Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls
type: Item
instance of: scholarly article
title: Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls
zbMATH Open document ID: 1483.16015
author: Truong Cong Quynh
author: M. Tamer Koşan
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2021-08-03T00:00:00Z
review text: A module \(M\) is called automorphism-invariant if \(M\) is invariant under any automorphism of the injective hull of \(M\). It is shown that for a ring \(R\), if every cyclic right \(R\)-module is automorphism-invariant, then \(R\) is stably-finite (i.e., \(\text{Mat}_n(R)\) is directly-finite for each positive integer \(n\)). The minimal automorphism-invariant essential extension of a module \(M\) is called the automorphism-invariant hull of \(M\). Further, a ring \(R\) is called right \(a\)-hypercyclic if each cyclic right \(R\)-module has a cyclic automorphism-invariant hull. Then the authors show that if a ring \(R\) is right \(a\)-hypercyclic and \(R_R\) has Krull dimension (in the sense of Rentschler and Gabriel), then \(R\) is right Artinian.
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 7377732
zbMATH Keywords: automorphism-invariant module
zbMATH Keywords: automorphism-invariant hull
zbMATH Keywords: \(a\)-hypercyclic ring
reviewed by: Jae Keol Park
MaRDI profile type: Publication
full work available at URL: s00200-021-00494-8
OpenAlex ID: W3127171398
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DOI: 10.1007/S00200-021-00494-8
description: scientific article; zbMATH DE number 7377732