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A characterization of Artinian rings. (English) Zbl 0637.16012

A well known theorem of B. L. Osofsky states that a ring R is semiprime Artinian if and only if every cyclic right R-module is injective [Pac. J. Math. 14, 645-650 (1964; Zbl 0145.266)]. A. W. Chatters [Q. J. Math., Oxf. II. Ser. 33, 65-69 (1982; Zbl 0443.16011)] has proved that a ring R is right Noetherian if and only if every cyclic right R-module is the direct sum of a projective module and a Noetherian module. The authors prove that a ring R is right Artinian if and only if every cyclic right R-module is the direct sum of an injective module and a finitely cogenerated (in particular, Artinian) module. The proof uses Osofsky’s Theorem. The remainder of this paper is concerned with hereditary Artinian rings, i.e. rings for which every ideal is a right Artinian ring.
Reviewer: P.F.Smith

MSC:

16P20 Artinian rings and modules (associative rings and algebras)
16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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References:

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