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

Viviane Beuter
  • Corresponding author
  • Departamento de MatemĂĄtica, Universidade Federal de Santa Catarina, FlorianĂłpolis, BR-88040-900; and Departamento de MatemĂĄtica, Universidade do Estado de Santa Catarina, Joinville, BR-89219-710, Brazil
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/ Daniel GonçalvesORCID iD: http://orcid.org/0000-0002-8149-9872 / Johan ÖinertORCID iD: http://orcid.org/0000-0001-8095-0820 / Danilo RoyerORCID iD: http://orcid.org/0000-0002-6554-0898
Published Online: 2018-11-22 | DOI: https://doi.org/10.1515/forum-2018-0160


Given a partial action π of an inverse semigroup S on a ring 𝒜, one may construct its associated skew inverse semigroup ring đ’œâ‹ŠÏ€S. Our main result asserts that, when 𝒜 is commutative, the ring đ’œâ‹ŠÏ€S is simple if, and only if, 𝒜 is a maximal commutative subring of đ’œâ‹ŠÏ€S 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 AR⁹(𝒱) associated with a Hausdorff and ample groupoid 𝒱.

Keywords: Skew inverse semigroup ring; Steinberg algebra; partial action; inverse semigroup

MSC 2010: 16S99; 16W22; 16W55; 22A22; 37B05


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About the article

Received: 2018-07-06

Revised: 2018-10-17

Published Online: 2018-11-22

Funding Source: Conselho Nacional de Desenvolvimento CientĂ­fico e TecnolĂłgico

Award identifier / Grant number: 304487/2017-1

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

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2018-0160.

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