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Commutativity and ideals in algebraic crossed products | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Commutativity and ideals in algebraic crossed products

Johan OINERT* and Sergei D. SILVESTROV

Centre for Mathematical Sciences, Lund University, Box 118, SE-22100 Lund, Sweden

*Corresponding Author:
Johan OINERT
Centre for Mathematical Sciences
Lund University, Box 118
SE-22100 Lund
Sweden
E-mails: [email protected] and [email protected]
 

Received date: October 15, 2007; Revised date: September 01, 2008



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Abstract

We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the coecient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the coecient subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the coecient subring, specially taking into account both the case of coecient rings without non-trivial zero-divisors and the case of coecient rings with non-trivial zero-divisors.

Introduction

The description of commutative subrings and commutative subalgebras and of the ideals in non-commutative rings and algebras are important directions of investigation for any class of non-commutative algebras or rings, because it allows one to relate representation theory, noncommutative properties, graded structures, ideals and subalgebras, homological and other properties of non-commutative algebras to spectral theory, duality, algebraic geometry and topology naturally associated with the commutative subalgebras. In representation theory, for example, one of the keys to the construction and classi cation of representations is the method of induced representations. The underlying structures behind this method are the semi-direct products or crossed products of rings and algebras by various actions. When a non-commutative ring or algebra is given, one looks for a subring or a subalgebra such that its representations can be studied and classi ed more easily, and such that the whole ring or algebra can be decomposed as a crossed product of this subring or subalgebra by a suitable action. Then the representations for the subring or subalgebra are extended to representations of the whole ring or algebra using the action and its properties. A description of representations is most tractable for commutative subrings or subalgebras as being, via the spectral theory and duality, directly connected to algebraic geometry, topology or measure theory.

If one has found a way to present a non-commutative ring or algebra as a crossed product of a commutative subring or subalgebra by some action on it of the elements from outside the subring or subalgebra, then it is important to know whether this subring or subalgebra is maximal abelian or, if not, to nd a maximal abelian subring or subalgebra containing the given subalgebra, since if the selected subring or subalgebra is not maximal abelian, then the action will not be entirely responsible for the non-commutative part as one would hope, but will also have the commutative trivial part taking care of the elements commuting with everything in the selected commutative subring or subalgebra. This maximality of a commutative subring or subalgebra and associated properties of the action are intimately related to the description and classifications of representations of the non-commutative ring or algebra.

Little is known in general about connections between properties of the commutative subalgebras of crossed product rings and algebras and properties of the action. A remarkable result

in this direction is known, however, in the context of crossed product C*-algebras. In the case of the crossed product C*-algebra C(X) equation Z of the C*-algebra of complex-valued continuous functions on a compact Hausdorff space X by an action of Z via the composition automorphism associated with a homeomorphism σ : X → X, it is known that C(X) sits inside the C*-crossed product as a maximal abelian C*-subalgebra if and only if for every positive integer n, the set of points in X having period n under iterations of  has no interior points [26, Theorem 5.4], [25, Corollary 3.3.3], [27, Proposition 4.14], [10, Lemma 7.3.11]. This condition is equivalent to the action of Z on X being topologically free in the sense that the non-periodic points of  are dense in X. In [24], a purely algebraic variant of the crossed product allowing for more general classes of algebras than merely continuous functions on compact Hausdorα spaces serving as coecient algebras in the crossed products was considered. In the general set theoretical framework of a crossed product algebra A equation of an arbitrary subalgebra A of the algebra CX of complexvalued functions on a set X (under the usual pointwise operations) by Z acting on A via a composition automorphism de ned by a bijection of X, the essence of the matter is revealed. Topological notions are not available here and thus the condition of freeness of the dynamics as described above is not applicable, so that it has to be generalized in a proper way in order to be equivalent to the maximal commutativity of A. In [24] such a generalization was provided by involving separation properties of A with respect to the space X and the action for signi cantly more arbitrary classes of coecient algebras and associated spaces and actions. The (unique) maximal abelian subalgebra containing A was described as well as general results and examples and counterexamples on equivalence of maximal commutativity of A in the crossed product and the generalization of topological freeness of the action.

In this article, we bring these results and interplay into a more general algebraic context of crossed product rings (or algebras) for crossed systems with arbitrary group actions and twisting cocycle maps [17]. We investigate the connections with the ideal structure of a general crossed product ring, describe the center of crossed product rings, describe the commutant of the coecient subring in a crossed product ring of a general crossed system, and obtain conditions for maximal commutativity of the commutant of the coefficient subring in terms of the action as well as in terms of intersection of ideals in the crossed product ring with the coecient subring, specially taking into account both the case of coefficient rings without non-trivial zero-divisors and the case of coefficient rings with non-trivial zero-divisors.

Preliminaries

In this section we recall the notation from [17], which is necessary for the understanding of the rest of this article. Throughout this article all rings are assumed to be associative rings.

Definition 1. Let G be a group with unit element e. The ring R is G-graded if there is a family {R σ}σεG of additive subgroups Rσ of R such that R =equationσεGRσ and RσRT ⊆ RσT (strongly G-graded if, in addition, ⊇ also holds) for every σT∈G.

De nition 2. A unital and G-graded ring R is called a G-crossed product if U(R)∩ Rσ 6= ; for every σ∈ G, where U(R) denotes the group of multiplication invertible elements of R. Note that every G-crossed product is strongly G-graded, as explained in [17, p.2].

De nition 3. A G-crossed system is a quadruple {A; G; σ,α} consisting of a unital ring A, a group G (with unit element e), a map σ: G → Aut(A) and a σ-cocycle map α : G * G → U(A) such that for any x; y; z ∈ G and a &isin A the following conditions hold:

(I)equation

(ⅱ)equation

(ⅲ)equation

Remark 1. Note that, by combining conditions (i) and (iii), we get σee(a)) = σe(a) for all a 2 A. Furthermore, σe : A → A is an automorphism and hence σe = idA. Also note that, from the de nition of Aut(A), we have σg(0A) = 0A and σg(1A) = 1A for any g ∈ G. From condition (i) it immediately follows that σ is a group homomorphism if A is commutative or if α is trivial.

Definition 4. Let equation be a copy (as a set) of G. Given a G-crossed system {A; G; σ α} we denote by equation

the free left A-module having equation as its basis and we de ne a multiplication on this set by

equation (2.1)

for all a1; a2 ∈ A and x ,y ∈ G. Each element of equation may be expressed as a sum equation where ag ∈ A and ag = 0A for all but a infinite number of g ∈ G. Explicitly, the addition and multiplication of two arbitrary elements equation is given by

equation (2.2)

Remark 2. The ring A is unital, with unit element 1A, and it is easy to see that equation is the multiplicative identity in equation

By abuse of notation, we shall sometimes let 0 denote the zero element in equation and sometimes the unit element in the abelian group (Z; +). The proofs of the two following propositions can be found in [17, Proposition 1.4.1, p.11] and [17, Proposition 1.4.2, pp. 12-13] respectively (see also [18], [19]).

Proposition 1. Let {A; G; σ; α) be a G-crossed system. Then equation is an associative ring (with the multiplication de ned in (2.1)). Moreover, this ring is G-graded, equation=equation and it is a G-crossed product.

Proposition 2. Every G-crossed product R is of the form equation for some ring A and some maps σ α

Remark 3. If k is a field and A is a k-algebra, then so is equation

The coefficient ring A is naturally embedded as a subring into equation via the canonical isomorphism equationdefined by equation We denote by equation the image of A under equation and by equation the fixed ring of A. If A is commutative we define Ann(r) = equation

Remark 4. Obviously, A is commutative if and only if equation is commutative.

Example. Let A be commutative and B =equation a crossed product. For x ∈ G and c; d ∈ A we may write equation

Let equation and f : B → B be a map defined by equation Then the above relation may be written as b a = f(a) b, which is a re-ordering formula frequently appearing in physical applications.

Commutativity in equation

From the de nition of the product in equation given by (2.2), we see that two elements equation and equation commute if and only if

equation

for each g ∈ G. The crossed product equation is in general non-commutative and in the following proposition we give a description of its center.

Proposition 3. The center of equation is

equation

Proof. Let equation be an element which commutes with every element of equation Then, in particular equation must commute with equation for every a ∈ A. From (3.1) we immediately see that this implies equation for every a ∈ A and s ∈ G. Furthermore, equation must commute with equation for any s ∈ G. This yields

equation

and hence, for each (s; t) ∈ G * G, we have equation

Conversely, suppose that equation is an element satisfying equation and equation for every a ∈ A and (s; t) 2 G*G. Let equation be arbitrary. Then

equation

and hence equation commutes with every element of equation

A few corollaries follow from Proposition 3, showing how a successive addition of restrictions on the corresponding G-crossed system, leads to a simpli ed description of Z equation

Corollary 1 (Center of a twisted group ring). If equation , then the center of equation is

equation

Corollary 2. If G is abelian and α is symmetric1, then the center of equation is

Corollary 3. If A is commutative, G is abelian and equationthen the center of equation is

equation

Remark 5. Note that in the proof of Theorem 3, the property that the image of α is contained in U(A) is not used and therefore the theorem is true in greater generality. Consider the case when A is an integral domain and let α take its values in equation In this case it is clear that equation for all a ∈ A equation for all a ∈ A equation for equation After a change of variable via x = s-1t the first condition in the description of the center may be written as equation for all (s,x) ∈ G * G. From this relation we conclude that rx = 0 if and only if rsxs-1 = 0, and hence it is trivially satis ed if we put rx = 0 whenever equation This case has been presented in [19, Proposition 2.2] with a more elaborate proof.

The final corollary describes the exceptional situation when Z(equation)coincides with equation that is when equation is commutative.

Corollary 4. equation is commutative if and only if all of the following hold:

(i) A is commutative

(ii) σs = idA for each s ∈ G

(iii) G is abelian

(iv) &alpha is symmetric

Proof. Suppose that Z(equation =equation Then, equation and hence (i) follows by Remark 4. By assumption, equation for any s ∈ G and by Proposition 3 we see that σs = idA for every s &isin G, and hence (ii). For any (x,y) ∈ G * G we have equation equation but α(x, y) ≠ 0A which implies xy = yx and also α(x, y) = α(y, x), which shows (iii) and (iv). The converse implication is easily verified.

The commutant of equation in equation

From now on we shall assume that G ≠ {e}. As we have seen, equation is a subring of equation and we define its commutant by Comm equation Theorem 1 tells us exactly when an element of equation lies in Comm(equation )

Theorem 1. The commutant of equation in equation is

equation

Proof. The proof is established through the following sequence of equivalences:

equation

equation

Here we have used the fact that α(s, e) = α(e, s) = 1A for all s ∈ G. The above equivalence can also be deduced directly from (3.1).

When A is commutative we get the following description of the commutant by Theorem 1.

Corollary 5. If A is commutative, then the commutant of equation in equation is

equation

When A is commutative it is clear that equation ⊆ Comm(equation ). Using the explicit description of Comm(equation ) in Corollary 5, we are now able to state exactly when equation is maximal commutative, i.e. Comm(equation ) = equation .

Corollary 6. Let A be commutative. equation is maximal commutative in equation if and only if, for each pair equation there exists a ∈ A such that σs(a) - ∉ 62 Ann(rs).

Example (The crossed product associated to a dynamical system). In this example we follow the notation of [24]. Let σ : X → X be a bijection on a non-empty set X, and A ⊆ CX an algebra of functions, such that if h ∈ A then h 0 σ ∈ A and h 0 σ-1 ∈ A. Let equation be defined by equation for f ∈ A. We now have a Z-crossed system (with trivial equation cocycle) and we may form the crossed product equation Recall the definition of the set Sepn A(X) =equation Corollary 6 is a generalization of [24, Theorem 3.5] and the easiest way to see this is by negating the statements. Suppose that A is not maximal commutative in equation . Then, by Corollary 6, there exists a pair equation such that equation for every g ∈ A, i.e. equation for every g ∈ A. In particular, this means that fn is identically zero on SepnA(X). However, equation is not identically zero on X and hence SepnA(X) is not a domain of uniqueness (as defined in [24, Definition 3.2]). The converse can be proved similarly.

Corollary 7. Let A be commutative. If for each equation it is always possible to nd some a ∈ A such that σs(a)-a is not a zero-divisor in A, then equation is maximal commutative in equation

The next corollary is a consequence of Corollary 6 and shows how maximal commutativity of the coefficient ring in the crossed product has an impact on the non-triviality of the action σ.

Corollary 8. If equation equation is maximal commutative in then σg ≠ idA for every equation

The description of the commutant Comm(equation ) from Corollary 5 can be further re ned in the case when A is an integral domain.

Corollary 9. If A is an integral domain2, then the commutant of equation in equation is

equation

where equation

2By an integral domain we shall mean a commutative ring with an additive identity 0A and a multiplicative identity 1A such that 0A ≠ 1A, in which the product of any two non-zero elements is always non-zero.

Corollary 10. Let A be an integral domain. equation is maximal commutative in equation if and only if σg ≠ idA for every equation

Corollary 10 can be derived directly from Corollary 8 together with either Corollary 7 or 9.

Remark 6. Recall that when A is commutative,  is a group homomorphism. Thus, to say that σg ≠ idA for all equation is another way of saying that ker(σ) = {e}, i.e. σ is injective.

Example. Let A = C[x1. . . . . xn] be the polynomial ring in n commuting variables x1. . . . . xn and G = Sn the symmetric group on n elements. An element  2 Sn is a permutation which maps the sequence (1. . . . . n) into ( (1). . . . .  (n)). The group Sn acts on C[x1. . . . . xn] in a natural way. To each T ∈ Sn we may associate a map A → A, which sends any polynomial f(x1. .... xn) 2 C[x1. . . . . xn] into a new polynomial g, de ned by g(x1. . . . . xn) = f(x(1). . . . . x(n)). It is clear that each such mapping is a ring automorphism on A. Let σ be the embedding equation Aut(A) and equation . Note that C[x1. . . . . xn] is an integral domain and that σ is injective. Hence, by Corollary 10 and Remark 6 it is clear that the embedding of C[x1. . . . . xn] is maximal commutative in C[x1. . . . . xn] equation

One might want to describe properties of the σ-cocycle in the case when equation is maximal commutative, but unfortunately this will lead to a dead end. The explaination for this is revealed by condition (iii) in the de nition of a G-crossed system, where we see that α(e,g) = α(g,e) = 1A for all g ∈ G and hence we are not able to extract any interesting information about α by assuming that equation is maximal commutative. Also note that in a twisted group ring A oα G, i.e. with equation , equation can never be maximal commutative (when G ≠ {e}), since for each g ∈ G, equation centralizes equation . If A is commutative, then this follows immediately from Corollary 8. We shall now give a sucient condition for Comm(equation ) to be commutative.

Proposition 4. If A is a commutive ring, G is an abelian group and α is symmetric, then Comm(equation ) is commutative.

Proof. Let equation andequation be arbitrary elements of Comm(equation ). By our assumptions and Corollary 5 we get

equation

This shows that Comm(equation ) is commutative.

This proposition is a generalization of [24, Proposition 2.1] from a function algebra to an arbitrary unital associative commutative ring A, from Z to an arbitrary abelian group G and from a trivial to a possibly non-trivial symmetric σ-cocycle α.

Remark 7. By using Proposition 4 and the arguments made in the previous example on the crossed product associated to a dymical system it is clear that Corollary 5 is a generalization of [24, Theorem 3.3]. Furthermore, we see that Corollary 3 is a generalization of [24, Theorem 3.6].

Ideals in equation

In this section we describe properties of the ideals in equationin connection with maximal commutativity and properties of the action σ.

Theorem 2. If A is commutative, then equation for every non-zero two-sided ideal I in equation

Proof. Let A be commutative. Then equation is also commutative. Let I ⊆ equation be an arbitrary non-zero two-sided ideal in equation.

Part 1: For each g ∈ G we define a map Tg :equationequation by equation Note that, for any g ∈ G, I is invariant3 under Tg. We have

equation

for every g ∈ G. It is important to note that if as ≠ 0A, then as α(s, g) ≠ 0A and hence this operation does not kill coefficients, it only translates and deformes them. If we have a non-zero element equation for which ae = 0A, then we may pick some non-zero coecient, say ap and apply the map Tp-1 to end up with

equation

This resulting element will then have the following properties:

equation

Part 2: For each a ∈ A we define a map Da : equationequation by

equation

Note that, for each a ∈ A, I is invariant under Da. By assumption A is commutative and hence the above expression can be simplified.

equation

The maps {Da}a∈A all share the property that they kill the coecient in front e. Hence, if ae ≠0A, then the number of non-zero coefficients of the resulting element will always be reduced by atleast one. Note that equation This means that for each non-zero equation in equation we may always choose some a ∈ A such that equation By choosing such an a we note that, using the same notation as above, we get equation

for each non-zero equation

Part 3: The ideal I is assumed to be non-zero, which means that we can pick some non-zero element equation.If equation then we are finished, so assume that this is not the case. Note that rs ≠ 0A for finitely many s ∈ G. Recall that the ideal I is invariant under Tg and Da for all g ∈ G and a ∈ A. We may now use the maps {Tg}g∈G and {Da}a∈A to generate new elements of I. More specifically, we may use the Tg:s to translate our element equation into a new element which has a non-zero coecient in front of equation (if needed) after which we use the map Da to kill this coecient and end up with yet another new element of I which is non-zero but has a smaller number of non-zero coecients. We may repeat this procedure and in a nite number of iterations arrive at an element of I which lies in Comm(equation ) n equation and if not we continue the above procedure until we reach an element which is of the form b e with some non-zero b ∈ A. In particular equation ⊆ Comm(equation ) and hence I ∩ Comm(equation ) ≠ {0}.

The embedded coefficient ring equation is maximal commutative if and only if equation = Comm(equation ) and hence we have the following corollary.

Corollary 11. If the subring equation is maximal commutative in equation then

I ∩ equation ≠ {0}

for every non-zero two-sided ideal I in equation

Proposition 5. Let I be a subset of A and define

equation

The following assertions hold:

(i) If I is a right ideal in A, then J is a right ideal in equation

(ii) If I is a two-sided ideal in A such that I ⊆ AG, then J is a two-sided ideal in equation

Proof. If I is a (possibly one-sided) ideal in A, then J is an additive subgroup of equation

(i). Let I be a right ideal in A. Then

equation

for arbitrary equation and equationand hence J is a right ideal.

(ii). Let I be a two-sided ideal in A such that I ⊆ AG. By (i) it is clear that J is a right ideal.

Let equation and equation be arbitrary. Then

equation

which shows that J is also a left ideal.

Theorem 3. Let σ : G → Aut(A) be a group homomorphism and N be a normal subgroup of G, contained in equation Let equation be the quotient group homomorphism and suppose that α is such that α(s, t) = 1A whenever s ∈ N or t ∈ N. Furthermore, suppose that there exists a map β : G/N *G/N → U(A) such that equation&alpha(s, t) for each (s, t) ∈ G * G. If I is an ideal in equation generated by an element equation for which the coecients (of which all but nitely many are zero) satisfy equation then

equation

Proof. Let equation be the ideal generated by an element equation which satisfies equation The quotient homomorphism equation satisfies equation By assumption, the map σ is a group homomorphism and σ(N) = idA. Hence by the universal property, see for example [7, p.16], there exists a unique group homomorphism ρ making the following diagram commute:

equation

By assumption there exists β such that equation for each equation One may verify that β is a ρ-cocycle and hence we can define a new crossed product equation Let T be a transversal to N in G and define to be the map equation

equation

which is a ring homomorphism. Indeed,equation is clearly additive and due to the assumptions, for any two elements equation and equation in equation , the multiplicativity of equation follows by

equation

equation

and hence equation defines a ring homomorphism. We shall note that the generator of I is mapped onto zero, i.e.

equation

and hence equation Furthermore, we see that

equation

and hence equation is injective. We may now conclude that if equation then equation and so necessarily c = 0. This shows that I∩equation = {0}.

If A is commutaive, then σ is automatically a group homomorphism and we get the following.

Corollary 12. Let A be commutative and equation a normal subgroup of G.Let equation be the quotient group homomorphism and suppose that &alpha is such that α(s, t) = 1A whenever s ∈ N or t ∈ N. Furthermore, suppose that there exists a map β : G/N * G/N → U(A) such that equation for each (s, t) ∈ G * G. If I is an ideal in equation generated by an element equation for which the coecients (of which all but finitely many are zero) satisfy equation then I ∩ equation = {0}.

When α ≡ 1A we need not assume that A is commutative, in order to make σ a group homomorphism. In this case we may choose β ≡ 1A and by Theorem 3 we have the following corollaries.

Corollary 13. Let α ≡ 1A and equation be a normal subgroup of G. If I is an ideal in equation generated by an element equation for which the coefficients (of which all but finitely many are zero) satisfy equation then I ∩ equation = {0}.

Corollary 14. If α≡1A then the following implication holds:

equation

(ii) For each g ∈ Z(G) ∩ σ-1(idA), the ideal Ig generated by the element equation for which equation has the property equation

Proof. Suppose that there exists a equation.Let equation be the ideal generated by equation where equation has the property equation

Proof. Suppose that there exists a equation Let equation be the ideal generated by equation where equation The element g commutes with each element of G and hence the cyclic subgroup N = < g > generated by g is normal in G and since σ is a group homomorphism equation.Hence equation by Corollary 13.

Corollary 15. If equation and G is abelian, then the following implication holds:

(i):equation for every non-zero two-sided ideal I in equation

(ii): equation for all g ∈ G / {e}

Proof by contrapositivity. Since G is abelian, G = Z(G). Suppose that (ii) is false, i.e. there exists equation such that equation Pick such a g and let equationbe the ideal generated by equation Then obviously Ig ≠ {0} and by Corollary 14 we get Ig \equation = {0} and hence (i) is false. This concludes the proof.

Example. We should note that in the proof of Corollary 15 one could have chosen the ideal in many di erent ways. The ideal generated by equation equation is contained in the ideal Ig, generated by equation and therefore it has zero intersection with equation if equation. Also note that for α≡1A we may always write

equation

and hence equation is a zero-divisor in equationwhenever g is a torsion element.

Example. We now give an example of how one may choose β as in Theorem 3. Let N ⊆σ-1(idA) be a normal subgroup of G such that for g ∈ N, α(s, g) = 1A for all s ∈ G and let α be symmetric. Since α is the σ-cocycle map of a G-system, we get

equation

for all (s,t) ∈ G * G. Using the last equality and the symmetry of we immediately see that equation

for all g; h ∈ N. The last equality means that α is constant on the pairs of right cosets which coincide with the left cosets by normality of N. It is therefore clear that we can define

equation

Theorem 4. If A is an integral domain, G is an abelian group and α ≡ 1A, then the following implication holds:

(i): equation for every non-zero two-sided ideal I in equation

(ii): equation is a maximal commutative subring in equation

Proof. This follows from Corollary 10 and Corollary 15.

Example (The quantum torus). Letequation and denote by equationthe twisted Laurent polynomial ring in two non-commuting variables under the twisting

y x = q x y (5.1)

The ring equation is known as the quantum torus. Now let equation equationfor n ∈ G and P(x) ∈ A, and let α(s, t) = 1A for all s; t ∈ G. It is easily verified that σ and α together satisfy conditions (i)-(iii) of a G-system and it is not hard to see thatequation In the current example, A is an integral domain, G is abelian, α ≡ 1A and hence all the conditions of Theorem 4 are satis ed. Note that the commutation relation (5.1) implies equation (5.2)

It is important to distinguish between two di erent cases:

Case 1 (q is a root of unity). Suppose that qn = 1 for some n ≠ 0. From equality (5.2) we note that equation and hence equation is not maximal commutative in equationThus, according to Theorem 4, there must exist some non-zero ideal I which has zero intersection withequation

Case 2 (q is not a root of unity). Suppose that qn ≠ 1 for all equation One can show that this implies that equation is simple. This means that the only non-zero ideal is equationitself and this ideal obviously intersects equation non-trivially. Hence, by Theorem 4, we conclude that equationis maximal commutative in equation

Ideals, intersections and zero-divisors

Let D denote the subset of zero-divisors in A and note that D is always non-empty since 0A ∈ D. By we denote the image of equation under the embedding .

Theorem 5. If A is commutative, then the following implication holds:

(i):equation for every non-zero two-sided ideal I in equation

(ii):equation i.e. the only zero-divisor that is fixed under all automorphisms is 0A

Proof by contrapositivity. Let A be commutative. Suppose that equation Then there exist some equation such that σs(c) = c for all s ∈ G. There is also some equation such that c.d = 0A. Consider the ideal Ann(c) = {a ∈ A a.c = 0A} in A. It is clearly non-empty since we always have 0A ∈ Ann(c) and d ∈ Ann(c). Let θ : A → A/ Ann(c) be the quotient homomorphism defined by a → a + Ann(c). Let us define a map ρ : G → Aut(A= Ann(c)) by ρs(a + Ann(c)) = σs(a) + Ann(c) for a + Ann(c) ∈ A= Ann(c) and s ∈ G. Note that Ann(c) is invariant under σs for every s ∈ G and thus it is easily verified that ρs is a well-de ned automorphism on A= Ann(c) for each s ∈ G. Define a map β : G * G → U(A= Ann(c)) by (s, t) → (θ0 α)(s, t). It is not hard to see that {A= Ann(c), G, &rho, β} is in fact a G-crossed system. Consider the map equation defined by equation For any two elements equation the additivity of equationfollows by

equation

and due to the assumptions, the multiplicativity follows by

equation

equation

where we have used that equationand equation for all bt ∈ A and s ∈ G. This shows that equation Now, pick some g ≠ e and let I be the ideal generated by equation.Clearly equation and Note that ker(θ) = Ann(c) and in particular equation implies a ∈ Ann(c): Take equation.Then equation and hence equation which is a contradiction. Thus, equationand by contrapositivity this concludes the proof.

Example (The truncated quantum torus). Let equation and consider the ring equation which is commonly referred to as the truncated quantum torus. It is easily verified that this ring is isomorphic to equation with equation for n ∈ G and P(x) ∈ A, and α(s, t) = 1A for all s; t ∈ G. One should note that in this case A is commutative, but not an integral domain. In fact, the zero-divisors in equation are precisely those polynomials where the constant term is zero, i.e. equation with equation such that a0 = 0. It is also important to remark that, unlike the quantum torus, equation is never simple (for m > 1). In fact we always have a chain of two-sided ideals

equation

independent of the value of q. Moreover, the two-sided ideal equation is contained in equation and contains elements outside of equation Hence we conclude that equation is not maximal commutative in equation.When q is a root of unity, with qn = 1 for some n < m, we are able to say more. Consider the polynomial p(x) = xn, which is a non-trival zero-divisor in equation For every equation. we see that p(x) = xn is fixed under the automorphism σs and therefore, by Theorem 5, we conclude that there exists a non-zero two-sided ideal in equation such that its intersection with equationis empty.

Comments to the literature

The literature contains several di erent types of intersection theorems for group rings, Ore extensions and crossed products. Typically these theorems rely on heavy restrictions on the coefficient rings and the groups involved. We shall now give references to some interesting results in the literature.

It was proven in [23, Theorem 1, Theorem 2] that the center of a semiprimitive (semisimple in the sense of Jacobson [6]) P.I. ring respectively semiprime P.I. ring has a non-zero intersection with every non-zero ideal in such a ring. For crossed products satisfying the conditions in [23, Theorem 2], it o ers a more precise result than Theorem 2 since Z(equation) ⊆ Comm(A~). However, every crossed product need not be semiprime nor a P.I. ring and this justi es the need for Theorem 2.

In [12, Lemma 2.6] it was proven that if the coecient ring A of a crossed product equation is prime, P is a prime ideal in equation such that P ∩ equation = 0 and I is an ideal in equation properly containing P, then I ∩ equation ≠ 0. Furthermore, in [12, Proposition 5.4] it was proven that the crossed product equation with G abelian and A a G-prime ring has the property that, if Ginn = {e} then every non-zero ideal in equation has a non-zero intersection with equation It was shown in [2, Corollary 3] that if A is semiprime and Ginn = {e},then every non-zero ideal in equationwith P ∩equation = 0 and if I is an ideal in equation properly containing P, then I ∩ equation ≠ 0. In [16, Proposition 2.6] it was shown that if A is a prime ring and I is a non-zero ideal in equation then equation In [16, Proposition 2.11] it was shown that for a crossed product equation with prime ring A, every non-zero ideal in equation has a non-zero intersection with equation if and only if equation is G-simple and in particular if equationthen every non-zero ideal in equation has a non-zero intersection with equation if and only if equation is prime.

Corollary 11 shows that if equation is maximal commutative in equation without any further conditions on the coefficient ring or the group, we are able to conclude that every non-zero ideal in equationhas a non-zero intersection with equation .

In the theory of group rings (crossed products with no action or twisting) the intersection properties of ideals with certain subrings have played an important role and are studied in depth in for example [3], [11] and [22]. Some further properties of intersections of ideals and homogeneous components in graded rings have been studied in for example [1], [14].

For ideals in Ore extensions there are interesting results in [4, Theorem 4.1] and [8, Lemma 2.2, Theorem 2.3, Corollary 2.4], explaining a correspondence between certain ideals in the Ore extension and certain ideals in its coefficient ring. Given a domain A of characteristic 0 and a non-zero derivation δ it is shown in [5, Proposition 2.6] that every non-zero ideal in the Ore extension R = A[x; δ] intersects A in a non-zero δ-invariant ideal. Similar types of intersection results for ideals in Ore extension rings can be found in for example [9] and [15]. The results in this article appeared initially in the preprint [20].

Acknowledgements. This work was supported by 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.

We are grateful to Marcel de Jeu, Christian Svensson, Theodora Theohari-Apostolidi and especially Freddy Van Oystaeyen for useful discussions on the topic of this article.

References

  1. s on the center of a ring with polynomial identity. Bull Amer MathSoc 79: 219-223.
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