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 S g , g ∈ G of R and isomorphisms α g : S g − 1 → S g such that (i) S 1 = R and α 1 is the identity mapping of R ; (ii) S ( gh ) − 1 ⊇ α − 1 h ( S h ∩ S g − 1 ), (iii) α g ◦ α h ( x ) = α gh ( x ), for any x ∈ α − 1 h ( S h ∩ S g − 1 ). The property (ii) easily implies that α g ( S g − 1 ∩ S h ) = S g ∩ S gh , for all g, h ∈ G . Also α g − 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 � a g u g , a g ∈ S g g ∈ G for every g ∈ G , where the addition is defined in the usual way and the multiplication is determined by ( a g u g )( b h u h ) = α g ( α g − 1 ( a g ) b h ) u gh . This algebra may be non-associative. It is an interesting question on whether it is associative. 2

Given a global action of a group G on an algebra T by automorphisms σ g , g ∈ G , and an ideal R of T , the restriction of the action on R is given by the following: Take S g = R ∩ σ g ( R ), for every g ∈ G , and define α g : S g − 1 → S g by α g ( x ) = σ g ( x ), for all g ∈ G . It is easy to see that this gives a partial action on R . 3

Given a partial action α of a group G on R , an enveloping action is an algebra T together with a global action β = { β g | g ∈ G } of G on T , where β g 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 β g ( R ) coincides with T and we have T = � g ∈ G β g ( R ); (ii) S g = R ∩ β g ( R ), for every g ∈ G ; (iii) α g ( x ) = β g ( x ), for all g ∈ G and x ∈ S g − 1 . 4

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 S g 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. 5

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 α g is the restriction of β g to the ideal S g − 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. 6

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 . g corresponding to α ∗ are the closure of the The ideals S ∗ ideals S g . 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. 7

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. 8

PARTIAL SKEW POLYNOMIAL RINGS Let R be an associative ring with an identity element 1 R , G an infinite cyclic group generated by σ and α = { α σ i : S σ − i → S σ i } 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 a i x i , where a i ∈ S σ i , 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 S σ i will be denoted simply by S i , i ∈ Z . 9

Assume that for all i , the ideal S i is generated by a central idempotent 1 i . 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 of 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 a i x i , a i ∈ S i , for every i ≥ 0. Thus the partial skew polynomial ring is an associative ring, contained in the skew polynomial ring T [ x ; σ ]. 10

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 > . 11

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 ≥ 1 S i x i , where Q is a prime ideal of R (i) P = Q ⊕ � i � =0 S i x i , where Q is a prime ideal of (resp. P = Q ⊕ � R with S j ⊆ Q , for any j � = 0). (ii) 1 i x i / ∈ P , for some i ≥ 1. The description of the prime ideals of the case (ii) is quite technical and we will omit here. 12

An ideal I of R is said to be an α -ideal if α σ i ( I ∩ S − i ) ⊆ I ∩ S i , for all i ≥ 0, and is said to be an α -invariant ideal if α σ i ( I ∩ S − i ) = I ∩ S i , for all i ∈ Z . If I is an α -ideal of R , then the set of all the i ≥ 0 a i x i , where a i ∈ I ∩ S i , is an ideal of polynomials � 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 1 j / ∈ Q and for any ideal I and α -ideal J of R , IJ ⊆ Q implies either that I ⊆ Q or J ⊆ Q . 13

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 1 i x i / ∈ 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 ] or P is maximal amongst the ideals N of R [ x ; α ] with N ∩ R = P ∩ R . 14

MAXIMAL IDEALS OF R [ x, α ] , by [ C, F ] The maximal ideals can be classified into two types: i ≥ 1 S i x i , (i) If M is a maximal ideal which contains � i ≥ 1 S i x i , where M ∩ R then we have M = ( M ∩ R ) ⊕ � is a maximal ideal of R and conversely. (ii) If M is a maximal with 1 i x i / ∈ M , for some i ≥ 1: Theorem. Assume that M is a prime ideal of R [ x ; α ] such that 1 i x i / ∈ M , for some i ≥ 0. Then M is maximal ′ of if and only if the corresponding prime ideal M ′ ∩ R [ x ; α ] = M is a maximal ideal. T [ x ; σ ] such that M ′ has an identity element. If this is the case T [ x ; σ ] /M 15

The α -pseudo radical ps α ( 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 α σ j ( a 1 − j ) = a 1 j , for all j ∈ Z . Definition An element a ∈ R is said to be α σ m -normalizing if for all r ∈ R we have ra = aα σ m ( r 1 − m ). 16

Theorem The following are equivalent: (i) There exists an R -disjoint maximal ideal ideal M of R [ x ; α ] such that 1 i x i / ∈ M , for some i ≥ 0. ′ of T [ x ; σ ] such (ii) There exists a T -disjoint ideal M ′ is simple with identity and that T [ x ; σ ] /M ′ . T [ x ; σ ] x � M (iii) T is σ -prime and ps σ ( T ) contains a non-zero element which is σ -invariant and σ m -normalizing, for some m ≥ 1. (iv) R is α -prime and ps α ( R ) contains a non-zero α -invariant element which is α m -normalizing, for some m ≥ 1. 17

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 1 i x i / ∈ M , for some i ≥ 0. (ii) R/Q is α -prime and ps α ( R/Q ) contains a non-zero α -invariant element which is α m -normalizing, for some m ≥ 1. 18

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 ] = � � ( U α ( R ) ∩ S i ) x i . U α ( R ) ∩ U ( R ) i ≥ 1 19