Semisimple rings of quotients in Morita contexts
type: Item
instance of: scholarly article
title: Semisimple rings of quotients in Morita contexts
zbMATH Open document ID: 0799.16023
DOI: 10.1016/0022-4049(94)90026-4
author: Zhengping Zhou
published in: Journal of Pure and Applied Algebra
publication date: 1994-11-23T00:00:00Z
review text: Let \((R,{_ R U_ S},{_ S V_ R},S)\) be a Morita context and \((\tau,\tau')\) a pair of corresponding Gabriel topologies, containing trace ideals \(I\) and \(J\). Conditions are investigated for the ring of quotients \(\widehat S\) of \(S\) with respect to \(\tau'\) to be semisimple artinian, simple artinian or a division ring. The main results are the following. 1) \(\widehat S\) is semisimple \(\Leftrightarrow \text{Sat}_ \tau(_ RU)\) satisfies a.c.c. and \(\tau\)- closed modules are injective \(\Leftrightarrow\text{Sat}_ \tau(_ RU)\) satisfies a.c.c. and essential submodules of a \(\tau\)-free module \(M\) are \(\tau\)-dense in \(M \Leftrightarrow\) the lattice \(L(_ RU)\) of submodules of \(_ RU\) is \(\tau\)-cofinally finite and \(\tau\)-closed modules are injective. 2) Suppose that \(d(U/T_ \tau(U)) < \infty\) and \(\widehat R\) is semisimple. Then \(\widehat R\) and \(\widehat S\) are Morita equivalent, hence \(\widehat S\) is semisimple. \(\widehat S\) is simple \(\Leftrightarrow \widehat R\) is simple. 3) Suppose that \(\widehat S\) is semisimple. \(\widehat S\) is simple \(\Leftrightarrow T_{\tau_ 1}(U) = T_ \tau(U)\) for any \(\tau_ 1 \geq\tau \Leftrightarrow \widehat{U} = \tau_ 1\widehat{U}\) for any \(\tau_ 1 \geq \tau\). 4) \(\widehat S\) is a division ring \(\Leftrightarrow{_ RU}\) is \(\tau\)-simple (i.e. \(_ RU\) has only two \(\tau\)-closed submodules: \(T_ \tau(M)\) and \(M\)). 5) Suppose the context is nondegenerate and \(\tau'\) is the dense topology on \(S\)-Mod. \(\widehat{S}\) (the maximal left quotient ring of \(S\)) is semisimple \(\Leftrightarrow Z(_ RU) = 0\) and \(d(_ RU) < \infty\).
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 561567
zbMATH Keywords: Morita context
zbMATH Keywords: Gabriel topologies
zbMATH Keywords: trace ideals
zbMATH Keywords: maximal left quotient ring
reviewed by: A. I. Kashu
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90026-4
OpenAlex ID: W1968727484
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description: scientific article; zbMATH DE number 561567