Volume 6, Issue 2 (Vol. 6, No. 2 2020)                   mmr 2020, 6(2): 271-276 | Back to browse issues page

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Modules with Copure Intersection Property. mmr 2020; 6 (2) :271-276
URL: http://mmr.khu.ac.ir/article-1-2771-en.html
Abstract:   (1756 Views)
Paper pages (271-276)
‎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.
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

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Type of Study: Original Manuscript | Subject: alg
Received: 2018/05/2 | Revised: 2020/09/13 | Accepted: 2018/11/28 | Published: 2020/01/28 | ePublished: 2020/01/28

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