The lattice of ideals of a polynomial semiring
type: Item
instance of: scholarly article
title: The lattice of ideals of a polynomial semiring
zbMATH Open document ID: 0798.16038
DOI: 10.1007/BF01188186
author: Francisco E. Alarcón
author: D. D. Anderson
published in: Algebra Universalis
publication date: 1994-03-10T00:00:00Z
review text: A semiring \((R,+,\cdot)\) is a nonempty set \(R\) where \((R,+)\) is a commutative monoid with additive identity 0 and the second binary operation satisfies the distributive laws \(a(b + c) = ab + ac\), \((b + c)a = ba + ca\) and \(a0 = 0 = 0a\). By \(R[X]\) we denote the semiring of polynomials over \(R\). Recall that a left ideal \(I\) of \(R[X]\) is said to be subtractive if \(f\in R[X]\) and \(f + g\), \(g\in I\) imply \(f\in I\). The authors prove that for a semiring \(R\), the following statements are equivalent: (1) \(R\) is a ring; (2) every left ideal of \(R[X]\) is subtractive; (3) the lattice of left ideals of \(R[X]\) is modular.
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 513026
zbMATH Keywords: modular lattice
zbMATH Keywords: subtractive left ideals
zbMATH Keywords: semiring of polynomials
zbMATH Keywords: lattice of left ideals
reviewed by: B. Pondělíček
MaRDI profile type: Publication
cites work: Q4023553
cites work: Q5605275
full work available at URL: https://doi.org/10.1007/bf01188186
OpenAlex ID: W2033402178
description: scientific article; zbMATH DE number 513026