NEARLY QUASI PRIME SUBMODULES

Nuhad S. ALMothafar and Adwia J. Abdil AL khalik. 1. Dept. of Mathematics, College of Science, University of Baghdad. 2. Dept. of Mathematics, College of Science, Al-Mustansiriya University. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

Let R be a commutative ring with identity and be a unitary of Rmodule. A proper submodule is called aquasi-prime if whenevera. b x ∈ ; a, b ∈ R , x ∈ , implies that either ∈ N or ∈N. In this paper we say that is a nearly quasi prime, if whenever ∈ N ; a, b ∈ R, x ∈ M, implies that either ∈ N + ( )or ∈ N + ( ), where ( )is the Jacobson radical of . Some of the properties of this concept will be investigated. Some characterizations of nearly quasi prime submodules will be given, and we show that under some assumptions quasi prime submodules and nearly quasi prime submodules are coincide.
Let , be two submodules of an Rmodule and  . If is a nearly quasi prime submodule of and ( ) ⊆ ( ), then is nearly quasi prime submodule of .
A submodule of a nearly quasi submodule need not be a nearly quasi prime submodule : For instance, = 3  is a Zsubmodule of the Zmodule 12 . It is clear that is a nearly quasi prime of 12 , but = 6  is not a nearly quasi prime submodule of since [ + ( ) ∶ ( 1 )] = [6  + 6  ∶ ( 1 ) ] = 6 , which is not a prime ideal of .
It is clear that every quasi prime submodule is a nearly quasi prime, but the converse is not true in general for examples: The submodule Z of the Z-module Q is a nearly quasi prime submodule since for each It is clear that every primesubmodule is a nearly quasi prime submodule. Proof: Let be a prime submodule of an Rmodule . Then is a quasi prime submodule of M by [3,remark(1)]. Therefore N is a nearly quasi prime submodule of.
An ideal I is a nearly quasi prime ideal of R if and only if I is a nearly quasi prime Rsubmodule of -module .
Consider the Zmodule, =  . The submodule = 2  0  is a nearly quasi prime submodule of since for any ∈ , [ 1. It is clear that every maximal submodule is a nearly quasi prime submodule . But the converse is not true in general for example. Let =  as Zmodule, = {0} is a nearly quasi prime submodule( since is a prime ). But N is not maximal by [19,example (2) , chapter ].
2. If is a simple Rmodule, then the zero submodule is a nearly quasi prime submodule of . Hence 0  is the only nearly prime submodule of ( p is prime number).
The following theorem gives some characterizations for nearly quasi prime submodule

Theorem (2.3):-
Let be a proper submodule of an Rmodule. Then the following are equivalent: 1.
is a nearly quasi prime submodule of .   Let be any ring. A subset of is called multiplicatively closed if 1 ∈ and ∈ for every , ∈ .We know that every proper ideal in is prime if and only if  is a multiplicatively closed subset of , [7, p.42 ] .
And if a submodule of anmodule and be a multiplicatively closed subset of , then ( ) = {  ∶  ∈ , such that ∈ }be a submodule of and ⊆ ( ) .
However, the following proposition gives a partial converse of corollary (2.4). 173 Proposition (

Proposition(2.7):-
Let and be two submodules of an Rmodule . Then N is a nearly quasi prime submodule of M if and only if ⊆ implies either ⊆ + ( )or ⊆ + ( ).
The converse is clear. The intersection of any collection of nearly quasi prime submodules of an Rmodule not necessarily nearly quasi prime submodules, 1 = 2  and 2 = 3  are nearly quasi prime submodules of 12 as a Z -module.But Recall that a ring R is said to be a good ring if ( ) = ( ) for each Rmodule , equivalently, R is a good ring if and only if ( ) = ( ) ∩ for each submodule of an Rmodule M, [8, p.236] . And a ring R is called a regular ring if each of its elements is regular, where an element  is said to be regular if  ∈ such that axa = a, [8,p.184]. It is know that if / ( ) is a regular ring, then is a good ring.
Hence ,we have the following Result:-Proposition(2.8):-Let be a good ring. If is a nearly quasi prime submodule of an -module and be a submodule of such that ( ) ⊆ and is not contained in , then ∩ is a nearly quasi prime submodule of .

Corollary(2.9):-
Let be a good ring. If is a nearly quasi prime submodule of an R-module and is a maximal submodule of is not contained in , then ∩ is a nearly quasi prime submodule of .

Corollary(2.10 ):-
Let / ( ) is a regular ring. If is a nearly quasi prime submodule of and is a maximal submodule of is not contained in , then ∩ is a nearly quasi prime submodule of .

Proposition (2.12):-
Let and M′ are Rmodules and : M → M′ is an epimorphism such that  ≪ . If is a nearly quasi submodule prime ofM′, then −1 ( ) is also a nearly quasi prime submodule of .

Proposition (2.13):-
Let : M ⟶ M′ be an R-epimorphism. If is a nearly quasi prime submodule of an R-module containing and  , then N is a nearly quasi prime submodule of M′ .
Recall that a submodule of an Rmodule is called small in ,if every submodule of with + = implies = notationally  [8,p.106].
An R -epimorphism∅ ∶ M → M′ is called small epimorphism if Ker ∅ ≪ , [8]. By using these concept ,we have the following: Let ϕ: M ⟶ M′ is small epimorphism. If is a nearly quasi primesubmodule of an R-module containing ∅, then ϕ N is a nearly quasi prime submodule of M′ .
Recall that an R-module is called hollow module if and only i f every submodule in M is small [9].

Corollary ( 2.15):-
Let be a hollow Rmodule and ϕ: M ⟶ M′ be an epimorphism . If is a nearly quasi prime submodule of an R-module containing ∅, then ϕ N is a nearly quasi prime submodule of M ′ .
Recall that an R-module M is called local if M has unique maximal submodule, [10].

Corollary ( 2.16):-
Let is a local Rmodule and ϕ: M ⟶ M′ be an epimorphism . If is a nearly quasi primesubmodule of an Rmodule containing ∅, then ϕ N is a -prime submodule of M ′ .

Corollary (2.17):-
Let be a submodule of an R-module and be a small submodule of contained in . Then / is a nearly quasi prime submodule of / if N is nearly quasi prime submodule of .

The Relation Between Nearly Quasi Prime Submodules And Other Submodules:-
We study in this section the relationships between nearly quasi prime submodules and other submodules such as quasi prime submodules, -semiprime, prime submodules.
As we have mentioned in section one, that quasi prime submoduleis nearly quasi prime submodule and the converse need not be true in general.
In the following proposition, we give a condition under which the converse is true.

Proposition (3.1):-
If is a nearly quasi prime submodule of an Rmodule M and ( ) ⊆ ,then is quasi prime submodule of M . Proof:-It is clear.
Recall that an Rmodule is called fully semiprime if for each proper submodule is semiprime.And is called an almost fully semiprime if each nonzero proper submodule is semiprime, [12] .

Proposition (3.2):-Let
is an almost fully semiprime -module which is not fully semiprime.If is a nearly quasi prime submodule , then is a quasi prime submodule of M . Recall that an Rmodule is called cosemisimple if each proper submodule of is an intersection of maximal submodules, [12].

Proposition (3.3):-
Let is an almost fully semiprime R-module which is not co-semisimple. If is nearly quasi primesubmodule of M, then N is a quasi prime submodule of M .
Recall that a ring R is called aring if every simplemodule is injective, [13 ].

Corollary (3.5):-
If is nearly quasi prime submodule of anmodule and is aring,then is quasi prime submoduleof . Next, anmodule is said to beregular if each submodule of is pure, [13]. By using this concept, we have the following:

Corollary (3.6):-
If is a nearly quasi primesubmodule of -regularmodule , then is a quasi prime submodule of .

Corollary (3.8):-
If is a nearly quasi prime submodule of -regularmodule ,then N is a quasi primesubmodule of M .

Now, because of the fact that if
is semisimplemodule,so ( ) = 0 by [8,theorem 9.2.1,p.218],then the following is a consequence of proposition (3.4 ) ,where an Rmodule is called semisimple if and if only every submodule of is a direct summand of , [8].

Corollary (3.9):-
If is a nearly quasi prime submodule of a semisimplemodule ,then is a quasi prime submodule of . Now, because of the fact that if is semisimple then every right and leftmodule is semisimple by [8, corollary (8.2.2), p.196], then the following is a consequence of corollary (3.9).

Corollary (3.10):-
If is nearly quasi prime submodule of anmodule and R is a semisimple ring,then isquasi prime submoduleof . Now, because of the fact that if is a pseudo regularmodule,so ( ) = 0 by [ 17,propostion11,p.4] ,then the following is a consequence of proposition (3.4), where an Rmodule is called pseudo regular if and if only every finitely generated submodule of M is a direct submodule, [17].

Corollary (3.11):-
If is a nearly quasi prime submodule of a pseudo regularmodule M,then is quasi prime submoduleof .
In the following result, we give another condition for which a nearly quasi Prime submodulebe a quasiprimesubmodule As anther consequence of proposition (3.4), we have the following result:

Corollary (3.13):-
If is a nearly quasi prime submodule of anmodule and is a good ring, then N is quasi prime submodule of .

Now, we can give the following:-Corollary (3.14):-
If is a nearly quasi prime submodule of anmodule and / ( ) is a regular ring,then is quasi prime submodule of .
Proof:-Since = ( ) , then R is a good ring.Hence the result follows by corollary (3.13).
By using this concept, we have the following:

Proposition (3.15):-
Let is and is a divisible Rmodule such that ( ) ≠ . If is a nearly quasi prime submodule, then is a quasi prime submodule of .

Corollary (3.17):-Let
be an essential pseudo -injective for any cyclic module A and ( ) ≠ .If is a nearly quasi-prime submodule,then is a quasi-primesubmoduleof .

Proof:-
Since be a essential pseudo -injective for any cyclic module , so is an injective by [ 18 ,corollary 1, p.4] and so is a divisible Rmodule and ( ) ≠ ,so ( ) = 0 by [25 ,prop.(1-4) ,p.12].Hence the result follows immediately from proposition (3.4). Now, we can give anther consequences of proposition (3.4). But first we need the following definition: Recall that anmodule is direct injective, if given any direct summand of , an injection: ∶  and everymonomorphism :  ,there is anendomorphism M such that  = [19].
Proof:-Since be a direct injective module, so is a divisible -module by [25].Hence the result follows immediately from proposition (3.4).
Recall that anmodule is called icpseudoinjective, if it is icpseudo -Minjective. Where an Rmodule is said to be ic -( pseudo)-injective,if for each icsubmodule of , everyhomomorphism ( monomorphism) from to can be extended to anhomomorphism from into .And asubmodule of is called icsubmodule,if is isomorphic to a closed submodule of , [21] .

Proof:-
Suppose is nearly quasi prime submodule of an Rmodule . If 2  for  and  , then by definition of nearly quasi prime submodule implies that  + ( ). Hence, issemi prime submodule of .
In the following proposition, we give a condition under which the two concepts are equivalent. But first we need the following definition:

Proposition (3.21):-Let is an irreducible submodule of anmodule and ( ) ⊆
. If is a -Semi prime, then is a nearly quasi prime submodule of M Proof:-Since ( ) ⊆ and is -semi prime submodule, then is a semi-prime submodule by [22]. But N is an irreducible submodule of M, so by [ 26 ] N is a prime submodule of M and hence N is quasi prime submodule of M Therefore N is nearly quasi prime submoduleof .
In the following proposition, we give other conditions under which the two concept are equivalent. But first we need the following definitions: A nonzeromodule is called secondary module provided that for every element ∈ , the endomorphism  is either surjective or nilpotent, [4].

Proposition (3.22):-
Let be a submodule of a secondary module and ( ) ⊆ . If is asemi prime, then is a nearly quasi prime submodule of .
As we have mentioned in section one that prime submodule is nearly quasi prime submodule and the converse need not be true in general.
In the following proposition, we give a condition under which the converse is true.

Proposition (3.23):-
Let is an irreducible submodule of anmodule and ( ) ⊆ . If is a nearly quasi prime submodule of , then is a prime submodule of .
Proof:-Since ( ) ⊆ and is a nearly prime submodule, then is a quasi prime submodule and so is a semiprimesubmodule of by [