Antiisomorphisms of endomorphism rings of modules close to free ones which are induced by Morita antiequivalences
type: Item
instance of: scholarly article
title: Antiisomorphisms of endomorphism rings of modules close to free ones which are induced by Morita antiequivalences
zbMATH Open document ID: 0917.16005
DOI: 10.1007/BF02355850
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-03-15T00:00:00Z
review text: Let the modules \(M_A\) and \(N_B\) be strong generators (i.e. they contain free cyclic direct summands) and let \(\alpha\colon\text{End}(M_A)\to\text{End}(N_B)\) be an antiisomorphism of their endomorphism rings. Suppose that \(M_A\) is a torsionless module (i.e. the canonical homomorphism \(M\to M^{**}\) is a monomorphism). In this situation some conditions are shown, which are equivalent to the following: 1. \(\alpha\) is induced by a Morita antiequivalence, i.e. by a category equivalence \(H\colon\text{Mod-}A^{op}\to\text{Mod-}B\). The conditions obtained are formulated using rank-1 idempotents of the rings studied. For example: 2. If \(e\in\text{End}(M_A)\) is a rank-1 idempotent (i.e. \(e^2=e\) and \(eM\) is a free cyclic module), then \((e^\alpha N)_B\) is a progenerator of \(\text{Mod-}B\); 3. There exist rank-1 idempotents \(e\in\text{End}(M_A)\) and \(u\in\text{End}(N_B)\) such that \((e^\alpha N)_B\) and \((u^{\alpha^{-1}} M)_A\) are generators.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1259437
zbMATH Keywords: strong generators
zbMATH Keywords: free cyclic direct summands
zbMATH Keywords: endomorphism rings
zbMATH Keywords: torsionless modules
zbMATH Keywords: Morita antiequivalences
zbMATH Keywords: category equivalences
zbMATH Keywords: idempotents
zbMATH Keywords: progenerators
zbMATH Keywords: generators
author: K. I. Beidar
author: A. V. Mikhalev
reviewed by: A. I. Kashu
MaRDI profile type: Publication
cites work: Isomorphisms between endomorphism rings of progenerators
cites work: Anti-isomorphisms of the endomorphism rings of a class of free modules
cites work: Anti-isomorphisms of the endo-morphism rings of torsion-free modules
cites work: Q3786586
cites work: Q4002547
cites work: Anti-isomorphisms of endomorphism rings of locally free modules
full work available at URL: https://doi.org/10.1007/bf02355850
OpenAlex ID: W2075990035
description: scientific article; zbMATH DE number 1259437