Fully idempotent semirings
type: Item
instance of: scholarly article
title: Fully idempotent semirings
zbMATH Open document ID: 0799.16035
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-11-23T00:00:00Z
review text: The paper deals with semirings \((A,+,\cdot)\) which are additively commutative and contain an absorbing zero. Such a semiring is called fully idempotent if \(I^ 2 = I\) holds for each ideal \(I\) of \((A,+,\cdot)\). As for rings, this is equivalent with \(I\cap J = IJ\) for all ideals \(I\) and \(J\), and the author shows that the lattice of all ideals of a fully idempotent semiring is Brouwerian and hence distributive. The main result states that \((A,+,\cdot)\) is fully idempotent iff each ideal \(I\) of \((A,+,\cdot)\) is the intersection of all prime ideals containing \(I\). Similar statements are proved for ideals of a semigroup \((S,\cdot)\).
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 556449
zbMATH Keywords: lattice of ideals
zbMATH Keywords: intersection of prime ideals
zbMATH Keywords: semirings
zbMATH Keywords: fully idempotent semiring
zbMATH Keywords: Brouwerian
zbMATH Keywords: distributive
author: Javed Ahsan
reviewed by: Hanns Joachim Weinert
MaRDI profile type: Publication
full work available at URL: pjaa.69.185
OpenAlex ID: W2066837225
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DOI: 10.3792/PJAA.69.185
description: scientific article; zbMATH DE number 556449