A strong Connes spectrum for finite group actions of simple rings

Osterburg, James; Peligrad, Costel

doi:10.1016/0021-8693(91)90317-2

Cited by 2 publications

(1 citation statement)

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“…The ideal structure of skew group rings has been studied in depth (see e.g. [4,5,7,11,12,17,18,19,26]). Nevertheless, necessary and sufficient conditions for simplicity of a general skew group ring are not known.…”

Section: Introductionmentioning

confidence: 99%

Simplicity of Skew Group Rings of Abelian Groups

Öinert

2013

Communications in Algebra

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Given a group G, a (unital) ring A and a group homomorphism σ : G → Aut(A), one can construct the skew group ring A ⋊ σ G. We show that a skew group ring A ⋊ σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. If G is abelian and A is commutative, then A ⋊ σ G is shown to be simple if and only if σ is injective and A is G-simple.As an application we show that a transformation group (X, G), where X is a compact Hausdorff space and G is abelian, is minimal and faithful if and only if its associated skew group algebra C(X) ⋊ σ G is simple. We also provide an example of a skew group algebra, of an (non-abelian) ICC group, for which the above conclusions fail to hold.

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Section: Introductionmentioning

confidence: 99%

Simplicity of Skew Group Rings of Abelian Groups

Öinert

2013

Communications in Algebra

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Non-commutative separability and group actions

Alfaro¹

1992

Publicacions Matemàtiques

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We give conditions for the skew group ring S * G to be strongly separable and H-separable over the ring S. In particular we show that the H-separability is equivalent to S being central Galois extension. We also look into the H-separability of the ring S over the fixed subring R under a faithful action of a group G. We show that such a chain: S * G H-separable over S and S H-separable over R cannot occur, and that the centralizer of R in S is an Azumaya algebra in the presente of a central element of trace one.In [A] we introduced the concept of subring-Galois extensions as a generalization of central Galois extensions and give a generalization of the correspondence theorem given by DeMeyer in [D] and Szeto in [SM] . Similar correspondence theorems were given by Sugano in [S] using Hseparability. Separability for non-commutative rings was introduced by Hirata, and the notions of H-separability and "strong" separability were introduced by Hirata in [HI] and MacMahon and Mewborn in [MM] respectively . Strong separability is a weaker notion than H-separability, but both are special cases of the general notion of separability of ring extensions.In the case of group actions we present here conditions for strong and H-separability of skew group rings and in particular we show that the skew group ring S * G is H-separable over S if and only if S is a central Galois extension . Furthermore, in this case S*G is a Z(S)-Galois

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A strong Connes spectrum for finite group actions of simple rings

Cited by 2 publications

References 5 publications

Simplicity of Skew Group Rings of Abelian Groups

Simplicity of Skew Group Rings of Abelian Groups

Non-commutative separability and group actions

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