Fully prime semirings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1018.16029
author: Francisco E. Alarcón
author: Dominika Polkowska
publication date: 2001-08-06T00:00:00Z
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1606252
zbMATH Keywords: commutative semirings
zbMATH Keywords: prime ideals
zbMATH Keywords: fully prime semirings
zbMATH Keywords: fully idempotent semirings
MaRDI profile type: Publication
title: Fully prime semirings
published in: Kyungpook Mathematical Journal
review text: Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero 0 and identity \(1\not=0\). An ideal \(P\not=S\) is prime if \(IJ\subseteq P\) implies \(I\subseteq P\) or \(J\subseteq P\) for all ideals \(I,J\) of \(S\). The semiring \(S\) is called fully prime if every ideal \(P\not=S\) is prime. It is shown that \(S\) is fully prime iff every ideal \(I\) of \(S\) is idempotent, i.e. \(II=I\), and the set \(\text{ideal}(S)\) of all ideals of \(S\) is totally ordered by inclusion. The authors also characterize those factor semirings \(\mathbb{N}_0/\kappa\) of the natural numbers \(\mathbb{N}_0\) which are fully prime, and they give a description of all \(k\)-ideals of \(\mathbb{N}_0/\kappa\).
reviewed by: Udo Hebisch
description: scientific article; zbMATH DE number 1606252