Simplicity of partial skew group rings and maximal commutativity

Abstract

Let R0 be a commutative associative ring (not necessarily unital), G a group and alpha a partial action by ideals that contain local units. We show that R0 is maximal commutative in the partial skew group ring R0*G if and only if R0 has the ideal intersection property in R0*G. From this we derive a criterion for simplicity of R0*G in terms of maximal commutativity and $G-$simplicity of R0 and apply this to two examples, namely to partial actions by clopen subsets of a compact set and to give a new proof of the simplicity criterion for Leavitt path algebras. A new proof of the Cuntz-Krieger uniqueness theorem for Leavitt path algebras is also provided.