On quasi-ideals in contracted semigroup rings
type: Item
instance of: scholarly article
title: On quasi-ideals in contracted semigroup rings
zbMATH Open document ID: 0584.20060
DOI: 10.1007/BF02573213
author: V. N. Dixit
author: P. K. Singhal
author: Hanns Joachim Weinert
published in: Semigroup Forum
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134823
review text: Let A be a ring with identity and S a semigroup with a zero 0 and an identity e. Then the contracted semigroup ring \(A_ 0[S]\) of S over A is the free unitary left A-module with \(S'=S\setminus \{0\}\) as a basis and a multiplication determined by that of S. Identifying A with the subring \(Ae=\{ae|\) \(a\in A\}\) of \(A_ 0[S]\), the latter is the ring- theoretical product \(<AS>=\{\sum a_ is_ i|\) \(a_ i\in A\), \(s_ i\in S\}\) of its subsets A and S. In this notation, each pair of quasi- ideals \(A_ 1\) of A and \(S_ 1\) of S determines a quasi-ideal \(Q=<A_ 1S_ 1>\). The latter are called formal quasi-ideals of \(A_ 0[S]\) and determine the corresponding \(A_ 1\) and \(S_ 1\) uniquely if \(Q\neq \{0\}\) holds. These formal quasi-ideals of \(A_ 0[S]\) are used to obtain several results on arbitrary quasi-ideals Q of S, in particular on minimal ones. For instance, each quasi-ideal Q of \(A_ 0[S]\) is contained in a unique smallest formal quasi-ideal, and if a minimal quasi-ideal Q of \(A_ 0[S]\) contains an element \(a\neq 0\) of A [s\(\neq 0\) of S], then Q is a formal quasi-ideal and \(S=G^ 0\) a 0-group [A a division ring].
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 3935341
zbMATH Keywords: contracted semigroup ring
zbMATH Keywords: formal quasi-ideals
zbMATH Keywords: minimal quasi-ideal
MaRDI profile type: Publication
cites work: Q5629854
cites work: Remarks on the concept of algebra
cites work: Q4187930
cites work: Quasi-ideals in rings
cites work: On quasi-ideals in rings
cites work: On 0-simple semirings, semigroup semirings, and two kinds of division semirings
OpenAlex ID: W1983456089
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description: scientific article; zbMATH DE number 3935341