On maximal submodules of a finite direct sum of hollow modules. I
type: Item
instance of: scholarly article
title: On maximal submodules of a finite direct sum of hollow modules. I
zbMATH Open document ID: 0543.16007
published in: Osaka Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let R be a right artinian ring with identity. It is assumed that every R- module M has finite composition length. For an R-module M the Jacobson radical of M is denoted by J(M). If M has as unique maximal submodule J(M), M is called hollow. In this paper the author studies the structure of right artinian rings R satisfying the following conditions i) \([J(R)]^ 2=0\) and ii) every submodule of a direct sum of hollow modules is also a direct sum of hollow modules. If R is either a commutative artinian ring or an algebra of finite dimension over an algebraically closed field, it is shown that \(| eR|\), the composition length of eR, is \(\leq 3\), where e is a primitive idempotent and R satisfies an additional condition on right ideals in eJ \((J=J(R))\) and dimensions. In the main theorem the author describes the structure of right artinian rings R with \(J^ 2=0\) under the assumptions that \(| eJ| \leq 4\) for every idempotent e. In this case R satisfies the above condition (ii) if and only if eJ has certain decompositions in terms of right ideals. Finally rings with \(| eJ| \geq 5\) are studied with \(J^ 2=0\). At the end of the paper the author illustrates his results giving examples.
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 16U99
zbMATH DE Number: 3863397
zbMATH Keywords: finite composition length
zbMATH Keywords: Jacobson radical
zbMATH Keywords: right artinian rings
zbMATH Keywords: direct sum of hollow modules
zbMATH Keywords: primitive idempotent
zbMATH Keywords: right ideals
author: Manabu Harada
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3863397