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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019
Abstract
Let be a ring and let be a unitary left -module. A proper submodule of an -
module is called 2-absorbing , if , where implies that either
or or [ ], and a proper submodule of an -module is called quasi-prime ,
if , where implies that either or . This led us to
introduce the concept pseudo quasi-2-absorbing submodule, as a generalization of both
concepts above, where a proper submodule of an -module is called a pseudo quasi-2-
absorbing submodule of , if whenever ,where implies that either
or or , where is socal of an
-module . Several basic properties, examples and characterizations of this concept are
given. Moreover, we investigate relationships between pseudo quasi-2-absorbing submodule
and other classes of submodules.
Keywords: Prime submodules, quasi-prime submodules, 2-absorbing submodules, quasi-2-
absorbing submodules, pseudo quasi-2-absorbing submodules.
1. Introduction and Preliminaries
Throughout this dissertation all ring is commutative with identity and all -modules are
left unitary. A proper submodule of an -module is called a prime submodule if
whenever , with implies that either or [ ] [ ]. Prime
submodules play an important role in the module theory over a commutative ring. There are
several generalizations of the notion of prime submodules such as, quasi prime submodule,
where a proper submodule of an -module is called a quasi-prime, if whenever
, with , implies that either or [ ]. WE-prime submodules
and WE-semi prime submodules which appear in [ ]. The concept of prime submodule was
generalized by Darani and Soheilnia to 2-absorbing submodule, where a proper submodule
of an -module is called 2-absorbing, if whenever , with implies
that either or or [ ][ ]. There are several generalizations of 2-
absorbing submodules such as WN-2-absorbing submodules and WNS- 2-absorbing
submodules which appear in [ ]. The concept of quasi-2-absorbing submodule, was
Pseudo Quasi-2-Absorbing Submodules and Some Related Concepts
omar.aldoori87@gmail.com
Ibn Al Haitham Journal for Pure and Applied Science
Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index
Article history: Received 27 December 2018, Accepted 20 January 2019, Publish May 2019
Omar A. Abdulla
Doi: 10.30526/32.2.2149
Department of Mathematics, College of Computer Science and Mathematics, University of Tikrit,
Tikrit, Iraq.
mohammadali2013@gmail.com
Haibat K. Mohammadali
mailto:omar.aldoori87@gmail.com
mailto:omar.aldoori87@gmail.com
mailto:mohammadali2013@gmail.com
mailto:mohammadali2013@gmail.com
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019
introduced in 2018 as a generalization of 2-absorbing submodule, where a proper submodule
of an -module is called a quasi-2-absorbing, if whenever , with
implies that either or or [ ] In this paper we establish
new concept called pseudo quasi-2-absorbing submodule as generalization of ( prime, quasi-
prime, 2-absorbing and quasi-2-absorbing ) submodules. Several basic properties examples,
and relationships of pseudo quasi-2-absorbing submodules, with other classes of submodules
are studied. Socle of a module denoted by defined to be the intersection of all
essential submodules of [ ]. Where a submodule of an -module is called essential, if
has non-zero intersection with every non-zero submodule of [ ]. Recall that a non-zero
proper ideal of is called 2-absorbing ideal of , if whenever and , then
or or [ ]. Recall that an -module is multiplication if every
submodule of is of the form for some ideal of [ ].
2. Pseudo quasi-2-Absorbing Submodules
In this section, we introduced the definition of a pseudo quasi-2-absorbing submodule
Definition (1)
A proper submodule of an -module is called a pseudo quasi-2-absorbing submodule,
if whenever , with , implies that either or
or . And a proper ideal of a ring is called a pseudo
quasi-2-absorbing, if is a pseudo quasi-2-absorbing submodule of an -module .
The following proposition gives characterization of a pseudo quasi-2-absorbing
submodules.
Proposition (2)
Let be an -module, and is a submodule of . Then is a pseudo quasi-2-absorbing
submodule of if and only if for every ideals of and submodule of with
, implies that either or or
.
Proof
Suppose that , where are ideals of , and is a submodule of
with and and . Thus, there
exists and and such that and
and . But , and K is a pseudo quasi-
2-absorbing submodule of , with , then we have
or . Again and , implies that either
or . Also, and
, implies that either or . Thus either
or or .
Suppose that , where then , so by
hypothesis, either or or
. Thus either or or . Hence
is a pseudo quasi-2-absorbing submodule of .
As a direct consequence of proposition (2) we get the following result.
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019
Corollary (3)
Let be an -module, and is a submodule of . Then is a pseudo quasi-2-
absorbing submodule of if and only if for each and for each submodule of
with , implies that either or or
.
Remarks and Examples (4)
1- It is clear that every quasi-prime submodule of an -module is a pseudo quasi-2-
absorbing submodule of , while the converse is not true in general. For the converse
consider the following example:
In the -module , the submodule 〈 ̅〉 is pseudo quasi-2-absorbing, but not quasi-
prime since ̅ 〈 ̅〉, but ̅ 〈 ̅〉. Since 〈 ̅〉, it is clear that for each
and , if 〈 ̅〉, implies that either 〈 ̅〉 or 〈 ̅〉 or
[〈 ̅〉 ].
2- It is clear that every prime submodule of an -module is a pseudo quasi-2-absorbing
submodule of , while the converse is not true in general. For the converse see the following
example:
In the -module , the submodule 〈 ̅〉 is pseudo quasi-2-absorbing, but not prime,
since ̅ , but ̅ and [ ].
3- It is clear that every 2-absorbing submodule of an -module is a pseudo quasi-2-
absorbing submodule of , while the converse is not true in general. For the converse see the
following example:
In the -module , the submodule 〈 ̅〉 is pseudo quasi-2-absorbing, but not 2-
absorbing, since ̅ , but ̅ and ̅ and [ ] . Since
〈 ̅〉, it is clear that for all and , if 〈 ̅〉, implies that
either 〈 ̅〉 or 〈 ̅〉 or 〈 ̅〉 .
4- It is clear that every quasi-2-absorbing submodule of an -module is a pseudo quasi-2-
absorbing submodule of , while the converse is not true in general. For the converse see the
following example:
In the -module , the submodule 〈 ̅〉 is pseudo quasi-2-absorbing, but not quasi-
2-absorbing, since ̅ , but ̅ and ̅ and ̅ . Since
〈 ̅〉, it is clear that is a pseudo quasi-2-absorbing submodule of .
Proposition (5)
Let be an -module, and is a proper submodule of , with [ ] is 2-
absorbing ideal of for each .Then is a pseudo quasi-2-absorbing submodule of .
Proof
Assume that , where . Since , implies
that [ ]. But [ ] is a 2-absorbing ideal of , then either
[ ] or [ ] or [ ]. That is either
or or . Hence is a pseudo quasi-
2-absorbing submodule of .
Proposition (6)
Let be an -module and is a pseudo quasi-2-absorbing submodule of , with
. Then [ ] is 2-absorbing (hence a pseudo quasi-2-absorbing) ideal of .
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Proof
Let [ ], where , then , it follows that for all .
But is a pseudo quasi-2-absorbing submodule of , implies that either
or or . But , implies that .
That is either or or for all . Hence either or
or . Thus either [ ] or [ ] or [ ]. That is [ ] is
2-absorbing ideal of , hence is a pseudo quasi-2-absorbing ideal of .
Recall that an -module is called faithful if [ ].
Before we give the converse of proposition (2.6), we recalled the following lemmas.
Lemma (7) [ ].
Let be faithful multiplication -module, then .
Proposition (8)
Let be a faithful multiplication -module and is a proper submodule of . If [ ] is
a pseudo quasi-2-absorbing ideal of , then is a pseudo quasi-2-absorbing submodule of
Proof
Let with , then . But is a multiplication, so
for some ideal of . Thus , it follows that [ ]. But [ ] is
a pseudo quasi-2-absorbing ideal of , then by corollary (3) either [ ] or
[ ] or [ ] . Hence either [ ]
or [ ] or [ ] . But [ ] and by
lemma (7) . Thus, either or or
. Therefore is a pseudo quasi-2-absorbing submodule of .
Recall that an -module is called singular module provided . At the other
extreme, we say that is non-singular module provided , where
{ }, where is the set of all essential right ideals of
the ring [ ].
Lemma (9) [ ]
If is a non-singular -module, then .
Proposition (10)
Let be a non-singular multiplication -module and is a proper submodule of . If
[ ] is pseudo quasi-2-absorbing ideal of , then is a pseudo quasi-2-absorbing
submodule of .
Proof
Similarly, as in proposition (6), by using lemma (9).
Lemma (11)[ ].
Let and be ideals of a ring and is a finitely generated multiplication -module.
Then if and only if .
Proposition (12)
Let be a faithful finitely generated multiplication -module. If is a pseudo quasi-2-
absorbing ideal of , then is a pseudo quasi-2-absorbing submodule of .
Proof
Let , where , that is . But is a multiplication,
then for some ideal of . Thus , and so by lemma (11),
, but is faithful, then , hence . But is a pseudo quasi-2-
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019
absorbing ideal of , then by corollary (3) either or or
. Thus either or or
. But by lemma (7) . Hence either or
or , it follows that or
or . Therefore is a pseudo quasi-2-absorbing
submodule of .
By using lemma (9) and lemma (11) we get the following result.
Proposition (13)
Let be a finitely generated multiplication non-singular -module. If is a pseudo
quasi-2-absorbing ideal of with , then is a pseudo quasi-2-absorbing
submodule of .
Proof
Similar steps of proposition (12).
Proposition (14)
Let be an -module and is a proper submodule of , with .Then is a
pseudo quasi-2-absorbing submodule of if and only if [ ] [ ] [ ]
[ ] for all .
Proof
Let [ ], implies that . But is a pseudo quasi-2-absorbing
submodule of , then either or or .
But , then , it follows that either or or
, implies that either [ ] or [ ] or [ ]. That is
[ ] [ ] [ ]. Clearly that [ ] [ ] [ ] [ ]. Thus
[ ] [ ] [ ] [ ].
Suppose that with , implies that [ ]
[ ] [ ] [ ], implies that either [ ] or [ ] or
[ ]. That is either or or
. Hence is a pseudo quasi-2-absorbing submodule of .
Proposition (15)
Let be an -module and is a proper submodule of , with .Then is a
pseudo quasi-2-absorbing submodule of if and only if [ ] [ ] [ ] for
all .
Proof
Let [ ], implies that . But is a pseudo quasi-2-absorbing
submodule of , and let , then either or
. But , then . Hence either or . Thus
either [ ] or [ ], it follows that [ ] [ ], hence
[ ] [ ] [ ]. Consequently [ ] [ ] [ ].
Suppose that with , with , then
[ ] [ ] [ ], implies that either [ ] or [ ], it
follows that or , hence either
or . Therefore is a pseudo quasi-2-absorbing submodule of .
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Proposition (16)
Let be an -module and is a pseudo quasi-2-absorbing submodule of , with
, then [ ] [ ] [ ] [ ] for all .
Proof
Let [ ], implies that . But is a pseudo quasi-2-absorbing
submodule of , then either or or
. But , then , it follows that either or
or . Hence [ ] or [ ] or [ ]. Therefore
[ ] [ ] [ ], hence[ ] [ ] [ ] [ ].
Consequently, the equality holds.
Proposition (17)
Let be an -module, and are submodules of such that and is an
essential submodule of . If is a pseudo quasi-2-absorbing submodule of , then is a
pseudo quasi-2-absorbing submodule of .
Proof
Let , with . Since is a pseudo quasi-2-absorbing submodule
of , then either or or . But is
essential submodule of , then by [ ]. we have . Hence either
or or . Therefore is a pseudo
quasi-2-absorbing submodule in .
Proposition (18)
Let be an -module, and are submodules of such that and
. If is a pseudo quasi-2-absorbing submodule of , then is a pseudo quasi-2-
absorbing submodule of .
Proof: Similar steps as proposition (17).
Remark (19)
The intersection of two pseudo quasi-2-absorbing submodules of an -module need not
to be pseudo quasi-2-absorbing submodule of , as the following example explains that:
In the -module , the submodules and are pseudo quasi-2-absorbing submodules of
, but is not a pseudo quasi-2-absorbing of , since , but
and .
Proposition (20)
Let be an -module, and are pseudo quasi-2-absorbing submodules of , with
and . Then is a pseudo quasi-2-absorbing submodule of .
Proof
Let , where , then and . But and
are pseudo quasi-2-absorbing submodules of , so either or
or and either or or
. But and , implies that ,
and hence , and
. Thus either or or
. Hence is a pseudo quasi-2-absorbing submodule of .
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Proposition (21)
Let be an -module and is a pseudo quasi-2-absorbing submodule of and is a
submodule of , with and is not contained in . Then is a pseudo
quasi-2-absorbing submodule of .
Proof
Since is not contained in , then is a proper submodule of . Let
, where , then and . But is a pseudo quasi-
2-absorbing submodule of , then either or or
. But , it follows that either or
or . By hypothesis , then
by [ ] we have either ( or (
or ( . But by [ ]
. Hence either or
or . Therefore is a pseudo quasi-2-absorbing
submodule of .
Proposition (22)
Let ́ be an -epimorphism and is a pseudo quasi-2-absorbing
submodule of ́. Then is a pseudo quasi-2-absorbing submodule of .
Proof
It is clear that is a proper submodule of . Let , where
, then , as is a pseudo quasi-2-absorbing submodule of ́, implies
that either ́ or ́ or ́ . That is
either ( ( ́)) or ( ( ́))
or ( ( ́)) , it follows that
either or or .
Therefore is a pseudo quasi-2-absorbing submodule of .
Proposition (23)
Let ́ be an -epimorphism and is a pseudo quasi-2-absorbing submodule
of , with . Then is a pseudo quasi-2-absorbing submodule of ́.
Proof
is a proper submodule of ́, if not that is ́. Let then ́
, then for some , hence , implies that ,
it follows that , hence contradiction. Let ́ , where and
́ ́, since is on to, then ́ for some , hence , it follows that
for some , that is , implies that ,
it follows that . But is a pseudo quasi-2-absorbing submodule of , then either
or or . It follows that either
( ) ́ or ( ) ́
or ( ) ́ . That is either ́ ́ or
́ ́ or ́ ́ . Therefore is a pseudo quasi-2-
absorbing submodule of ́.
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Proposition (24)
Let be an -module, where are -modules and be a
submodule of , where are submodules of respectively, with . If
is a pseudo quasi-2-absorbing submodule of , then and are pseudo quasi-2-absorbing
submodules of respectively.
Proof
Let , where , then . But is a pseudo quasi-
2-absorbing submodule of , then either or
or . But , implies that and
. If , implies that
. If , implies that
, also in similar way we get . Therefore is a pseudo
quasi-2-absorbing submodule of .
Similarly, is a pseudo quasi-2-absorbing submodule of .
Proposition (25)
Let be two -modules and . Then
a) is a pseudo quasi-2-absorbing submodule of , with and is a semi
simple if and only if is a pseudo quasi-2-absorbing submodule of .
b) is a pseudo quasi-2-absorbing submodule of , with and is a semi
simple if and only if is a pseudo quasi-2-absorbing submodule of .
Proof
a) Let , where and , implies that
and . Since is a pseudo quasi-2-absorbing submodule of , and
, then either or
or . Since is a semi simple, then by [ ]
, then
. If
.
If .
Hence is a pseudo quasi-2-absorbing submodule of .
Let , where , then for each ,
. But is a pseudo quasi-2-absorbing submodule of , so either
or or
. If
( )
. It follows that .
Similarly, if , implies that:
, it follows that .
Similarly, we get . Therefore is a pseudo quasi-2-absorbing
submodule of .
The proof of (b) is similarly.
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3. Conclusion
In this research we introduce and study the concept pseudo quasi-2-absorbing submodules
as a generalization of quasi-prime and 2-absorbing submodules. The main results of this study
are the following:
1- A proper submodule of an -module is a pseudo quasi-2-absorbing submodule of if
and only if for every ideals of and submodule of with , implies
that either or or .
2- Let be a faithful finitely generated multiplication -module and is a pseudo quasi-2-
absorbing ideal of , then is a pseudo quasi-2-absorbing submodule of .
3- A proper submodule of an -module , with is a pseudo quasi-2-
absorbing submodule of if and only if [ ] [ ] [ ] [ ] for all
.
4- A proper submodule of an -module , with is a pseudo quasi-2-
absorbing submodule of if and only if [ ] [ ] [ ] for all
.
5- The intersection of two pseudo quasi-2-absorbing submodules of an -module need not
to be pseudo quasi-2-absorbing. This explain by example see Remark (19). But under
certain conditions the intersection are satisfies see Proposition (20).
6- The inverse image and homomorphism image of pseudo quasi-2-absorbing submodule is
pseudo quasi-2-absorbing see Proposition (22), (23).
7- The direct summand of pseudo quasi2-absorbing submodule is a pseudo quasi-2-
absorbing submodule see Proposition (24).
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