On uniform dimensions of ideals in right nonsingular rings
type: Item
instance of: scholarly article
title: On uniform dimensions of ideals in right nonsingular rings
zbMATH Open document ID: 0931.16001
DOI: 10.1016/S0022-4049(97)00188-6
author: S. K. Jain
author: André Leroy
author: T. Y. Lam
published in: Journal of Pure and Applied Algebra
publication date: 1999-12-07T00:00:00Z
review text: A new bimodule invariant is defined, applied to ideals in a right nonsingular ring \(R\), and used to relate the ideal structure of \(R\) to that of its maximal right quotient ring \(Q\). Given a bimodule \(M\), the authors define \(d(M)\) to be the supremum of those positive integers \(n\) such that \(M\) contains an essential right submodule which is a direct sum of \(n\) nonzero sub-bimodules. For example, if \(R\) is semiprime and \(M\) is an ideal in \(R\), then \(d(M)\) equals the uniform dimension of \(_RM_R\). This equality is also verified in a number of other cases; for instance, it holds for ideals in the family \({\mathcal F}(R)\) defined below in case the symmetric maximal quotient ring of \(R\) is semiprime. The invariant \(d\) is particularly well suited to the study of those ideals \(I\) of \(R\) for which there exists an ideal \(J\) with \(I\oplus J\) essential in \(R_R\); this family of ideals is denoted \({\mathcal F}(R)\). Various characterizations of \({\mathcal F}(R)\) are given, and the subfamily of right essentially closed ideals in \({\mathcal F}(R)\) is shown to form a complete Boolean algebra isomorphic to the Boolean algebra of central idempotents in \(Q\). One characterization of \({\mathcal F}(R)\) is that an ideal \(I\) lies in \({\mathcal F}(R)\) if and only if the injective hull \(E(I)\) is an ideal of \(Q\); in this case, it is known that \(E(I)=eQ\) for some central idempotent \(e\in Q\), and so \(E(I)\) is a unital ring. For \(I\in{\mathcal F}(R)\), the authors prove that the ring \(E(I)\) is a direct product of prime rings if and only if there exist ideals \(I_j\) in \(R\), with \(d(I_j)=1\) for all \(j\), such that \(\bigoplus_j I_j\) is an essential submodule of \(I_R\). There is an improved criterion for the case of finite direct products: \(E(I)\) is a direct product of \(n\) prime rings if and only if \(d(I)=n\).
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 1308267
zbMATH Keywords: bimodules
zbMATH Keywords: right nonsingular rings
zbMATH Keywords: right ideals
zbMATH Keywords: maximal right quotient rings
zbMATH Keywords: direct sums
zbMATH Keywords: essential right submodules
zbMATH Keywords: semiprime rings
zbMATH Keywords: uniform dimensions
zbMATH Keywords: symmetric maximal quotient rings
zbMATH Keywords: central idempotents
zbMATH Keywords: direct products of prime rings
reviewed by: Kenneth R. Goodearl
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00188-6
OpenAlex ID: W2139450342
description: scientific article; zbMATH DE number 1308267