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On semilocal modules and rings. (English) Zbl 0927.16012

A module \(M\) is called semilocal if \(M/\text{Rad}(M)\) is semisimple. When \(M\) is a ring, this gives the usual definition of a semilocal ring. In the present paper, the author obtains several new characterizations of semilocal modules, in terms of weakly supplemented modules and finite dual Goldie dimension. Following H. Zöschinger [Math. Ann. 237, 193-202 (1978; Zbl 0371.13009)], a module \(M\) is called weakly supplemented if every submodule \(N\) of \(M\) has a weak supplement. It is shown in this paper that if a module \(M\) is weakly supplemented, then \(M\) is semilocal, and if \(M\) is finitely generated, then \(M\) is semilocal if and only if \(M\) is weakly supplemented, if and only if \(M\) has finite dual Goldie dimension. As applications, the author provides module-theoretic proofs to several new and known characterizations of semilocal rings, including a deep result of R. Camps and W. Dicks [Isr. J. Math. 81, No. 1-2, 203-211 (1993; Zbl 0802.16010)] that was crucial for their proof of the interesting theorem that the endomorphism ring of any Artinian module is semilocal.

MSC:

16L30 Noncommutative local and semilocal rings, perfect rings
16D80 Other classes of modules and ideals in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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