Modules with Copure Intersection Property

Paper pages (271-276)

Introduction

‎Throughout this paper‎, will denote a commutative ring with‎ ‎identity and will denote the ring of integers.

Let be an -module‎. A submodule of is said to be pure if for every ideal of . has the copure sum property if the sum of any two copure submodules is again copure‎. is said to be a comultiplication module if for every submodule of there exists an ideal of such that . satisfies the double annihilator conditions if for each ideal of , we have . is said to be a strong comultiplication module if is a comultiplication R-module which satisfies the double annihilator conditions. A submodule of is called fully invariant if for every endomorphism ,.

In [5]‎, ‎H‎. ‎Ansari-Toroghy and F‎. ‎Farshadifar introduced the dual notion of pure submodules (that is copure submodules) and investigated the first properties of this class of modules‎. ‎A submodule of is said to be copure if for every ideal of .

Material and methods

We say that an -modulehas the copure intersection property if the intersection of any two copure submodules is again copure‎. In this paper, we investigate the modules with the copure intersection property and obtain some related results.

Conclusion

The following conclusions were drawn from this research.

Every distributive -module has the copure intersection property.
Every strong comultiplication -module has the copure intersection property.
An -module has the copure intersection property if and only if for each ideal of and copure submodules of we have



If is a , then an -module has the copure intersection property if and only if has the copure sum property.
Let , where is a submodule of . If has the copure intersection property, then each has the has the copure intersection property. The converse is true if each copure submodule of is fully invariant../files/site1/files/62/12Abstract.pdf