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PI-rings with Artinian proper cyclics are Noetherian. (English) Zbl 1301.16023

Let \(R\) be an associative ring. The main result of this paper asserts that if \(R\) is a PI integral domain in which every proper cyclic right \(R\)-module is Artinian, then \(R\) is a right Noetherian ring. The other results are for group algebras. Let \(KG\) be a group algebra of a solvable group \(G\). If each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian, then \(KG\) is Noetherian. Moreover, let \(G\) be an arbitrary group. If each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian and \(K\)-finite dimensional, then \(KG\) is also Noetherian. The author does not discuss the conditions under which each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian.

MSC:

16P20 Artinian rings and modules (associative rings and algebras)
16P40 Noetherian rings and modules (associative rings and algebras)
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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