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# 6 - M. Behboodi, A generalization of Bear's lower nilradical for modules, J. Algebra Appl. 6(2) (2007), 337-353. (ISI)

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals A,B ⊆ R and for all submodules N ⊆ M, ABN ⊆ P implies that AN ⊆ P or BN ⊆ P.We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.radR(M)] and Baer’s lower nilradical for a module [denoted by Nil∗(RM)]. For a module RM, cl.radR(M) is defined to be the intersection of all classical prime submodules of M and Nil_∗(M) 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.radR(M)=Nil∗(RM) and, for any moduleM over a left Artinian ring R, cl.radR(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.radR(M)=Nil_∗(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim(R)≤1 (or over a commutative domain R with dim(R)≤1), every semiprime

submodule of any module is an intersection of classical prime submodules.