COMMUNICATIONS IN ALGEBRA, vol.34, no.3, pp.841-856, 2006 (Journal Indexed in SCI)
In this article, we consider the module theoretic version of I-semiperfect rings R for an ideal I which are defined by Yousif and Zhou (2002). Let M be a left module over a ring R , N is an element of sigma[M], and tau(M) a preradical on sigma[M]. We call N tau(M)-semiperfect in sigma[M] if for any submodule K of N , there exists a decomposition K = A circle plus B such that A is a projective summand of N in sigma[M] and B <= tau(M) (N). We investigate conditions equivalent to being a tau(M)-semiperfect module, focusing on certain preradicals such as Z(M) , Soc , and delta(M) . Results are applied to characterize Noetherian QF-modules (with Rad (M) <= Soc(M)) and semisimple modules. Among others, we prove that if every R-module M is Soc-semiperfect, then R is a Harada and a co-Harada ring.