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Authors: Dilel Pusat-Yilmaz and Patrick Smith
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Summary  

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 P, we have AN P or BN 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) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 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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Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.

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