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Paper   IPM / M / 8114
School of Mathematics
  Title:   On weakly prime radical of modules and semi-compatible modules
  Author(s):  M. Behboodi
  Status:   Published
  Journal: Acta Math. Hungar.
  Vol.:  113
  Year:  2006
  Pages:   243-254
  Supported by:  IPM
Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN\subseteqq P, we have AN \subseteqq P or BN \subseteqq P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R)\leqq 1. Also, we show that over a commutative domain R with dim (R) \leqq 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module RR is a semi-compatible module, then R is a Bezout domain.

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