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Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals and for all submodules N ⊆ M, implies that or . We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad_R(M)] and Baer's lower nilradical for a module [denoted by Nil_*(_RM)]. For a module _RM, cl.rad_R(M) is defined to be the intersection of all classical prime submodules of M and Nil_*(_RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad_R(M) = Nil_*(_RM) and, for any module M over a left Artinian ring R, cl.rad_R(M) = Nil_*(_RM) = Rad(M) = Jac(R)M. In particular, if R is a commutative Noetherian domain with dim(R) ≤ 1, then for any module M, we have cl.rad_R(M) = Nil_*(_RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general …