PARTIAL ACTIONS OF GROUPS ON ALGEBRAS Miguel Ferrero, with D. - - PDF document

PARTIAL ACTIONS OF GROUPS ON ALGEBRAS

Miguel Ferrero, with D. Bagio, W. Cort´ es, M. Dokuchaev,

• J. Lazzarin, H. Marubayashi, A. Paques

Universidade Federal do Rio Grande do Sul - Brazil

Let G be a group and R a unital k-algebra, k a ring. A partial action of G on R is a collection of ideals Sg, g ∈ G of R and isomorphisms αg : Sg−1 → Sg such that (i) S1 = R and α1 is the identity mapping of R; (ii) S(gh)−1 ⊇ α−1

h (Sh ∩ Sg−1),

(iii)αg ◦ αh(x) = αgh(x), for any x ∈ α−1

h (Sh ∩ Sg−1).

The property (ii) easily implies that αg(Sg−1 ∩ Sh) = Sg ∩ Sgh, for all g, h ∈ G. Also αg−1 = α−1

g , for every g ∈ G. 1

Let α be a partial action of G on R. The partial skew group ring R ⋆α G is defined as the set of all finite formal sums

• g∈G

agug, ag ∈ Sg for every g ∈ G, where the addition is defined in the usual way and the multiplication is determined by (agug)(bhuh) = αg(αg−1(ag)bh)ugh. This algebra may be non-associative. It is an interesting question on whether it is associative.

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Given a global action of a group G on an algebra T by automorphisms σg, g ∈ G, and an ideal R of T, the restriction of the action on R is given by the following: Take Sg = R ∩ σg(R), for every g ∈ G, and define αg : Sg−1 → Sg by αg(x) = σg(x), for all g ∈ G. It is easy to see that this gives a partial action on R.

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Given a partial action α of a group G on R, an enveloping action is an algebra T together with a global action β = {βg | g ∈ G} of G on T, where βg is an automorphism of T, such that the partial action is given by restriction of the global action. This means that we may consider R as an ideal of T and the following holds: (i) the subalgebra of T generated by

g∈G βg(R)

coincides with T and we have T =

g∈G βg(R);

(ii) Sg = R ∩ βg(R), for every g ∈ G; (iii) αg(x) = βg(x), for all g ∈ G and x ∈ Sg−1.

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We will always assume that R is a unital algebra. An important result by M. Dokuchaev and R. Exel shows:

• Theorem. The partial action α has an enveloping

action if and only if all the ideals Sg are unital algebras (i.e., they are generated by central idempotents of R). As a consequence, when α has an enveloping action, then the partial skew group ring R ⋆α G is associative.

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Another type of enveloping action can be defined. We say that (T, β) is a weak enveloping action of α if T is a ring which contains R, β is a global action of G on T by automorphisms, and for any g ∈ G the map αg is the restriction of βg to the ideal Sg−1 of R. The following results was already proved:

• Theorem. [F] If R is a semiprime ring, then any partial

action α on R possesses a weak enveloping action. As a consequence, for any semiprime ring the partial skew group ring R ⋆α G is associative.

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The construction of the weak enveloping action in the above result is doing in the following way: first we consider the Martindale ring of quotients Q of R and we extend the partial action to a partial action α∗ of Q. The ideals S∗

g corresponding to α∗ are the closure of the

ideals Sg. Then they are closed ideals and so generated by central idempotent elements of Q. Then we consider the enveloping action (T, β) of α∗. This is the weak enveloping action of α. For the corresponding partial skew group rings we have: R ⋆α G ⊆ Q ⋆α∗ G ⊆ T ⋆ G.

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The enveloping action is defined by an universal

• property. So it is unique, unless equivalence.

Question It is an open problem to find a general definition of weak enveloping action in order to have

• uniqueness. Until now I did not solve this problem.

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PARTIAL SKEW POLYNOMIAL RINGS Let R be an associative ring with an identity element 1R, G an infinite cyclic group generated by σ and α = {ασi : Sσ−i → Sσi} a partial action of G on R. The partial skew group ring R ∗α G can be identified with the set of all the finite sums m

i=−n aixi, where

ai ∈ Sσi, for any integer number i, where the addition and the multiplication are defined as above. We denote R ∗α G by R < x; α >. The ideal Sσi will be denoted simply by Si, i ∈ Z.

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Assume that for all i, the ideal Si is generated by a central idempotent 1i. In this case α has an enveloping action which will be denoted by (T, σ), where σ is an automorphism of T. The skew group ring T ⋆ G is the skew Laurent polynomial ring T < x; σ > and R < x; α > is a subring

• f T < x; σ >.

We define the partial skew polynomial ring R[x; α] as the subring of R < x; α > whose elements are the polynomials n

i=0 aixi, ai ∈ Si, for every i ≥ 0. Thus

the partial skew polynomial ring is an associative ring, contained in the skew polynomial ring T[x; σ].

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PRIME IDEALS OF R < x; α > AND R[x; α]

• Proposition. [C, F] (i) There is a one-to-one

correspondence, via contraction, between the set of all prime ideals of R[x; α] and the set of all prime ideals of T[x; σ] which do not contain R. (ii) There is a one-to-one correspondence, via contraction, between the set of all prime ideal of R < x; α > and the set of all prime ideals of T < x; σ >. Using this and the known results about prime ideals in T[x; σ] and T < x; σ > we can obtain a complete description of prime ideals of R[x; α] and R < x; α >.

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• Proposition. [C, F] Let P be a prime ideal of R[x; α]

(resp. R < x; α >). Then we have one of the following possibilities: (i) P = Q ⊕

i≥1 Sixi, where Q is a prime ideal of R

(resp. P = Q ⊕

i=0 Sixi, where Q is a prime ideal of

R with Sj ⊆ Q, for any j = 0). (ii) 1ixi / ∈ P, for some i ≥ 1. The description of the prime ideals of the case (ii) is quite technical and we will omit here.

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An ideal I of R is said to be an α-ideal if ασi(I ∩ S−i) ⊆ I ∩ Si, for all i ≥ 0, and is said to be an α-invariant ideal if ασi(I ∩ S−i) = I ∩ Si, for all i ∈ Z. If I is an α-ideal of R, then the set of all the polynomials

i≥0 aixi, where ai ∈ I ∩ Si, is an ideal of

R[x; α]. Similar for an α-invariant ideal of R < x, α >. Let Q be an α-invariant ideal of R. (i) Q is said to be α-prime if IJ ⊆ Q, for α-invariant ideals I and J of R, implies that either I ⊆ Q or J ⊆ Q. (ii) Q is said to be strongly α-prime if for any m ≥ 1 there exists j ≥ m such that 1j / ∈ Q and for any ideal I and α-ideal J of R, IJ ⊆ Q implies either that I ⊆ Q or J ⊆ Q.

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• Corollary. Let P be an ideal of R < x; α >. Then P is

prime if and only of P ∩ R is α-prime and either P = (P ∩ R) < x; α > or P is maximal amongst the ideals N of R < x; α > such that N ∩ R = P ∩ R.

• Corollary. Let P be an ideal of R[x; σ] such that

1ixi / ∈ P, for some i ≥ 1. Then P is prime if and only if P ∩ R is strongly α-prime and either P = (P ∩ R)[x; α]

• r P is maximal amongst the ideals N of R[x; α] with

N ∩ R = P ∩ R.

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MAXIMAL IDEALS OF R[x, α], by[C, F] The maximal ideals can be classified into two types: (i) If M is a maximal ideal which contains

i≥1 Sixi,

then we have M = (M ∩ R) ⊕

i≥1 Sixi, where M ∩ R

is a maximal ideal of R and conversely. (ii) If M is a maximal with 1ixi / ∈ M, for some i ≥ 1:

• Theorem. Assume that M is a prime ideal of R[x; α]

such that 1ixi / ∈ M, for some i ≥ 0. Then M is maximal if and only if the corresponding prime ideal M

′ of

T[x; σ] such that M

′ ∩ R[x; α] = M is a maximal ideal.

If this is the case T[x; σ]/M

′ has an identity element.

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The α-pseudo radical psα(R) of R is defined as the intersection of all non-zero α-prime ideals of R. Definition An element a ∈ R is said to be α-invariant if ασj(a1−j) = a1j, for all j ∈ Z. Definition An element a ∈ R is said to be ασm-normalizing if for all r ∈ R we have ra = aασm(r1−m).

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Theorem The following are equivalent: (i) There exists an R-disjoint maximal ideal ideal M of R[x; α] such that 1ixi / ∈ M, for some i ≥ 0. (ii) There exists a T-disjoint ideal M

′ of T[x; σ] such

that T[x; σ]/M

′ is simple with identity and

T[x; σ]x M

′.

(iii) T is σ-prime and psσ(T) contains a non-zero element which is σ-invariant and σm-normalizing, for some m ≥ 1. (iv) R is α-prime and psα(R) contains a non-zero α-invariant element which is αm-normalizing, for some m ≥ 1.

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Corollary Assume that α is a partial action on R and Q is an α-invariant ideal. Then the following conditions are equivalent: (i) There exists a maximal ideal ideal M of R[x; α] such that M ∩ R = Q and 1ixi / ∈ M, for some i ≥ 0. (ii) R/Q is α-prime and psα(R/Q) contains a non-zero α-invariant element which is αm-normalizing, for some m ≥ 1.

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• Theorem. Under the same assumptions as above, the

Brown-McCoy radical of a partial skew polynomial ring can be obtained: U(R[x; α]) = U(T[x; σ]) ∩ R[x; α] = Uα(R) ∩ U(R)

i≥1

(Uα(R) ∩ Si)xi.

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PARTIAL SKEW POLYNOMIAL RINGS OF SEMIPRIME RINGS Assume that R is semiprime and α is arbitrary, i.e., the ideals Si do not necessarily have identity element. In this case we define the partial skew Laurent polynomial ring as: R < x; α >= {

• −n≤i≤m

aixi|ai ∈ Si}, where the addition and the multiplication are defined as

• above. So we have

R < x; α >⊆ Q < x; α∗ >⊆ T < x; σ >, where (T, σ) denotes the enveloping action of (Q, α∗).

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Then R < x; α > is an associative ring and R[x; α] can be naturally defined as above: R[x; α] = {

• 0≤i≤n

aixi} ⊆ R < x; α > . The essential difference in studying the first case and the second one is that here we have to go from R[x; α] to Q[x; α∗] and then from this to T[x; σ], and come back.

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PARTIAL SKEW POLYNOMIAL RINGS AND GOLDIE RINGS Recall that a ring R is said to be a right Goldie ring if R has finite right uniform dimension and satisfies ACC

• n right annihilators.

The right singular ideal Z(R) of R is the set of all the elements a ∈ R such that the right annihilator rAnnR(a) = {x ∈ R|ax = 0} is an essential right ideal. A semiprime ring R is right Goldie if and only if R has finite right uniform dimension and the right singular ideal of R is zero (R is right non-singular).

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It is well-known that if T is a semiprime ring and σ is an automorphism of T, then T is right Goldie if and

• nly if T < x; σ > is right Goldie if and only if T[x; σ] is

right Goldie. In addition, T < x; σ > and T[x; σ] are also semiprime. For partial skew Laurent polynomial rings and partial skew polynomial rings over semiprime rings we proved the following results [C, F, M] Proposition 1. If R is a semiprime right non-singular ring, then R < x; α > and R[x; α] are also right non-singular.

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Proposition 2. If R is a semiprime ring, then the right uniform dimension of R, R < x; α > and R[x; α] are equal. So we have the following:

• Theorem. If R is a semiprime ring, then the following

conditions are equivalent: (i) R is right Goldie; (ii) R < x; α > is right Goldie; (iii) R[x; α] is right Goldie. We also proved that in this case R < x; α > is semiprime, but R[x; α] is not necessarily semiprime.

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There is natural question related with these results: Is the weak enveloping action T of α also right Goldie when R is right Goldie? The answer is NO in general. To have an affirmative answer we must have a rather special partial action. Assume that R is semiprime. Definition We say that the partial action α on R is of finite type if for any uniform (two-sided) ideal I of R there exist and ideal H of R and a positive integer i such that rAnnR(I) = rAnnR(H) and H ⊆ S−i. The above definition works well for skew polynomial rings, but now we have a more general definition and some equivalences.

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• Theorem. Assume that R is semiprime right Goldie

ring and α, (Q, α∗) and (T, σ) are as above. Then the following conditions are equivalent: (i) α is of finite type; (ii) There exists N > 0 such that σN(Q) = Q; (iii) T has an identity element; (iv) T has a finite number of central idempotent elements; (v) The right uniform dimension of T is finite. Under the above situation T, T < x; σ > and T[x; σ] are also right Goldie and R[x; α] is semiprime.

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PARTIAL ACTIONS OF FINITE TYPE Recall that an ideal H of R is said to be essential if for any non-zero ideal I we have I ∩ H = 0 Definition A partial action α = {αg : Sg−1 → Sg} on R is said to be of finite type if there exist g1, ...gn ∈ G such that for any g ∈ G we have that

1≤i≤n Sggi is an

essential ideal of R. This condition is a natural one to have good finiteness

• conditions. When α has an enveloping action (T, β) we

have the following form of the condition.

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Theorem The following conditions are equivalent: (i) α is of finite type; (ii) There exist g1, ..., gn ∈ G such that

• 1≤i≤n Sggi = R, for any g ∈ G;

(iii) There exist g1, ..., gn ∈ G with T =

1≤i≤n βgi(R);

(iv) T has an identity element. Using a result by M. Dokuchaev and R. Exel we have Corollary Assume that α is a partial action of finite type on R and there exists an enveloping action (T, β). Then R ⋆α G and T ⋆ G are Morita equivalent.

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In the case R is semiprime the above definition have the following equivalences. Theorem Assume that R is semiprime, Q is the Martindale ring of right quotients of R and (T, β) is the enveloping action of α∗, the extension of α to Q. The following conditions are equivalent: (i) α is of finite type. (ii) α∗ is of finite type; (iii) T has an identity element.

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We give an example to see that the definition is natural. Theorem Assume that R is a right noetherian ring, G is a polycyclic by finite group, α is a partial action of G

• n R and (T, β) is an enveloping action of α. Then the

following conditions are equivalent: (i) α is of finite type; (ii) T is right noetherian; (iii) T has a finite number of central idempotents; (iv) {βg(R)|g ∈ G} is a finite set.

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The converse of the above is also true. Theorem Assume that T is a right noetherian ring, G is a polycyclic by finite group and (βg)g∈G is an action

• f G on T. If e is a central idempotent of T and

R = Te, then the induced partial action of β on R is of finite type.

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FIXED RINGS AND THE TRACE MAP For a finite group G and action β of G on T the fixed subring of T under the action is defined by T G = {x ∈ R | βg(x) = x, ∀g ∈ G} Similarly, for any partial action α of G on R we define the invariant subring of α by Rα = {x ∈ R | αg(xa) = xαg(a), ∀g ∈ G ∀a ∈ Sg−1} Assume that every ideal Sg has an identity element 1g. Then we easily have Rα = {x ∈ R | αg(x1g−1) = x1g, ∀g ∈ G}

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For a global actions of a finite group G on T the trace map is defined by tr(x) =

g∈G βg(x).

It is easy to see that tr(T) ⊆ T G. When the action α on R is partial and it has enveloping action, denoting as above the identities of the ideals by 1g, g ∈ G, we have also a trace map: trα(x)

• g∈G

αg(x1g−1) and as in the global case we have trα(R) ⊆ Rα

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PARTIAL SKEW GROUP RINGS, by [F, L] In this section we assume that G is a finite group and also that every ideal Sg has an identity element 1g. Proposition Assume that R is right noetherian (artinian). Then R ⋆α G is right noetherian (artinian). Theorem (Maschke Theorem 1) If R is a semisimple ring and |G|−1 ∈ R, then R ⋆α G is semisimple. We can prove another version of Maschke’s Theorem:

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Theorem (Maschke Theorem 2) Assume that trα(1R) is invertible in R. If M is any left module over R ⋆α G and N is a R ⋆α G-submodule of M which is a direct summand as an R-submodule, then N is a direct summand as R ⋆α G-submodule. So, as in the classical case we have Corollary If R is semisimple and trα(1R) is invertible in R, then R ⋆α G is semisimple. The assumptions in the above two Maschke’s theorem are independent.

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Denote by J(R) the Jacobson radical of R. We have Proposition For any group G (not necessarily finite) and partial action α of G on R we have J(R ⋆α G) ∩ R ⊆ J(R). An ideal I of R is said to be α-invariant if αg(I ∩ Sg−1) = I ∩ Sg, for any g ∈ G. If I is an α-invariant ideal of R, then I ⋆α G =

g∈G(I ∩ Sg)ug is

an ideal of R ⋆α G It is easy to see that J(R) is α-invariant. So J(R) ⋆α G is well defined and an ideal of R ⋆α G.

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Now assume that G is finite of cardinality n. We have the following which generalizes a well-known result. Theorem J(R ⋆α G)n ⊆ J(R) ⋆α G ⊆ J(R ⋆α G) Corollary Under the same assumption as above we have: (i) J(R ⋆α G) ∩ R = J(R). (ii) J(R) is nilpotente if and only if J(R ⋆α G) is nilpotente.

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Another result we can prove is the following (again |G| = n). Proposition If R is semiprime and has no (additive) n-torsion, then R ⋆α G is semiprime. Recall that a ring R is said to be von Neumman regular if for any a ∈ R there exists b ∈ R such that bab = b. Theorem Assume that R is von Neumman regular and either n or trα(1R) is invertible in R. Then R ⋆α G is von Neumman regular.

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FIXED SUBRING, by [F, L] There are many results given relations between a ring T and the invariant subring T G. We generalizes several of this results. In the following α is a partial action of G on R which has an enveloping action (T, G). One main result is the following Theorem If G is a finite group, then Rα = T G1R. In particular, Rα and T G are isomorphic. For infinite groups we have Rα ⊂ T G1R. We can have an strict inclusion.

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Corollary In G is a finite group and γ is any radical of rings, then γ(Rα) = γ(T G)1R. Corollary If R is semiprime and do not have (additive) n-torsion, then Rα is semiprime. Corollary If R is semisimple and n is invertible in R, then Rα is semisimple. Theorem If R is semiprime and trα(1R) is not a zero divisor in R, then Rα is semiprime and for any non-zero α-invariant left (right) ideal I of R, trα(I) = 0.

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Assume that I is a left ideal of R which is α-invariant and put K = Rα ∩ I. Theorem If trα(1R) is invertible in R we have: (i) R/I is artinian (noetherian) as a left R-module, then Rα/K is artinian (noetherian) as a left Rα-module. (ii) If the left R-module R/I has a composition series of length l, then the left Rα-module Rα/K has a composition series of length ≤ l. Corollary If R is left artinian (noetherian) and trα(1R) is invertible in R, then Rα is left artinian (noetherian).

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We proved also results on radicals. Theorem If n = |G| is invertible in R, then J(Rα) = J(R) ∩ Rα. In particular, if R is J-semisimple, then Rα is J-semisimple. Theorem For n = |G| we have nJ(Rα) ⊆ J(R). In particular, if R is J-semisimple and do not have (additive) n-torsion, then Rα is J-semisimple. Corollary If R is semisimple, do not have (additive) n-torsion and trα(1R) is invertible in R, then Rα is semisimple.

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We denote the prime radical of R by P(R). We have Theorem For any group G, P(R) ∩ Rα ⊆ P(Rα). Theorem If G is finite and either trα(1R) is not a zero divisor in R or R do not have (additive) n-torsion, then P(Rα) = P(R) ∩ Rα.

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We say that α have non-degenerated partial trace if Rα is semiprime and trα(I) = 0 for any non-zero α-invariant left ideal I of R. Theorem If α have non-degenerated partial trace and R is a left Goldie ring, then Rα is a left Goldie ring. If in the above result we assume, in addition, that R do not have (additive) n-torsion, then we have: (i) Rα is a left Goldie ring if and only if R is a left Goldie ring. (ii) Rα is semisimple if and only if R is semisimple.

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INVERSE SEMIGROUPS AND ACTIONS ON ALGEBRAS A semigroup H (always with identity) is said to be an inverse semigroup if for any x ∈ H there exists x∗ ∈ H such that xx∗x = x and x∗xx∗ = x∗. The set of all partially defined bijective maps on a set X is an inverse semigroup, which we will denote by I(X). The composition of maps is given by composition of partial maps in the largest domain where it makes sense. Definition An action of an inverse semigroup H on X is a homomorphism of inverse semigroups φ : H → I(X).

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For a group G, R. Exel defined a universal inverse semigroup S(G), associated to G. Then he proved that there is a one-to-one correspondence between partial action of a group G on X and actions of the inverse semigroup S(G) on X. We consider an algebraic version of this result. Let R ⊆ S be a ring extension and IR(S) the inverse semigroup of all R-isomorphisms between ideals of S which are generated by central idempotents. It is easy to see that IR(S) is an inverse semigroup and AutR(S) ⊆ IR(S).

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In a recent paper we study a ring extension under the action of an inverse semigroup H. Assume that H is an inverse subsemigroup of IR(S). We denote by eh the central idempotent of S such that Seh is the domain of h−1 The fixed subring is defined as above: SH = {s ∈ S | h(seh−1) = seh, ∀ h ∈ H}

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It seems to be clear that when considering partial actions of groups on a ring extension R ⊆ S (with identity elements) the inverse semigroup IR(S) is as relevant as the group AutR(S) is when leading with global actions. So the computation of IR(S) is of interest. In our paper we compute this semigroup as function of the group of automorphisms. For any central idempotent e ∈ S and automorphism σ ∈ AutR(S) we define a partial isomorphism by he,σ = σ|sσ−1(e) : Sσ−1(e) → Se

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In the set of all the pairs (e, σ), as above, we can define an structure of inverse semigroups by (e, σ).(f, τ) = (eσ(f), στ) Question Determine conditions under which any action

• f an inverse semigroup on S can be obtained as

restriction of an inverse semigroup of pairs (e, σ), as above. We proved the following:

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Theorem [B, C, F, P] Assume that S is a finitely generated and projective algebra over R and H is a finite inverse subsemigroup of IR(S) with SH = R. Then (i) For any h ∈ IR(S) there exists σ ∈ AutR(S) and a central idempotent e of S such that h = σ|Sσ−1(e). (ii) There exists a finite subgroup G of AutR(S) such that SG = R.

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MAIN REFERENCES Dokuchaev, M., Exel, R.; Associativity of crossed products by partial actions, enveloping actions and partial representations; Trans. Amer. Math. Society 357, n. 5 (2005), 1931-1952. Exel, R.; Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), 3481-3494.

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RECENT PAPERS Bagio, D., Cortes, W., Ferrero, M., Paques, A.; Actions

• f inverse semigroups on algebras; to appear.

Cortes, W., Ferrero, M.; Partial skew polynomial rings: prime and maximal ideals; Comm. Algebra, to appear. Cortes, W., Ferrero, M., Marubayashi, H.; Partial skew polynomial rings and Goldie rings; to appear.

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Dokuchaev, M., Ferrero, M., Paques A.; Partial actions and Galois theory; J. Pure Appl. Algebra, to appear. Lazzarin, J.; A¸ coes parciais de grupos sobre an´ eis: o skew anel de grupo parcial e o subanel dos elementos invariantes; Tese de Doutorado, UFRGS.

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