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| Paper IPM / M / 8114 |
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| Abstract: | |
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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 R ⊕R is a semi-compatible module, then
R is a Bezout domain.
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