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Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

Viviane Beuter, Daniel Gonçalves ORCID logo, Johan Öinert ORCID logo and Danilo Royer ORCID logo
From the journal Forum Mathematicum

Abstract

Given a partial action π of an inverse semigroup S on a ring 𝒜, one may construct its associated skew inverse semigroup ring 𝒜πS. Our main result asserts that, when 𝒜 is commutative, the ring 𝒜πS is simple if, and only if, 𝒜 is a maximal commutative subring of 𝒜πS and 𝒜 is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra AR(𝒢) associated with a Hausdorff and ample groupoid 𝒢.


Communicated by Karl-Hermann Neeb


Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award Identifier / Grant number: 304487/2017-1

Funding statement: The second author was partially supported by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico, under grant number 304487/2017-1.

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Received: 2018-07-06
Revised: 2018-10-17
Published Online: 2018-11-22
Published in Print: 2019-05-01

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