Rings in which annihilators form a sublattice of the lattice of ideals
type: Item
instance of: scholarly article
title: Rings in which annihilators form a sublattice of the lattice of ideals
zbMATH Open document ID: 0538.16009
DOI: 10.1007/BF00970321
author: M. Ya. Finkel'shtejn
published in: Siberian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: An element of a ring R is said to be regular, if \(a=axa\) for some \(x\in R\). Regular elements of R constitute a semigroup if and only if R is an AC-ring, i.e. left ideals of R generated by single idempotents constitute a sublattice of the lattice of left ideals of R. The equivalence of the following properties of a ring R is stated: (1) R is a left Rickart AC- ring; (2) R is a left Rickart ring and the sum of every two left ideals generated by single idempotents is a left annihilator; (3) R/M(R), where M(R) is the regular radical of R [\textit{B. Brown}, \textit{N. H. McCoy}, Proc. Am. Math. Soc. 1, 165-171 (1950; Zbl 0036.297)], is a left Rickart ring without nilpotent elements and the left ideal Ra is projective for each \(a\in R\) such that \(a+M(R)\) is a non zero-divisor of R/M(R). A ring R is a left Rickart ring with \(r(I\cap J)=r(I)+r(J)\) for every left ideals of R (r(X) means the right annihilator of X) if and only if \(R=Q\oplus B\), where Q is a left self-injective ring and B is a Baer ring with regular maximal ring of quotients. A ring R is a left Rickart ring with \(r(\cap I_ i)=\sum r(I_ i)\) for each set of left ideals of R if and only if \(R=S\oplus T\), where S is a direct sum of integral domains and T is an Artinian semisimple ring.
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 3855312
zbMATH Keywords: AC-ring
zbMATH Keywords: idempotents
zbMATH Keywords: lattice of left ideals
zbMATH Keywords: left Rickart AC-ring
zbMATH Keywords: left annihilator
zbMATH Keywords: regular radical
zbMATH Keywords: left self-injective ring
zbMATH Keywords: Baer ring
zbMATH Keywords: maximal ring of quotients
zbMATH Keywords: direct sum of integral domains
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3855312