On the least multiplicative nucleus of a ring
type: Item
instance of: scholarly article
title: On the least multiplicative nucleus of a ring
zbMATH Open document ID: 0791.16019
DOI: 10.1016/0022-4049(92)90113-T
author: Shu-Hao Sun
published in: Journal of Pure and Applied Algebra
publication date: 1993-01-16T00:00:00Z
review text: Let \(R\) be a ring with identity and \(\text{Idl } R\) be the ideal lattice of \(R\). A function \(k\) on \(\text{Idl } R\) is a closure operator if \(k\) is inflationary, order-preserving, and idempotent. A multiplicative nucleus on \(\text{Idl } R\) is a closure operator \(k\) on \(\text{Idl } R\) such that, for all ideals \(I\) and \(J\) of \(R\), (1) \(k(I)\cap k(J)= k(I\cap J)= k(IJ)\) and (2) if \(k(I)=R\), then \(I=R\). It is known that the least multiplicative nucleus of a commutative ring is the Levitzki radical. In this paper, the author shows the existence of the least multiplicative nucleus for any ring by defining, through transfinite induction, a new kind of radical ideal. In commutative rings, this radical ideal is the Levitzki radical. If \(a\in R\), let (a) be the principal ideal generated by \(a\). A ring \(R\) is called an \(m^*\)-ring if for each \(a\in R\) and each natural number \(n\), \((a)^ n\) is finitely generated. The author shows that the prime ideal theorem for distributive lattices implies that each semisimple ideal is an intersection of prime ideals for \(m^*\)-rings.
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 13A10
zbMATH DE Number: 89556
zbMATH Keywords: ideal lattices
zbMATH Keywords: closure operators
zbMATH Keywords: least multiplicative nucleus
zbMATH Keywords: Levitzki radical
zbMATH Keywords: radical ideals
zbMATH Keywords: principal ideals
zbMATH Keywords: prime ideal theorem
zbMATH Keywords: distributive lattices
zbMATH Keywords: semisimple ideals
zbMATH Keywords: prime ideals
zbMATH Keywords: \(m^*\)-rings
MaRDI profile type: Publication
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full work available at URL: 0022-4049(92)90113-t
OpenAlex ID: W2069699906
description: scientific article; zbMATH DE number 89556