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| Paper IPM / M / 8344 |
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| Abstract: | |
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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 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 sub modules
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) = 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.
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