Rings over which each module possesses a maximal submodule
type: Item
instance of: scholarly article
title: Rings over which each module possesses a maximal submodule
zbMATH Open document ID: 0919.16004
DOI: 10.1007/BF02355415
published in: Mathematical Notes
publication date: 1999-09-08T00:00:00Z
review text: Let \(A\) be an associative ring with nonzero identity element. \(A\) is said to be a right Bass ring if any nonzero right module over \(A\) has a maximal submodule. By \(C(A)\) is denoted the center of the ring \(A\). The ring \(A\) is said to be algebraic over its center if for every element \(a\in A\), there is a polynomial \(f\in C(A)[x]\) such that \(f(a)=0\) and the leading coefficient of the polynomial \(f(x)\) is a regular element of the ring \(A\). The main result established by the author is: Assume that any prime quotient ring of the ring \(A\) is algebraic over its center. Then the following conditions are equivalent: (1) \(A\) is a right perfect ring; (2) \(A\) is a right Bass ring that contains no infinite set of orthogonal idempotents. Also, a series of other results on right Bass rings is proved.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 1238067
zbMATH Keywords: right modules
zbMATH Keywords: maximal submodules
zbMATH Keywords: centers
zbMATH Keywords: regular elements
zbMATH Keywords: prime quotient rings
zbMATH Keywords: rings algebraic over center
zbMATH Keywords: right perfect rings
zbMATH Keywords: orthogonal idempotents
zbMATH Keywords: right Bass rings
author: A. A. Tuganbaev
reviewed by: Iuliu Crivei
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1238067