CLASSICAL PRIME RADICAL OF MODULES AND A GENERATION OF BEAR'S LOWER NILRADICAL FOR MODULES

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 for a modul [denoted by cl.adR(M)] and Baers lower nilradical for a module [denoted by Nil8(RM)]. for a module RM, cl.radR(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.radR(M) = Nil*(RM) and, for any module M over a left Artinian ring R, cl.radR(M) = Rad(M) = Jac (R)M. In Particular, if R is a commutative Noetherian dimian 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 modulse. Moreover, over an FBN prime ring R with dim (R) <1), every semiprime submodule of any module is an intersection of classical prim submodules.

کلیدواژه ها:

Classical prime module ، Baer-McCoy radical ، Baers lower nileradical