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On essentially left-Artinian rings. (Russian) Zbl 0714.16014

The ring R is named essentially left Artinian if it contains some essential Artinian left ideal. A pretorsion r of the category \({}_ R{\mathfrak M}\) is named: 1) ideal, if its filter F(r) is closed relative to arbitrary intersections; 2) coessential, if from \(r\cap t=\epsilon\) follows \(t=\epsilon\), where \(\epsilon (M)=M\) for every \(M\in_ R{\mathfrak M}\). A ring R is essentially left Artinian iff Goldie’s pretorsion z in \({}_ R{\mathfrak M}\) is ideal and coessential and every pretorsion \(r>z\) in \({}_ R{\mathfrak M}\) is ideal. As a corollary is obtained the following description of semisimple Artinian rings: semiprime rings R for which all pretorsions in \({}_ R{\mathfrak M}\) are ideal.
Reviewer: A.I.Kashu

MSC:

16P20 Artinian rings and modules (associative rings and algebras)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16N60 Prime and semiprime associative rings
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