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Rings with restricted descending chain conditions. (English) Zbl 0882.16015

The authors prove that if \(R\) is a ring all of whose left primitive factor rings are Artinian and \(R\) satisfies the descending chain condition on essential left ideals then every prime ideal of \(R\) is maximal and \(R\) is strongly \(\pi\)-regular. Applications to PI-rings are given.

MSC:

16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
16D25 Ideals in associative algebras
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