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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) = ∩ _i=1 …