Chain conditions on semirings
type: Item
instance of: scholarly article
title: Chain conditions on semirings
zbMATH Open document ID: 0845.16047
DOI: 10.1155/S0161171296000452
author: T. K. Mukherjee
author: M. K. Sen
author: Shamik Ghosh
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1996-09-11T00:00:00Z
full work available at URL: https://eudml.org/doc/47490
review text: Let \(S=(S,+, \cdot)\) be an additively commutative semiring with an absorbing zero and \(M=(_SM,+)\) a (left) semimodule over \(S\). An \(S\)-subsemimodule \(U\) of \(M\) is called a \(k\)-subsemimodule \([h\)-subsemimodule] if \(u+x=v\) [\(u+x+z=v+z]\) for \(u,v \in U\) and \(x,z \in M\) imply \(x \in U\), and \(M\) is called artinian [\(k\)-artinian, \(h\)-artinian] if \(M\) satisfies the descending chain condition on \(S\)-subsemimodules [\(k\)-subsemimodules, \(h\)-subsemimodules]. Note that the left ideals of \((S,+, \cdot)\) are the \(S\)-subsemimodules of \((_SS,+)\) and let \(S^n=(_SS^n,+)\) be the direct product of \(n\) copies of \((_SS,+)\). Since the semiring \(M_n(S)\) of all \(n \times n\)-matrices over an artinian semiring \(S\) (with identity) need not be artinian, the following result is of interest: \(M_n(S)\) is artinian [\(k\)-artinian, \(h\)-artinian] iff \(S^n\) is. Further: If \(S\) has a ring of differences \(D(S)\), then \(M_n (S)\) is \(k\)-artinian iff \(D(S)\) is artinian. Other results include the ring \(D(S/\kappa)\) in these considerations, where \(\kappa\) is the least additively cancellative congruence on \((S,+, \cdot)\). In a similar way noetherian [\(k\)-noetherian, \(h\)-noetherian] semirings are defined. One result is: If \(S\) has a ring of differences \(D(S)\), then the polynomial semiring \(S[x]\) is \(k\)-noetherian if \(D(S)\) is noetherian. This generalizes a result of \textit{H. E. Stone} [Trans. Am. Math. Soc. 233, 339-353 (1977; Zbl 0371.16016)], where \(D(S)\) was assumed to have an ideality.
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 874316
zbMATH Keywords: \(k\)-subsemimodules
zbMATH Keywords: semirings of \(n\times n\)-matrices
zbMATH Keywords: \(h\)-subsemimodules
zbMATH Keywords: descending chain condition on \(S\)-subsemimodules
zbMATH Keywords: left ideals
zbMATH Keywords: Artinian semirings
zbMATH Keywords: ring of differences
zbMATH Keywords: additively cancellative congruences
zbMATH Keywords: polynomial semirings
reviewed by: Hanns Joachim Weinert
MaRDI profile type: Publication
OpenAlex ID: W2117175482
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description: scientific article; zbMATH DE number 874316