On Multiplication Modules And Comultiplication Modules

Let R be a commutative ring with identity and M an R-module. In this paper, some properties of weak primary submodules of multiplication modules and comultiplication modules are studied. It is shown that: (1) If M is a multiplication module and N is a weak primary submodule of M, then N is a prime submodule of M ; (2) Let M be a multiplication module, L≤Mand f :M→Lan epimorphism. If N is a weak primary submodule of L, then f~1 ( N)is a weak primary submodule of M ; (3) If M is a comultiplication module and N is a copure submodule of M , then M N is a comultiplication module; (4) If M is a comultiplication module satisfying the DAC and N≤M, then N≤eMif and only if there exists I?R such that N = (0:M I); (5) If M is a comultiplication module satisfying the DAC , then M is a finitely cogenerated module; (6) If M is a comultiplication module satisfying the DAC , N≤M. Then P is a maximal second submodule of N if and only if there exists a minimal prime ideal I containing AnnR ( N )such that P = (0:M I)≠0; (7) This paper gives a partial answer on the posed question that let R be a commutative ring and M a cocyclic R-module, is M a comultiplicatin R-module.