On the prime radical and Baer’s lower nilradical of modules

Abstract

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 \( \sqrt[p]{N} \):= { m e M: every m-system containing m meets N}. It is shown that \( \sqrt[p]{N} \) is the intersection of all prime submodules of M containing N. We define radR(M) = \( \sqrt[p]{{(0)}} \). This is called Baer-McCoy 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) = ∩i=1n\( \mathcal{P}_i M \) for some maximal ideals \( \mathcal{P}_1 , \ldots ,\mathcal{P}_n \) of R. Also, Baer’s lower nilradical of M [denoted by Nil* (RM)] is defined 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.