Simplicity of algebras via epsilon-strong systems
2020 (English)In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 162, no 2, p. 279-301Article in journal (Refereed) Published
Abstract [en]
We obtain sufficient criteria for simplicity of systems, that is, rings R that are equipped with a family of additive subgroups R-s for s is an element of S, where S is a semigroup satisfying R = Sigma (s is an element of S) R-s and RsRt subset of R-st for s, t is an element of S. These criteria are specialized to obtain sufficient criteria for simplicity of what we call s-unital epsilon-strong systems, that is, systems where S is an inverse semigroup, R is coherent, in the sense that R-s subset of R-t for all s, t is an element of S with s <= t and for each s is an element of S, the RsRs*-Rs*Rs -bimodule R-s is s-unital. As an application, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings by Beuter, Goncalves, Oinert and Royer, and then for Steinberg algebras over non-commutative rings by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michell.
Place, publisher, year, edition, pages
2020. Vol. 162, no 2, p. 279-301
Keywords [en]
graded ring; skew inverse semigroup ring; Steinberg algebra; partial action
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:hv:diva-15808DOI: 10.4064/cm7887-9-2019ISI: 000556329800008Scopus ID: 2-s2.0-85090404042OAI: oai:DiVA.org:hv-15808DiVA, id: diva2:1466883
Note
CORRIGENDUM 2024 10.4064/cm9319-3-2024
The results that are stated in [Colloq. Math. 162 (2020), 279–301] hold true but due to an oversight by the author in the publication process, the definition of a system being left/right non-degenerate was stated incorrectly. The purpose of this note is to provide the correct definition. We recall the notation from [1]: N := 1, 2, 3, . . . and Z≥0 := N ∪ 0; S denotes a semigroup, that is, a nonempty set equipped with an associative binary operation; E(S) denotes the set of idempotents of S; R denotes a system, that is, a ring equipped with additive subgroups Rs, for s ∈ S, such that R = P s∈S Rs and for all s, t ∈ S the inclusion RsRt ⊆ Rst holds. The function d : R → Z≥0 is defined in the following way. If r = 0, then put d(r) = 0. Now suppose that r ̸= 0. Take n ∈ N, s1, . . . , sn ∈ S and ri ∈ Rsi , for i = 1, . . . , n, such that r = Pn i=1 ri. Amongst all such representations of r, choose one with n minimal and put d(r) = n. If I is a nonzero ideal of R and r ∈ R, then r is called I-minimal if r ∈ I and (Formula presented). In order to make the proof of [1, Prop. 3.5] correct as it stands, [1, Def. 3.2] has to be replaced by the following: Definition 1. We say that R is left [right] non-degenerate if for every nonzero deal I of R and every I-minimal element (Formula presented).
2020-09-142020-09-142025-02-24Bibliographically approved