Volume 6, Issue 2 (Vol. 6, No. 2 2020)
 Mathematical Researches 2020, 6(2): 271-276 Back to browse issues page
Modules with Copure Intersection Property
Abstract:   (456 Views)
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

Keywords: Pure submodule, copure submodule, copure intersection property.