Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics

摘要

Let R-0 be a commutative and associative ring (not necessarily unital), G a group and alpha a partial action of G on ideals of R-0, all of which have local units. We show that R-0 is maximal commutative in the partial skew group ring R-0 G if and only if Ro has the ideal intersection property in R-0 (sic)(alpha) G. From this we derive a criterion for simplicity of R-0 (sic)(alpha) G in terms of maximal commutativity and G-simplicity of R-0. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz-Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set. (C) 2014 Elsevier Inc. All rights reserved.