10 - M. Behboodi, On the prime radical and Baer's lower nilradical of modules, Acta Math. Hungar. 122 (3) (2009), 293-306. (ISI)
Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define ppN := fm 2 M: every m-system containing m meets Ng. It is shown that ppN is the intersection of all prime submodules of M containing N. We define radR(M) = pp(0). This is called BaerMcCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M= radR(M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M= radR(M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either radR(M) = M or radR(M) = Tn i=1 PiM for some maximal ideals P1; : : : ;Pn of R. Also, Baer's lower nilradical of M [denoted by Nil¤(RM)] is dened to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, radR(M) = Nil¤(RM) and, for any module M over a left Artinian ring R, radR(M) = Nil¤(RM) = Rad (M) = Jac (R)M.