Isomorphisms between infinite matrix rings
type: Item
instance of: scholarly article
title: Isomorphisms between infinite matrix rings
zbMATH Open document ID: 0574.16020
DOI: 10.1016/0024-3795(85)90079-5
author: Michael L. Bolla
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: In [J. Algebra 87, 261-281 (1984; Zbl 0554.16009)] the author obtained a categorical description of the isomorphisms between endomorphism rings of progenerators. As a special situation he described isomorphisms between matrix rings (over arbitrary rings) from a categorical point of view. In this paper the author succeeds in gaining a categorical description for the isomorphisms between endomorphism rings of countably infinitely generated free modules (over arbitrary rings), i.e. he proves the following main theorem: Let R, S be arbitrary associative rings with identity. Let \(U=R^{(N)}\), \(V=S^{(N)}\), where N is a countably infinite set. Let \(\Phi\) : End\({}_ R(U){\tilde \to}End_ S(V)\) be a ring isomorphism. Then there exists a unique (up to natural isomorphism) category equivalence \(F_{\Phi}: M_ R\to M_ S\) such that \(F_{\Phi}(U)=V\), and \(F_{\Phi}(f)=\Phi (f)\) for all \(f\in End_ R(U)\). - The author also shows that a semilinear description can be recovered from the categorical description in some particular cases.
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 3916502
zbMATH Keywords: isomorphisms between endomorphism rings of progenerators
zbMATH Keywords: isomorphisms between matrix rings
zbMATH Keywords: infinitely generated free modules
zbMATH Keywords: category equivalence
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90079-5
OpenAlex ID: W2029276197
cites work: Isomorphisms between endomorphism rings of progenerators
cites work: Morita Equivalence and Infinite Matrix Rings
cites work: Ultrafunctor
cites work: Endomorphism rings of infinitely generated projective modules
cites work: Q4061092
cites work: Isomorphisms of the endomorphism ring of a free module over a principal left ideal domain
description: scientific article; zbMATH DE number 3916502