Simplicity of Partial Skew Group Rings of Abelian Groups

Abstract

Abstract. Let A be a ring with local units, E a set of local units for A, G an abelian group and α a partial action of G by ideals of A that contain local units. We show that A⋆ α G is simple if and only if A is G-simple and the center of the corner eδ 0 (A⋆ α G)eδ 0 is a field for all e ∈ E. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Simple semigroup graded rings

“…In this section, we apply Theorem 2 to partial skew group rings. We generalize a recent result by D. Gonçalves [18] to partial skew group rings by hypercentral groups over rings with local units (see Theorem 17). At the end of this section, we also point out how E. Jespers' result immediately implies a generalization of a simplicity result, recently obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by twisted partial actions (see Remark 18).…”

Partial crossed products as equivalence relation algebras

“…Also, conditons for simplicity of skew group rings and applications to topological dynamics (and hence to the associated C*-algebras) have been studied in [16,15]. Some of these results have recently been generalized to partial skew group rings, with applications to partial actions on compact sets (see [9,12]).…”

Recent developments around partial actions

“…More recent developments on the ring theoretic properties include the study of the simplicity of the partial crossed products in [44,183,184,245] and [246], and the chacarterization of the Leavitt path algebras as partial group rings [186], which permitted one to obtain alternative proofs for the simplicity criterion and for the Cuntz-Krieger uniqueness theorem for Leavitt path algebras (see [184,186]). Furthermore, in [35] the relations between the module properties over A, A α G and A α were studied, where A α denotes the subring of the α-invariants, whereas in [107] the related structure of the partial skew power series ring was recently considered with respect to the distributive and Bezout duo properties.…”