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- *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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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 classication 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 classied 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) 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 of an arbitrary subalgebra A of the algebra C^{X} of complexvalued
functions on a set X (under the usual pointwise operations) by Z acting on A via a
composition automorphism dened 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 signicantly
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.

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 =_{σεG}R_{σ} and R_{σ}R_{T} ⊆ R_{σT} (strongly
G-graded if, in addition, ⊇ also holds) for every σ*T*∈G.

Denition 2. A unital and G-graded ring R is called a *G-crossed product* if U(R)∩ R_{σ} 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].

Denition 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)

(ⅱ)

(ⅲ)

**Remark 1**. Note that, by combining conditions (i) and (iii), we get σ_{e}(σ_{e}(a)) = σ_{e}(a) for all
a 2 A. Furthermore, σ_{e} : A → A is an automorphism and hence σ_{e} = id_{A}. Also note that, from
the denition of Aut(A), we have σ_{g}(0A) = 0A and σ_{g}(1_{A}) = 1_{A} 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 be a copy (as a set) of G. Given a G-crossed system
{*A*; *G*; σ α}
we denote by

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

(2.1)

for all a_{1}; a_{2} ∈ A and x ,y ∈ G. Each element of may be expressed as a sum where a_{g} ∈ A and a_{g} = 0_{A} for all but a infinite number of g ∈ G. Explicitly, the addition and
multiplication of two arbitrary elements is given by

(2.2)

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

By abuse of notation, we shall sometimes let 0 denote the zero element in 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 is an associative ring
(with the multiplication dened in (2.1)). Moreover, this ring is G-graded, = and it is a G-crossed product. *

**Proposition 2.** *Every G-crossed product R is of the form 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

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

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

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

Let and f : B → B be a map defined by 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 **

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

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

**Proposition 3.** The center of is

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

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

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

and hence commutes with every element of

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

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

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

**Corollary 3**. *If A is commutative, G is abelian and then the center of is *

**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 In this case it is clear that for all a ∈ A for all a ∈ A for After a change of variable via x = s^{-1}t the first condition
in the description of the center may be written as for all
(s,x) ∈ G * G. From this relation we conclude that r_{x} = 0 if and only if r_{sxs}-1 = 0, and hence
it is trivially satised if we put r_{x} = 0 whenever This case has been presented in
[19, Proposition 2.2] with a more elaborate proof.

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

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

*(i) A is commutative*

*(ii) σ _{s} = id_{A} for each s ∈ G*

*(iii) G is abelian*

*(iv) &alpha is symmetric*

**Proof.** Suppose that Z( = Then, and hence (i)
follows by Remark 4. By assumption, for any s ∈ G and by Proposition 3 we
see that σ_{s} = id_{A} for every s &isin G, and hence (ii). For any (x,y) ∈ G * G we have but *α(x, y) ≠ 0 _{A}* which implies

**The commutant of in **

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

**Theorem 1.** The commutant of in is

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

Here we have used the fact that *α(s, e) = α(e, s) = 1 _{A}* 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 in is *

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

**Corollary 6.** *Let A be commutative. is maximal commutative in if and only if, for
each pair there exists a ∈ A such that σ _{s}(a) - ∉ 62 Ann(r_{s}). *

**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 ⊆ C^{X} an
algebra of functions, such that if h ∈ A then h 0 σ ∈ A and h 0 σ^{-1} ∈ A. Let be defined by for f ∈ A. We now have a Z-crossed system (with trivial cocycle) and we may form the crossed product Recall the definition of the set Sep^{n} _{A}(X) = 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 . Then, by Corollary 6, there exists a pair such that for every g ∈ A, i.e. for every g ∈ A. In particular, this means that f_{n} is identically zero on Sep^{n}_{A}(X). However, is not identically zero on X and hence Sep^{n}_{A}(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 it is always possible to nd some
a ∈ A such that σ _{s}(a)-a is not a zero-divisor in A, then is maximal commutative in *

*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 is maximal commutative in
then σg ≠ id _{A} for every *

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

**Corollary 9.*** If A is an integral domain ^{2}, then the commutant of in is *

where

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

**Corollary 10.*** Let A be an integral domain. is maximal commutative in if and only
if σ _{g} ≠ id_{A} for every*

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 σ_{g} ≠ id_{A} for all is another way of saying that ker(σ) = {e}, i.e. σ is injective.

Example. Let A = C[x_{1}. . . . . x_{n}] be the polynomial ring in n commuting variables x_{1}. . . . . x_{n} 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[x_{1}. . . . . x_{n}] in a natural way.
To each T ∈ S_{n} we may associate a map A → A, which sends any polynomial f(x_{1}. .... x_{n}) 2
C[x_{1}. . . . . x_{n}] into a new polynomial g, dened by g(x_{1}. . . . . x_{n}) = f(x(1). . . . . x(n)). It is
clear that each such mapping is a ring automorphism on A. Let σ be the embedding Aut(A) and . Note that C[x_{1}. . . . . x_{n}] is an integral domain and that σ is injective.
Hence, by Corollary 10 and Remark 6 it is clear that the embedding of C[x_{1}. . . . . x_{n}] is maximal
commutative in C[x_{1}. . . . . x_{n}]

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

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

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

This shows that Comm( ) 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 **

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

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

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

**Part 1:** For each g ∈ G we define a map T_{g} : → by Note that, for any g ∈ G, I is invariant^{3} under T_{g}. We have

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

This resulting element will then have the following properties:

**Part 2:** For each a ∈ A we define a map D_{a} : → by

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

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

for each non-zero

**Part 3:** The ideal I is assumed to be non-zero, which means that we can pick some non-zero element .If then we are finished, so assume that this is not the
case. Note that r_{s} ≠ 0_{A} for finitely many s ∈ G. Recall that the ideal I is invariant under T_{g} and D_{a} for all g ∈ G and a ∈ A. We may now use the maps {T_{g}}_{g}∈G and {D_{a}}_{a}∈A to generate
new elements of I. More specifically, we may use the T_{g}:s to translate our element into a new element which has a non-zero coecient in front of (if needed) after which we
use the map D_{a} 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( ) n 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 ⊆ Comm( ) and hence I ∩ Comm( ) ≠ {0}.

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

Corollary 11. If the subring is maximal commutative in then

I ∩ ≠ {0}

for every non-zero two-sided ideal I in

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

The following assertions hold:

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

(ii) If I is a two-sided ideal in A such that I ⊆ A^{G}, then J is a two-sided ideal in

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

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

for arbitrary and and hence J is a right ideal.

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

Let and be arbitrary. Then

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

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

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

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

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

and hence Furthermore, we see that

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

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

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

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

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

**Corollary 14.*** If α≡1 _{A} then the following implication holds:*

(ii) For each g ∈ Z(G) ∩ σ^{-1}(id_{A}), the ideal I_{g} generated by the element for which has the property

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

**Proof.** Suppose that there exists a Let be the ideal
generated by where 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 .Hence by Corollary 13.

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

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

(ii): for all g ∈ G / {e}

**Proof by contrapositivity.** Since G is abelian, G = Z(G). Suppose that (ii) is false, i.e. there
exists such that Pick such a g and let be the ideal generated
by Then obviously I_{g} ≠ {0} and by Corollary 14 we get I_{g} \ = {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 dierent ways. The ideal generated by is contained in the ideal I_{g}, generated by and therefore it
has zero intersection with if . Also note that for α≡1_{A} we may always write

and hence is a zero-divisor in whenever 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}(id_{A}) be a normal subgroup of G such that for g ∈ N, α(s, g) = 1_{A} for all s ∈ G and let α
be symmetric. Since α is the *σ-cocycle* map of a G-system, we get

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

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

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

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

(ii): is a maximal commutative subring in

Proof. This follows from Corollary 10 and Corollary 15.

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

*y x = q x y *(5.1)

The ring is known as the quantum torus. Now let for n ∈ G and P(x) ∈ A, and let α(s, t) = 1_{A} 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
that In the current example, A is an integral domain, G is abelian,
α ≡ 1_{A} and hence all the conditions of Theorem 4 are satised. Note that the commutation
relation (5.1) implies (5.2)

It is important to distinguish between two dierent cases:

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

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

**Ideals, intersections and zero-divisors**

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

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

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

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

**Proof by contrapositivity.** Let A be commutative. Suppose that Then there
exist some such that σ_{s}(c) = c for all s ∈ G. There is also some such
that c.d = 0_{A}. Consider the ideal Ann(c) = {a ∈ A a.c = 0_{A}} in A. It is clearly non-empty
since we always have 0_{A} ∈ 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 σ_{s} for every s ∈ G and thus it is easily verified that ρ_{s} is a well-dened
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 defined by For
any two elements the additivity of follows by

and due to the assumptions, the multiplicativity follows by

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

Example (The truncated quantum torus). Let and consider the ring which is commonly referred to as the truncated quantum torus. It is easily verified
that this ring is isomorphic to with for n ∈ G and P(x) ∈ A, and α(s, t) = 1_{A} 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 are precisely
those polynomials where the constant term is zero, i.e. with such
that a0 = 0. It is also important to remark that, unlike the quantum torus, is never
simple (for m > 1). In fact we always have a chain of two-sided ideals

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

The literature contains several dierent 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 oers a more precise result than Theorem 2 since Z() ⊆ Comm(A~). However, every crossed product need not be semiprime nor a P.I. ring and this justies the need for Theorem 2.

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

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

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.

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