On Semilocal Modules and Rings

Details On Semilocal Modules and Rings On Semilocal Modules and Rings

1998-07-21

arxiv.org/abs/math/9807113v1

Communications in Algebra, 27(4), 1921-1935 (1999)

Mathematics

Rings and Algebras

to appear in Communications in Algebra

Scientific paper

It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R as a left (or as a right) R-module is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).

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