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Paper   IPM / M / 8954
School of Mathematics
  Title:   On the prime radical and Baer's lower nilradical of modules
  Author(s):  M. Behboodi
  Status:   Published
  Journal: Acta Math. Hungar.
  Vol.:  122
  Year:  2009
  Pages:   293-306
  Supported by:  IPM
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}:={mM: 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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