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Paper   IPM / M / 8344
School of Mathematics
  Title:   A generalization of Baer's lower nilradical for modules
  Author(s):  M. Behboodi
  Status:   Published
  Journal: J. Algebra Appl.
  Vol.:  6
  Year:  2007
  Pages:   337-353
  Supported by:  IPM
Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals A,BR and for all submodules NM, ABNP implies that ANP 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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