On Semilocal Modules and Rings

Abstract

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).