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• W2024977214 abstract "An R-module M has Artinian prime factors if M/PM is an Artinian module for each prime ideal P of R. For commutative rings R it is shown that Noetherian modules with Artinian prime factors are Artinian. If R is either commutative or a von Neumann regular V-ring then the endomorphism ring of a module with Artinian prime factors is a strongly 7r-regular ring. A ring R with 1 is left T-regular if for each a E R there is an integer n > 1 and b E R such that an = a + I'b. Right 7T-regular is defined in the obvious way, however a recent result of F. Dischinger [5] asserts the equivalence of the two concepts. A ring R is 7T-regular if for any a E R there is an integer n > 1 and b E R such that an = anban. Any left 7T-regular ring is so-regular but not conversely. Because of this, we say that R is strongly 7T-regular if it is left (or right) 7T-regular. In [2, Theorem 2.5] it was established that if R is a (von Neumann) regular ring whose primitive factor rings are Artinian and if M is a finitely generated R-module then the endomorphism ring EndR(M) of M is a strongly 7T-regular ring. Curiously enough, the same is not true for finitely generated modules over strongly 7T-regular rings, as Example 3.1 of [2] shows. Obviously, it also fails for arbitrary regular rings. These observations lead one to consider conditions on finitely generated modules which ensure that the endomorphism ring is strongly 7T-regular. A natural one seems to be that of having Artinian prime factors. In fact, we establish that such modules have strongly g7-regular endomorphism ring whenever the base ring is either commutative or a regular V-ring. Consider a finitely generated module M over a ring R. If R is commutative then R is g7-regular if and only if its prime ideals are maximal [11]. Accordingly, when R is commutative and 7T-regular, M/PM is an Artinian module for all primes P. This observation serves as a starting point. THEOREM 1. Suppose R is a commutative ring. For a finitely generated R-module M the following conditions are equivalent. (a) M has Artinian prime factors. (b) S = EndR(M) is a strongly T-regular ring. (c) R/AnnR(M) is a T-regular ring. Received by the editors September 19, 1978 and, in revised form, April 17, 1979. AMS (MOS) subject classifications (1970). Primary 16A30, 16A46, 16A64." @default.
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• W2024977214 title "Modules with Artinian prime factors" @default.
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