.s][[sigma].sub.s]([b.sub.d(s)]))[u.sub.s]. In the [A.sub.e] component of this sum we have [b.sub.e][a.sub.e] - [a.sub.e][b.sub.e] = 0 since [b.sub.e] [member of] Z([A.sub.e]). Thus, the summand vanishes for s = e, and hence we get, by the assumption on N, that x" = 0. Since a [member of] Z(A) was arbitrarily chosen, we get that x' [member of] C which is a contradiction. Therefore N = 0 and hence S = [??] which in turn implies that x = 0. Since x [member of] I was arbitrarily chosen, we finally get that I = {0}.

Corollary 5. If A [[??].sup.[sigma].sub.[alpha]] G is a groupoid crossed product with A maximal commutative and for every s [member of] G, [alpha](s, [s.sup.-1]) is not a zero-divisor in [A.sub.c(s)], then every intersection of a nonzero two-sided ideal of A [[??].sup.[sigma].sub.[alpha]] G with A is nonzero.

Proof. This follows immediately from Theorem 2. []

Now we examine conditions under which the converse statement of Corollary 5 is true. To this end, we recall some notions from category theory that we need in the sequel (for the details see e.g. [15]). Let G be a category. A congruence relation R on G is a collection of equivalence relations [R.sub.a,b] on hom(a, b), a, b [member of] ob(G),chosen so that if (s, s') [member of] [R.sub.a,b] and (t, t') [member of] [R.sub.b,c], then (ts, t's') [member of] [R.sub.a,c] for all a, b, c [member of] ob(G). Given a congruence relation R on G we can define the corresponding quotient category G/R as the category having as objects the objects of G and as arrows the corresponding equivalence classes of arrows from G. In that case there is a full functor [Q.sub.R] : G [right arrow] G/R which is the identity on objects and sends each morphism of G to its equivalence class in R. We will often use the notation [s] : = [Q.sub.R](s), s [member of] G. Suppose that H is another category and that F : G [right arrow] H is a functor. The kernel of F, denoted ker(F), is the congruence relation on G defined by letting (s, t) [member of] ker[(F).sub.a, b], a, b [member of] ob(G), whenever s,t [member of] hom(a,b) and F(s) = F(t). In that case there is a unique functor [P.sub.F] : G/ker(F) [right arrow] H with the property that [P.sub.F] [Q.sub.ker(F)] = F. Furthermore, if there is a congruence relation R on G contained in ker(F), then there is a unique functor N : G/R [right arrow] G/ker(F) with the property that N [Q.sub.R] = [Q.sub.ker(F)]. In that case there is therefore always a factorization F = [P.sub.F] N [Q.sub.R]; we will refer to this factorization as the canonical one.

Proposition 6. Let {A, G, [sigma], [alpha]} and {A, H, [tau], [beta]} be crossed systems with ob(G) = ob(H). Suppose that there is a functor F : G [right arrow] H satisfying the following three criteria: (i) F is the identity map on objects; (ii) [[tau].sub.F(s)] = [[sigma].sub.s], for s [member of] G; (iii) [beta](F(s),F(t)) = [alpha](s, t), for (s,t) [member of] [G.sup.(2)]. Then there is a unique A-algebra homomorphism A [[??].sup.[sigma].sub.[alpha]] G [right arrow] A [[??].sup.[tau].sub.[beta]] H, denoted [??], satisfying [??]([u.sub.s]) = [u.sub.F(s)], for s [member of] G.

Proof. Take x := [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G where [a.sub.s] [member of] [A.sub.c(s)], for s [member of] G. By A-linearity we get that [??](x) = [[summation].sub.s[member of]G] [a.sub.s][??]([u.sub.s]) = [[summation].sub.s[member of]G] [a.sub.s][u.sub.F(s)]. Therefore [??] is unique. It is clear that [??] is additive. By (i), [??] respects the multiplicative identities. Now we show that [??] is multiplicative. Take another y := [[summation].sub.s[member of]G] [b.sub.s][u.sub.s] in A [[??].sup.[sigma].sub.[alpha]] G where [b.sub.s] [member of] [A.sub.c(s)], for s [member of] G. Then, by (ii) and (iii), we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] []

Remark 7. Suppose that {A, G, [sigma], [alpha]} is a crossed system. By abuse of notation, we let A denote the category with the rings [A.sub.e], for e [member of] ob(G), as objects and ring homomorphisms [A.sub.e] [right arrow] [A.sub.f], for e, f [member of] ob(G), as morphisms. Define a map [sigma] : G [right arrow] A on objects by [sigma](e) = [A.sub.e], for e [member of] ob(G), and on arrows by [sigma](s) = [[sigma].sub.s], for s [member of] G. By Eq. (4) it is clear that [sigma] is a functor if the following two conditions are satisfied: (i) for all (s, t) [member of] [G.sup.(2)], [alpha](s, t) belongs to the centre of [A.sub.c(s)]; (ii) for all (s, t) [member of] [G.sup.(2)], [alpha](s, t) is not a zero-divisor in [A.sub.c(s)].

Proposition 7. Let A [[??].sup.[sigma].sub.[alpha]] G be a category crossed product with [sigma] : G [right arrow] A a functor. Suppose that R is a congruence relation on G with the property that the associated quadruple {A, G/R, [sigma]([x]), [alpha]([x], [x])} is a crossed system. If I is the two-sided ideal in A [[??].sup.[sigma].sub.[alpha]] G generated by an element [[summation].sub.s[member of]G] [a.sub.s][u.sub.s], where [a.sub.s] [member of] [A.sub.c(s)], for s [member of] G, satisfying [a.sub.s] = 0 if s does not belong to any of the classes [e], for e [member of] ob(G), and [[summation].sub.s[member of][e]] [a.sub.s] = 0, for e [member of] ob(G), then A [intersection]I = {0}.

Proof. By Proposition 6, the functor [Q.sub.R] induces an A-algebra homomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] G/R. By the definition of [a.sub.s], for s [member of] G, we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that [[??].sub.R](I) = {0}. Since [[??].sub.R|A] = [id.sub.A], we therefore get that I [intersection] A = ([[??].sub.R|A])(A [intersection] I) [subset or equal to] [[??].sub.R](I) = {0}.

Let G be a groupoid and suppose that we for each e [member of] ob(G) are given a subgroup [N.sub.e] of [G.sub.e]. We say that N = [[union].sub.e[member of]ob(G)] [N.sub.e] is a normal subgroupoid of G if [sN.sub.d(s)] = [N.sub.c(s)]s for all s [member of] G. The normal subgroupoid N induces a congruence relation ~ on G defined by letting s ~ t, for s, t [member of] G, if there is n in [N.sub.d(t)] with s = nt. The corresponding quotient category is a groupoid which is denoted G/N. For more details, see e.g. [4]; note that our definition of normal subgroupoids is more restrictive than the one used in [4].

Proposition 8. Let A [[??].sup.[sigma].sub.[alpha]] G be a groupoid crossed product such that for each (s, t) [member of] [G.sup.(2)], [alpha](s, t) [member of] Z(Ac(s)) and [alpha](s, t) is not a zero-divisor in [A.sub.c(s)]. Suppose that N is a normal subgroupoid of G with the property that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for n [member of] N, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if s [member of] N or t [member of] N. If I is the two-sided ideal in A [[??].sup.[sigma].sub.[alpha]] G generated by an element [[summation].sub.s[member of]G] [a.sub.s][u.sub.s], with as [member of] [A.sub.c(s)], for s [member of] G, satisfying [a.sub.s] = 0 if s does not belong to any of the sets [N.sub.e], for e [member of] ob(G), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for e [member of] ob(G), then A [intersection] I = {0}.

Proof. By Remark 7, [sigma] is a functor G [right arrow] A and ~ [subset or equal to] ker([sigma]). Therefore, by the discussion preceding Proposition 6, there is a well-defined functor [sigma][x] : G/N [right arrow] A. Now we show that the induced map [alpha]([x], [x]) is well-defined. By Eq. (3) with s = n [member of] [N.sub.c(t)] we get that [alpha](n, t)[alpha](nt, r) = [[sigma].sub.n]([alpha](t, r))[alpha](n, tr). By the assumptions on [alpha] and [sigma] we get that [alpha](nt, r) = [alpha](t, r). Analogously, by Eq. (3) with t = n [member of] [N.sub.d(r)], we get that [alpha](s, t) = [alpha](s, tn). Therefore, [alpha]([x], [x]) is well-defined. The rest of the claim now follows immediately from Proposition 7.

Proposition 9. Let A [[??].sup.[sigma]] G be a skew category algebra. Suppose that R is a congruence relation on G contained in ker([sigma]). If I is the two-sided ideal in A [[??].sup.[sigma]] G generated by an element [[summation].sub.s[member of]G] [a.sub.s][u.sub.s], where [a.sub.s] [member of] [A.sub.c(s)], for s [member of] G, satisfying [a.sub.s] = 0 if s does not belong to any of the classes [e], for e [member of] ob(G), and [[summation].sub.s[member of][e]] [a.sub.s] = 0, for e [member of] ob(G), then A [intersectioin] I = {0}.

Proof. By Remark 7 and the discussion preceding Proposition 6, there is a well-defined functor [sigma][x] :G/R [right arrow] A. The claim now follows immediately from Proposition 7.

Proposition 10. Let A [[??].sup.[sigma]] G be a skew groupoid ring with all [A.sub.e], for e [member of] ob(G), equal integral domains and each [G.sub.e], for e [member of] ob(G), an abelian group. If every intersection of a nonzero two-sided ideal of A [[??].sup.[sigma]] G and A is nonzero, then A is maximal commutative in A [[??].sup.[sigma]] G.

Proof. We show the contrapositive statement. Suppose that A is not maximal commutative in A [[??].sup.[sigma]] G. By the second part of Corollary 4, there is e [member of] ob(G) and a nonidentity s [member of] [G.sub.e] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [N.sub.e] denote the cyclic subgroup of [G.sub.e] generated by s. Note that since [G.sub.e] is abelian, [N.sub.e] is a normal subgroup of [G.sub.e]. For each f [member of] ob(G), define a subgroup [N.sub.f] of [G.sub.f] in the following way. If [G.sub.e,f] [not equal to]] [??], then let [N.sub.f] = [sN.sub.e][s.sup.-1], where s is a morphism in [G.sub.e, f]. If, on the other hand, [G.sub.e, f] = [??], then let [N.sub.f] = {f}. Note that if [s.sub.1], [s.sub.2] [member of] [G.sub.e, f], then [s.sup.-1.sub.2][s.sup.-1] [member of] [G.sub.e] and hence [s.sub.1][N.sub.e][s.sup.-1.sub.1] = [s.sub.2][s.sup.- 1.sub.2][s.sub.1] [N.sub.e] [([s.sup.-1.sub.2][s.sub.1]).sup.-1] [s.sup.-1.sub.2] = [s.sub.2][N.sub.e][s.sup.-1.sub.2]. Therefore, [N.sub.f] is well-defined. Now put N = [[universal].sub.f[member of]ob(G)] [N.sub.f]. It is clear that N is a normal subgroupoid of G and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let I be the nonzero two-sided ideal of A [[??].sup.[sigma]] G generated by [u.sub.e] - [u.sub.s]. By Proposition 8 (or Proposition 9) it follows that A [intersection] I = {0}.

Remark 8. Proposition 2, Corollary 5, and Propositions 7-10 together generalize Theorem 2, Corollary 11, Theorem 3, Corollaries 2-15, and Theorem 4 in [18] from groups to categories.

By combining Theorem 2 and Proposition 10, we get the following result.

Corollary 6. If [[??].sup.[sigma]] G is a skew groupoid ring with all [A.sub.e], for e [member of] ob(G), equal integral domains and each [G.sub.e], for e [member of] ob(G), an abelian group, then A is maximal commutative in A [[??].sup.[sigma]] G if and only if every intersection of a nonzero two-sided ideal of A [[??].sup.[sigma]] G and A is nonzero.

doi: 10.3176/proc.2010.4.13

ACKNOWLEDGEMENTS

The first author was partially supported by The Swedish Research Council, The Crafoord Foundation, The Royal Physiographic Society in Lund, The Swedish Royal Academy of Sciences, The Swedish Foundation of International Cooperation in Research and Higher Education (STINT) and "LieGrits", a Marie Curie Research Training Network funded by the European Community as project MRTN-CT 2003-505078.

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(1) The term groupoid has various meanings in the literature, e.g. in [8], a set with a binary operation is referred to as a groupoid.

Johan Oinert (a) * and Patrik Lundstrom (b)

(a) Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden Current address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark

(b) University West, Department of Engineering Science, SE-46186 Trollhattan, Sweden

* Corresponding author, oinert@math.ku.dk

Received 13 April 2009, accepted 2 June 2009
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