“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
Paper IPM / M / 8954  


Abstract:  
Let M be a left Rmodule. In this paper a generalization of the notion of
msystem set of rings to modules is given. Then for a submodule N of M,
we define ^{p}√{N}:={m ∈ M: every msystem containing m meets
N}. It is shown that ^{p}√{N} is the intersection of all
prime submodules of M containing N. We define rad_{R}(M)=^{p}√{(0)}.
This is called BaerMcCoy radical or prime radical of M.
It is shown that if M is an Artinian module over a PIring (or an FBNring) R, then M/rad_{R}(M) is a
Noetherian Rmodule. Also, if M is a Noetherian module over
a PIring (or an FBNring) R such that every prime submodule of
M is virtually maximal, then M/rad_{R}(M) is an Artinian
Rmodule. This yields if M is an Artinian module over a
PIring R, then either rad_{R}(M) = M or
rad_{R}(M)=∩_{i=1}^{n}P_{i}M for some maximal
ideals P_{1},…,P_{n} of R. Also, Baer's
lower nilradical of M [denoted by Nil_{*}(_{R}M)] is defined to be
the set of all strongly nilpotent elements of M. It is shown
that, for any projective Rmodule M, rad_{R}(M)=Nil_{*}(_{R}M)
and, for any module M over a left Artinian ring R,
rad_{R}(M)=Nil_{*}(_{R}M)=Rad(M)=Jac(R)M.
Download TeX format 

back to top 