Pure Maximal Submodules and Related Concepts

In this work we discuss the concept of pure-maximal denoted by (Pr-maximal) submodules as a generalization to the type of R-maximal submodule, where a proper submodule of an R-module is called Pr-maximal if < 𝐻 ≤ 𝑊 ,for any submodule of W is a pure submodule of W, We offer some properties of a Pr-maximal submodules, and we give Definition of the concept, near-maximal, a proper submodule of an R-module is named near (N-maximal) whensoever is pure submodule of such that then K= .Al so we offer the concept Pr-module, An R-module W is named Pr-module, if every proper submodule of is Pr-maximal. A ring is named Pr-ring if whole proper ideal of is a Pr-maximal ideal, we offer the concept pure local (Pr-local) module an R-module is named pure local (Pr-local) module. If it has only a Pr-maximal submodule which includes all proper submodule of . A ring is named pure local (Pr-local) ring, if is a Pr-local R-module. We give some relatio among Pr-maximal submodules and others related concept.


Introduction
In this work is commutative ring with identity, and all R-modules are left until.A proper submodule of an R-module is named a pure submodule "if for every ideal of , [1].A proper submodule of an R-module is named maximal in [3] "if whenever is a submodule of R-Module with Implies .Abduljaleel and Yaseen in [2] offer the concept of large maximal submodules as a generalization of the concept maximal submodules, "where a proper submodule of an Rmodule is named large-maximal(L-maximal) if implies is an essential submodule of , where a submodule of R-module is named essential, if for every non-zero submodule of , [3].Many authors studies module and submodule for example see [4] and [5].In [11] B.H.AL-Bahrani generalization of the type of a purely extending modules, defined using Y-closed submodules, In,this discuss, we introduce, the concept of pure-maximal (Pr-maximal) submodules as a generalization of maximal submodule , where a proper submodule of an R-module is named Pr-maximal, if , for any submodule of , implies is a pure a submodule of , In section two we give several properties of this type of submodules as every multiplication module contains a Pr-maximal submodule.Also, if N, K are non-zero submodule of such that  ≤  if N is Pr-maximal in W then K is Pr-maximal in W and if N is Pr-maximal submodule of an Rmodule W and I be ideal of R, if [  : ] is a proper submodule of W then [  : I] is Pr-maximal submodule.we study the relation, among Pr-maximal (submodules and other related module), In section three we study Pr-maximal submodule, under the multiplication module and we check some condition under which Pr-maximal submodules and maximal submodules are equivalent.Every multiplication module contains a Pr-maximal submodule.Also, we have every cyclic Rmodule has Pr-maximal submodule.We found if W is a F-regular module then every submodule of W is Pr-maximal.

Preliminaries
This section is going to review some well-known definitions in a algebraic theory.

Definition 2.1 [3]
A proper submodule of an R-module i s named maximal if such that namely = Definition 2.

[1]
A submodule of an -module is named pure -submodule if for each ideal of Definition 2.3 [6] "A submodule of an R-module is called weak maximal if is F-regular R-module".

Lemma (2.4) [7]
"If : is an epimorphism and is pure submodule of , then ( ) is pure in ."Definition 2.5 [8] "An R-module is named pure simple if and it has no pure, submodule except and Definition 2.6 [4] An R-module is called faithful if = Definition 2.7 [9,10] "An R-module is said to be multiplication if for each submodule of there exists an ideal of such that = ".Equivalently is a multiplication R-module if and only if for each submodule of , = [ ] .

Proposition 2.8 [10]
If is an R-module and has an, unique maximal submodule , then is called local module.

Pr-Maximal Submodules:
In this section, the basic definitions and facts related to this work are recalled, which starts with the following definition.

Definition (3.1)
A sound R-submodule of an R-module is named pure-maximal ( -maximal) submodule of if there exists with then is pure -submodule of W ( ≤  ).

Remark and Example (3.2)
1. Every maximal submodule of R-module is Pr-maximal.Proof: Impose maximal R-submodule of R-module there exist 0 H submodule of such that since is maximal then but is pure of therefore is Pr-maximal the convers is not true as the following example in W= + as a Z-module Let, =2 4 and H= such that 2 since H= is a summand of = , hence is pure in , then is Pr-maximal submodule of but is not maximal since .2. A subset of Pr-maximal -submodule need not be Pr-maximal -submodule as the breech example in as Z-module impose 3 12 <  12 ≤  12 implies 3 is Pr-maximal submodule of since , but 6 is not Pr-maximal submodule of since 6 and is not pure in .3. 4 as Z-module we have {0 ̅ , 2 ̅ }is Pr-maximal since 4 is pure of 4 and {0 ̅ ,2 ̅ }< 4≤ 4.
is simple, then is Pr-maximal.
Proof: clearly since is simple implies is maximal then assist remark (2.2) every, is Prmaximal.
. Let and are nonzero submodule of such that if is Pr-maximal of and is Pr-maximal of then is not Pr-maximal of for example let =2 24 and =6 , =2 , is Pr-maximal in since 62 and 2 2 and is Pr-maximal in since is maximal but is not Pr-maximal in since 6 2 and2 is not pure in .7. If = as, Z-module and =4 a -submodule of ,then is not Pr-maximal in since 4 2 and 2 is not pure in .8. If is a semi simple R-module, thence every sound -submodule of is -maximal if and only if it is Pr-maximal for example 2 and 3 are Pr-maximal submodule of as Z-module.
. Every proper submodule contain in a pure submodule is, Pr-maximal.

Definition (3.3)
A proper submodule of an R-module is named near-maximal(N-maximal) whenever is pure submodule of such that then

Proposition (3.5)
If is an F-regular module then every submodule of is, Pr-maximal.Proof Let be submodule of such that since is F-regular then respective submodule of is pure hence is pure submodule of then is Pr-maximal.

Remark (3.6)
If is simple -module thence every -submodule of W is Pr-maximal.Proof: Since every simple -module is regular, thence every maximal is Pr-maximal.Proposition (3.7) Proof: ) Suppose, is a Pr-module and impose is a proper ideal of .Since is a multiplication R-module, Then, .But is Pr-module, hence is a Prmaximal submodule of .Assists Theorem (3.3), is an ideal, Pr-maximal of ) Assume that R is a Pr-ring and let be a proper submodule of M. Since M is R-module a multiplication, thence , for some ideal of .Since is a Pr-maximal ideal, then assists Theorem

Conclusions
We will try to generalize the concept of Pr-maximal R-submodules to some other concepts in future works.
In this study, the concepts of Pr-maximal R-submodules is studied as a generalization of maximal submodule and some properties of this concept are investigated such as: 1. Every maximal submodule of R-module is Pr-maximal.2. A subset of Pr-maximal -submodule need not be Pr-maximal -submodule.

3.
If is an F-regular module then every submodule of is Pr-maximal.

If
are non-zero submodule of such that if is Pr-maximal in then is Pr-maximal in . 5. Let : be an epimorphism, where be R-modules.If is Pr-maximal submodule of , then ( ) is Pr-maximal R-submodule of .6. Every cyclic R-module breakpoint Pr-maximal submodule.7. Every local R-module breakpoint Pr-maximal R-submodule.

8 )
regular module then any pure submodule of is Pr-maximal.Proof Let H is a pure R-submodule of W such that H and since   is F-regular then is pure submodule of   since is pure of W then is pure in [4] implies is Pr-maximal submodule of .Corollary (3.If is a weak R-maximal submodule of an R-module thence every pure submodule of is r-maximal.Proposition (3.9)If are non-zero submodule of such that if is Pr-maximal in then is Prmaximal in .Proof Let submodule of such that since  and N is pr-maximal of W then is pure namely is pr-maximal assist definition (3.1).Corollary (3.10)If are sound submodules of R-module and is pr-maximal of , then both N and are pr-maximal submodule of .Proof Since and is Pr-maximal of , thence assist proposition (3.9) is Pr-maximal similarity we prove is Pr-maximal.Corollary (3.11)If ,  are nonzero submodules of R-module if or are Pr-maximal, then is Prmaximal.Proof: Since is, Pr-maximal and K ≤W assist proposition (3.9) implies is Prmaximal.Corollary (3.12)If is, Pr-maximal submodule of an R-module and I is an ideal of R, if [  : ] is a Proper submodule of then, [  : ] is pr-maximal -submodule of .Then [[  : ] is Pr-maximal submodule of W.
(4.3) is Pr-maximal submodule of .thus is Pr-module.Remark (4.7)Not all, finite R-module is local for example = { , , , , } is not local since 2 6 and 3 6 are maximal submodules this not local but  4 R-module is local since 2 4 is the only proper maximal submodule of  4 Proposition (4.8)Every local R-module has Pr-maximal submodule.Proof: Let is local R-module, then has only one maximal R-submodule implies has Prmaximal submodule assist remark (2.2).Remark (4.9)The convers of proposition (4.8) is not true for example has Pr-maximal submodules but not local.We need to give the following concept Definition (4.10)An R-module is named pure local (Pr-local) module.If it has only one Pr-maximal submodule which contains all proper submodule of .A ring is called pure local (Pr-local) ring, if is a Pr-local R-module Remark and Examples (4.11) 1.The Z-module 3 24 is a Pr-local module, since it has only one R-submodule Pr-maximal that is 6 24 .2. Every local is, Pr-local but not conversely for example: The Z-module 2 12 in  12 as Z-module is Pr-local since 4 12 is the only Pr-maximal submodule of 2 12 .But not local since 4 12 and 6 12 are maximal submodules of 2 12 .3.If  is a non-zero multiplication and Pr-local R-module and  is a non-zero Pr-maximal submodule of , thence  not necessary, pure submodule for example: If = 4 as Z-module,  is non-zero multiplication and Pr-local Z-module and 2 4 is non-zero, submodule, Pr-maximal of  4 but not pure.Remark (4.12): description of R-module.