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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
URL: http://mmr.khu.ac.ir/article-1-2771-en.html
Abstract: (2462 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.
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
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
Received: 2018/05/2 | Revised: 2020/09/13 | Accepted: 2018/11/28 | Published: 2020/01/28 | ePublished: 2020/01/28
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