Partial Actions and Quotient Rings

In this paper we study the Martindale ring of α−quotients Q associated with the partial action (R,α). Among other results we extend the partial action to Q and prove that it can be identified with an ideal of Q, the Martindale ring of β-quotients of T , where (T, β) denotes the enveloping action of (R,α). We prove that, in general, (Q, β) is not the enveloping action of (Q, α) and study the relationship between the rings R, Q, T and Q. Finally, we establish some properties related to the center of Q and the extended α−centroid of R. ∗The results presented in this paper are part of the PhD. thesis of the author presented to Universidade Federal do Rio Grande do Sul (Brazil). The paper was partially supported by CNPq (Brazil).


Introduction
Partial actions of groups have been considered in many contexts.This theory were introduced in the theory of operator algebras (see [6], [7] and the literature quoted therein).The partial actions on algebras in a purely algebraic framework were introduced by Dokuchaev and Exel in [6].They are a powerful tool in the generalization of known results of global actions in several areas as partial Galois theory, skew polynomial rings, skew group rings, fixed rings, Hopf algebras and entwining structures.
According to [9], Martindale's theory was originally introduced for prime rings and was primarily intended to deal with applications to rings satisfying a polynomial identity.The generalization to semiprime rings was due to Amitsur.These rings of quotients associated with semiprime rings have since proved to be useful not only for the theory of rings with polynomial identities, but also for the Galois theory of noncommutative rings and for the study of prime ideals under ring extensions in general.
The Martindale ring of quotients has been used successfully applied by several authors to the study of partial actions.For example, Ferrero ([7]) proved that any proper partial action α on a semiprime ring R possesses a weak enveloping action.Also, Cortes et al. ( [5]) proved that R is right Goldie if and only if R[x; α] is right Goldie if and only if Rhx; αi is right Goldie, where R is a semiprime ring, α is a partial action on R, R[x; α] is the partial skew polynomial ring and Rhx; αi is the partial skew Laurent polynomial ring.More recently, the Martindale ring of α−quotients Q for a partial action (R, α) has been introduced in [2] to study the correspondence between all closed (R-disjoint prime) ideals of R α G and all closed (Qdisjoint prime) ideals of Q α G. So, the ring Q is shown to be a suitable environment to generalize many well-known results of global actions.
Q has been only employed as a tool for studying prime ideals in some ring extensions.The main goal of this paper is to present several properties of Q and study the relationship between all the rings of the diagram where R is an α-prime ring, (T, β) is the enveloping action of (R, α), Q the ring of α-quotients of R, (E, β) the enveloping action of (Q, α) and Q the ring of β-quotients of T .
In Section 1 we present partial actions and some notions related to enveloping actions and α−invariant ideals, which will be used in other sections.In Section 2 we sketch the construction of the ring Q and extend the partial action (R, α) to a partial action on Q.In addition, we prove that the ring Q can be identified with an ideal of Q, where Q is the ring of βquotients of the enveloping action (T, β) of (R, α).Moreover, we determine under which assumptions (Q, β) is the enveloping action of (Q, α).Among other results in Section 3 we prove some properties related to the center of Q and the extended α−centroid of R.

Preliminaries
In this section we present the basic theory of partial actions related to enveloping actions and α−invariant ideals.More details can be found in [6].
Definition 1.1.Let G be a group and R a unital k-algebra, where k is a commutative ring.A partial action α of G on R is a collection of ideals S g , g ∈ G of R and isomorphisms of (non-necessarily unital) k-algebras α g : S g −1 → S g such that for all g, h ∈ G the following statements hold: 1. S 1 = R and α 1 is the identity mapping of R.
Natural examples of partial actions can be obtained by restricting a global action to an ideal.More precisely, suppose that the group G acts on an algebra T by automorphisms β g : T → T and let R be an ideal of T .Set S g = R ∩ β g (R) and let α g be the restriction of β g to S g −1 , for every g ∈ G. Then it is easy to see that α = {α g : S g −1 → S g | g ∈ G} is a partial action of G on R. In this case, we say that α is the restriction of β to R. In additon, if T is generated by In what follows, we present the definition of equivalent partial actions and enveloping action ( [6], Definitions 4.1 and 4.2 respectively).

Definition 1.2. Given the partial actions
respectively, we say that α and α 0 are equivalent if there exists an algebra isomorphism ϕ : R → R 0 such that for every g ∈ G the following conditions hold: Definition 1.
3. An action β of G on an algebra T is said to be an enveloping action for the partial action α of G on R if α is equivalent to an admissible restriction of β to an ideal of T .
In other words, β is an enveloping action for α if there exists an algebra isomorphism ϕ of R onto an ideal of T such that for all g ∈ G the following properties hold:

T is generated by
Thus it is natural to ask when a given partial action can be obtained as the restriction of a global action.The solution to this problem was given in [6] (Theorem 4.5) as follows.
Theorem 1.4.Le R be an unital algebra.Then a partial action α of a group G on R admits an enveloping action β if and only if each ideal S g , g ∈ G is a unital algebra.Moreover, if such an enveloping action exists, it is unique up equivalence.
In this paper we assume that each ideal S g , g ∈ G is generated by a central idempotent of R, denoted by 1 g .This condition guarantees the existence of the enveloping action (T, β) for (R, α) (Theorem 1.4).This means that there exists an algebra T together with a global action β = {β g | g ∈ G} of G on T , where each β g is an automorphism of T , such that the partial action is given by restriction of the global action (Definition 1.3).Then we may consider that R is an ideal of T , T = Let (R, α) be a partial action of the group G on R.An ideal I of R is said to be α-invariant if α g (I ∩S g−1 ) = I ∩S g , for all g ∈ G.We will denote this by I α R. The ring R is said to be α-prime if the condition AB = 0, where A and B are α-invariant ideals of R, implies either A = 0 or B = 0 and it is said to be α-semiprime if the condition A 2 = 0, where A is an α-invariant ideal of R, implies A = 0.In the global case, similar notions are defined.

The Martindale ring of α−quotients Q of R
In this section we assume that α is a partial action of a group G on R and R is an α-semiprime ring.If H is an α−invariant ideal of R then the left (right) annihilator of H, An l (H) (An r (H)), is an α−invariant ideal of R and An l (H) = An r (H).We consider the filter where An(H) denotes the annihilator of H in R.
As in the global case, given U ∈ F (R) and the homomorphism of left Rmodules f : R U → R R, either uf or (u)f denotes f applied to u, for u ∈ U .Moreover, note that each element x ∈ R induces an R−homomorphism of R via right multiplication, which will be denoted by x r . Let On the set T we define an equivalence relation by (U, f ) ∼ (V, g) if and only if . This ring is called the (Martindale) ring of α-quotients of R, which will be denoted by Q.
It is easy to see that R is a subring of Q via right multiplication (Proposition 3.1 (1)).Then, each element x ∈ R can be identified with the coset [R, x r ] and so we can assume that the identity element of Q is 1 R .
Recall that in the global case, for T a β−prime ring, the action of G was naturally extended to the ring of β−quotients Q of T ([10], Section 1).In the following theorem we extend the partial action of G on R to a partial action on Q.For each ideal S g , g ∈ G of R, we define S g = {q ∈ Q | there exists H ∈ F (R) such that Hq ⊆ S g } .
Theorem 2.1.Let R be an α-semiprime ring.The following statements hold: 1.The set S g , g ∈ G is an ideal of Q, which is generated by a central idempotent.

The set α
defines a partial action of G on Q. Proof.
1.If q ∈ S g and p ∈ Q, then there exist H, J ∈ F (R) such that Hq ⊆ S g and Jp ⊆ R. Thus, (HJ)pq ⊆ H(Jp)q ⊆ Hq ⊆ S g .So, pq ∈ S g .Symmetrically we obtain that qp ∈ S g .
Since each 1 g is a central idempotent of R, then it is clear that each 1 g is a central idempotent of Q.Given q ∈ Q, there exists H ∈ F (R) such that Hq ⊆ R and so Hq1 g ⊆ S g and q1 g = 1 g q ∈ S g .If p ∈ S g , then Jp ⊆ S g for some J ∈ F (R). Thus, for all j ∈ J we have that jp1 g = jp which implies that J(p1 g − p) = 0. Hence, p1 g = p.
Let p = [U, η] ∈ S g −1 and H ∈ F (R) such that Hp ⊆ S g −1 .Then, Thus, by symmetry we have that α g , g ∈ G is an isomorphism of rings.3. It is analogous to that of Theorem 3.1 of [7]. 2 Each isomorphism α g , g ∈ G of the above theorem will be denoted by α g .In the following theorem we show that the ring Q can be identified with an ideal of the ring of β−quotients Q of T , where (T, β) is the enveloping action of (R, α).In addition, we find an enveloping action for (Q, α), which exists by virtue of Theorems 1.4 and 2.1.

Let
h H , f i ∈ Im ϕ and 0 6 = [J, j] ∈ Q, where J ∈ F (T ) and j : J → T is a homomorphism of left T −modules.Then, Then, Im ϕ is a right ideal of Q.
Similarly we have that ϕ We must prove 1, 2 and 3 of Definition 1.3.3 is consequence of item 3. To prove 2, let q = [A, η] ∈ S g −1 .Then there exists H ∈ F (R) such that To prove 1, let q ∈ S g .Then, ϕ(q) ∈ ϕ(Q) and For the other inclusion, we can identify . Hence, T ⊆ E because (T, β) is the enveloping action of (R, α) and (E, β) the enveloping action of (Q, α).In conclusion, we obtain the following diagram of ring extensions Thus it is natural to ask when (Q, β) is the enveloping action of (Q, α), or equivalently, in which case E = Q.In the following proposition we give a necessary and sufficient condition to solve this problem.First, we recall that the partial action (R, α) with enveloping action (T, β) is said to be of finite type if T is a ring with identity 1 T (see [8], Proposition 1.2 for equivalences).Proposition 2.3.Let R be an α-semiprime ring.(Q, β) is the enveloping action of (Q, α) iff α is a partial action of finite type.
Proof.If (Q, β) is the enveloping action of (Q, α), then Q = P g∈G β g (Q).In particular, 1 Q ∈ P n i=1 β g i (Q) for some g i ∈ G i ∈ {1, ..., n}.Since β g i (1 R ) is the identity of β g i (Q) and each β g i (Q) is an ideal of Q for i ∈ {1, ..., n}, we have that 8], Proposition 1.10).Hence, T is a ring with identity 1 T = 1 Q and thus α is of finite type.Conversely, if α is of finite type then T is a ring with identity 1 T .Thus, 1 T = 1 Q and since T ⊆ E we have that 1 Q ∈ E. Finally, since E is an ideal of Q we conclude that E = Q and so the result follows. 2

Other properties
In this section we present some properties of the ring Q related to the center of Q and the extended α−centroid of R, which extend to the partial case well-known results in the theory of quotient rings.For the sake of completeness, we start with the following proposition ([2], Proposition 1.1).Proposition 3.1.Let R be an α-semiprime ring.The following statements hold: 1. R is a subring of Q, via right multiplication.
2. If (U, f ) ∈ T , then for every u ∈ U we have that u r • f = (uf ) r , where xu r = xu, for all x ∈ R.