On special \(Q_ r\)-filters of semigroups, semirings and rings
type: Item
instance of: scholarly article
title: On special \(Q_ r\)-filters of semigroups, semirings and rings
zbMATH Open document ID: 0532.16028
DOI: 10.1007/BF02573449
author: Udo Hebisch
published in: Semigroup Forum
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134715
review text: In the theory of semigroups, right quotient filters \((Q_ r\)-filters) of a semigroup S are used to construct the \(Q_ r\)-semigroup \(Q_ r(S,{\mathcal F},{\mathcal K})\) of S, depending on two of those \(Q_ r\)-filters \({\mathcal F}\) and \({\mathcal K}\). In this paper, we collect some basic properties and characterizations of special \(Q_ r\)-filters of a semigroup and define a corresponding concept for seminearrings and hence for semirings. To include rings as a subcase, we also deal with k-closed \(Q_ r\)-filters of seminearrings. It is shown that nearly all basic statements about special \(Q_ r\)- filters of semigroups are transferable to the semiring-case and, via k- closed \(Q_ r\)-filters, also to the ring-case. We use this analogy to unify the notations and to treat all three cases simultaneously. As an application of our concepts we show that the commutativity of an additively subcommutative semiring (a ring) S transfers to the semiring (ring) of right quotients \(Q_ r(S,{\mathcal F},{\mathcal K})\) of S with respect to (k-closed) \(Q_ r\)-filters \({\mathcal F}\) and \({\mathcal K}\) of S, if \({\mathcal K}\) is special. The corresponding result for semigroups is already known. Denoting by \(S^*\) the image of \(\sigma\) (S) of S under the natural homomorphism \(\sigma:S\to Q_ r(S,{\mathcal F},{\mathcal K})\) of S into a \(Q_ r\)-semigroup \((Q_ r\)-semiring, \(Q_ r\)-ring), we obtain that the image \(\sigma\) (K) of each \(K\in {\mathcal K}\) is a dense right ideal of \(S^*\), if \({\mathcal K}\) is special. From this fact we derive an embedding theorem for \(Q_ r(S,{\mathcal F},{\mathcal K})\) into a McMorris \(Q_ r\)-semigroup \((Q_ r\)-semiring, \(Q_ r\)-ring) of \(S^*\), which really contains \(S^*\) as a subalgebra.
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 3843961
zbMATH Keywords: semiring of right quotients
zbMATH Keywords: McMorris semigroup
zbMATH Keywords: right quotient filters
zbMATH Keywords: seminearrings
zbMATH Keywords: commutativity
zbMATH Keywords: subcommutative semiring
MaRDI profile type: Publication
cites work: The Ring of Quotients of R[S]; R a Commutative Ring and S a R<scp>EES</scp> Matrix Semigroup over a Semigroup
cites work: Q3329653
cites work: Generalized Semigroups of Quotients
cites work: The generalized translational hull of a semigroup
cites work: Q3663530
cites work: The maximal quotient semigroup
cites work: Q4049223
cites work: Q3900262
cites work: Q3669590
OpenAlex ID: W2088071809
description: scientific article; zbMATH DE number 3843961