Study of Noetherian \(\Gamma\)-semirings via operator semirings
type: Item
instance of: scholarly article
title: Study of Noetherian \(\Gamma\)-semirings via operator semirings
zbMATH Open document ID: 1010.16050
author: T. K. Dutta
author: Sujit Kumar Sardar
published in: Southeast Asian Bulletin of Mathematics
publication date: 2002-12-01T00:00:00Z
review text: The authors study various ascending chain conditions on \(\Gamma\)-semirings by considering the equivalent concepts for the operator semirings. All \(\Gamma\)-semirings considered in this paper possess left and right unities. Central to their investigation is Theorem 2.9 which states that a \(\Gamma\)-semiring \(S\) is right (\(k\)-, \(h\)-) Noetherian if and only if its left operator semiring \(L\) is right \((k\)-, \(h\)-) Noetherian. Cohen's theorem is extended to \(\Gamma\)-semirings, i.e. it is shown that \(S\) is Noetherian if and only if every prime ideal of \(S\) is finitely generated. Various decomposition theorems are then proved for ideals. In particular, it is shown that if \(S\) is right Noetherian, then every proper right ideal of \(S\) can be represented as the intersection of finitely many irreducible right ideals of \(S\).
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1836954
zbMATH Keywords: Noetherian \(\Gamma\)-semirings
zbMATH Keywords: ascending chain conditions
zbMATH Keywords: operator semirings
zbMATH Keywords: prime ideals
zbMATH Keywords: decomposition theorems
zbMATH Keywords: irreducible right ideals
MaRDI profile type: Publication
full work available at URL: s100120200004
OpenAlex ID: W2053713942
DOI: 10.1007/S100120200004
Recommended article: On the operator semirings of a \(\Gamma\)-semiring.
Recommended article: Q4344989
Recommended article: An introduction to operator seminearrings of a near-\(\Gamma\)-semiring.
Recommended article: Q3496798
Recommended article: Study of additively regular Γ-semirings and derivations
Recommended article: \(\Gamma\)-semirings. II
Recommended article: Q4981631
Recommended article: A study on ideal elements in ordered Gamma-semirings
Recommended article: Noetherian Semigroup Algebras and Beyond
Recommended article: Q4762683
description: scientific article; zbMATH DE number 1836954