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| Paper IPM / M / 8954 |
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| Abstract: | |
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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 p√{N}:={m ∈ M: every m-system containing m meets
N}. It is shown that p√{N} is the intersection of all
prime submodules of M containing N. We define radR(M)=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/radR(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/radR(M) is an Artinian
R-module. This yields if M is an Artinian module over a
PI-ring R, then either radR(M) = M or
radR(M)=∩i=1nPiM for some maximal
ideals P1,…,Pn of R. Also, Baer's
lower nilradical of M [denoted by Nil*(RM)] is defined to be
the set of all strongly nilpotent elements of M. It is shown
that, for any projective R-module M, radR(M)=Nil*(RM)
and, for any module M over a left Artinian ring R,
radR(M)=Nil*(RM)=Rad(M)=Jac(R)M.
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