Applications of epigroups to graded ring theory
type: Item
instance of: scholarly article
title: Applications of epigroups to graded ring theory
zbMATH Open document ID: 0824.16031
DOI: 10.1007/BF02573530
published in: Semigroup Forum
publication date: 1995-11-02T00:00:00Z
full work available at URL: https://eudml.org/doc/135391
review text: By an epigroup the author means a semigroup \(S\) such that a power of every element \(s \in S\) lies in a subgroup of \(S\). The paper is concerned with rings \(R = \bigoplus_{s \in S} R_ s\) graded by an epigroup \(S\), with an emphasis on classical finiteness conditions and on nil properties of such rings. The main results are in the form of two equivalent conditions (i) and (ii), where (i) is a ring property satisfied by all \(S\)-graded rings and condition (ii) gives a description of the semigroup \(S\). Condition (i) exhibits the derivation of the given property of \(R\) from the corresponding property for the group-graded subrings \(R_ G = \bigoplus_{g \in G} R_ g\) for subgroups \(G\) of \(S\). Here is a sample result: for a semigroup \(S\) the following conditions are equivalent: (i) each \(S\)-graded ring \(R\) is nilpotent provided that the subrings \(R_ G\) are nilpotent for all maximal subgroups \(G\) of \(S\), (ii) \(S\) is an epigroup with a finite number of idempotents and every nil factor of \(S\) is nilpotent. The paper contains a list of problems and an extensive bibliography.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 763799
zbMATH Keywords: epigroups
zbMATH Keywords: finiteness conditions
zbMATH Keywords: semigroup-graded rings
zbMATH Keywords: group-graded subrings
zbMATH Keywords: nilpotent rings
zbMATH Keywords: maximal subgroups
zbMATH Keywords: idempotents
Wikidata QID: Q56675570
author: A. V. Kelarev
reviewed by: Jan Okniński
MaRDI profile type: Publication
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Applications of epigroups to graded ring theory (cites work)
cites work: Semigroup Rings and Semilattice Sums of Rings

Applications of epigroups to graded ring theory (OpenAlex ID)
OpenAlex ID: W1993842880