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 ε 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 rad _R (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/rad _R (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/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ ⁿ _i=1 $\mathcal{P}_i M$ for some maximal ideals $\mathcal{P}_1 , \ldots ,\mathcal{P}_n$ of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M.