On P-Essential Submodules

Let be a commutative ring with identity and let be an R-module. We call an R-submodule of as P-essential if for each nonzero prime submodule of and 0 . Also, we call an R-module as P-uniform if every nonzero submodule of is P-essential. We give some properties of P-essential and introduce many properties to P-uniform R-module. Also, we give conditions under which a submodule of a multiplication R-module becomes P-essential. Moreover, various properties of P-essential submodules are considered.


1-Introduction
Let be a commutative ring with unity and let be a unitary R-module. A non-zero submodule of is called essential if for each non-zero submodule of [1]. is called uniform if every non-zero submodule of is essential [1]. In (2019), Ahmad and Ibrahiem studied a new concept, which is named H-essential submodules [2]. Ali and Nada [3] introduced the concept of semi-essential submodules as a generalization of the class of essential submodules. They stated that a nonzero submodule of is called semi-essential , if for each nonzero prime submodule of . In section two, we introduce a Pessential submodule concept as a generalization of the essential submodule concept. We call an R-submodule of as P-essential if for each nonzero prime submodule of and 0 . Our main concerns in this section are to give characterizations for P-

ISSN: 0067-2904
Shahad and Al-Mothafar Iraqi Journal of Science, 2021, Vol. 62, No. 12, pp: 4916-4922 7114 essential submodules and generalize some known properties of essential submodules to Pessential submodules. In section three, we give conditions under which a submodule of a faithful multiplication R-module becomes P-essential. In section four, we present the Puniform module concept as a generalization of the uniform concept. We also generalize a characterization and some properties of uniform modules to P-uniform modules.

2-P-Essential Submodules
Recall that a non-zero submodule of an R-module is called essential if for each submodule of [1].

Definition(2-1)
Let be an R-module and be a non-zero prime submodule of . A submodule of is said to be P-essential, written as , if for every proper submodule of then Or, a non-zero submodule of is called P-essential , if . Remarks and Examples(2-2) 1-Every essential submodule is P-essential submodule, but the converse is not true in general . For example, consider as Z-module ,

2-Let
be a Z-module and the prime submodules of are : ̅ ̅ It follows that ̅ ̅ are P 1 -essential and P 2 -essential resp. in , but are not essential in , since ̅ ̅ ̅ , but ̅ ̅ . 3-A submodule of a P-essential submodule needs not to be P-essential. For example, let be a Z-module ,

4-If and
are P-essential submodules of , then needs not be to P-essential of . For example, let and let ̅ , ̅ ̅ be P-essential of , but ̅ ̅ ̅ is not P-essential of Z 24 . 5-The sum of two P-essential submodules of an R-module is also P-essential submodule. Proof: Let be R-module and let and be two P-essential submodules of . Note that , since , implies that . 6-A semi-essential submodule needs not to be P-essential submodule, as we see in the following example: Consider Z 12 as Z-module .

Proposition (2-3)
Let be an R-module, be a prime submodule of and be any submodule of . If , then if and only if . Proof : Suppose that . Let be a prime submodule of and let such that , implies that ( ) . Since , then . By hypothesis, , thus which implies that . The converse is obvious.

Poposition (2-4)
A non-zero submodule of is P-essential if and only if for each non-zero submodule of a submodule , such that , where is a prime submodule of . The proof is easy and hence is omitted.

Shahad and Al-Mothafar
Iraqi Journal of Science, 2021, Vol. 62, No. 12, pp: 4916-4922 7114 Let be an R-module and let , be submodules of such that . If is Pessential submodule of , then is a P-essential submodule of .

Proof
Let be a prime submodule of , . By using proposition (2)(3)(4), . Since , then then implies that . The converse of prop. (2)(3)(4)(5) is not true in general; for example : Consider Z 24 as a Z-module and ̅ is a submodule of ̅ . By remarks and example

Corollary(2-6)
Let and be submodules of . If is P-essential submodule of , then and are P-essential.

Proof
By using proposition (2)(3)(4)(5), since and , so . In the same way, . The converse of the previous corollary is not true in general, as shown in remarks and examples(2-2)(5).

Proposition(2-7)
Let be an R-module and let and be submodules of . If is an essential submodule of and is a P-essential submodule of , then is also P-essential submodule of . Proof Let be prime submodule of and let submodule of . Since is P-essential submodule of , thus . And since is an essential submodule of , then ( ) ( ) . This implies that is P-essential submodule of .

Proposition(2-8)
Let and be R-modules and let be an epimorphism. If is a P-essential submodule of , then ( ) is a ( )-essential of .

Proof
We know that if is a prime submodule of then ( ) is a prime submodule of [4 ]. Let ( ) ( ) . To prove that , then ( ) . Since is P-essential in and ( ) , then ( ) Proof : Let P be a prime submodule of ́ and let be a non-zero submodule of . Since is an epimorphism, then ( ) is a submodule of ( ) which is prime submodule of by [4]. But , then ( ) . On the other hand, is a monomorphism, thus ( ) . This completes the proof.

Proposition(2-10)
If is a submodule of an  (2-12) is not true in general, as the following example shows: Consider Z 24 as Z-module , the submodule ̅ is P-essential of Z 24 , by remarks and examples . But ̅ is not P-essential submodule of ̅ where ̅ . Recall that an R-module is fully prime, if every proper submodule of is a prime submodule [2].

Proposition(2-13)
Let be a fully prime R-module where and are submodules of , and let . Then is P-essential of if and only if is a P-essential submodule of and is a P-essential submodule of . Proof ( ) Since is a fully prime module, then by [5] is an essential submodule of and by [6, proposition(5-20)], is an essential submodule and is an essential submodule of . But since every essential submodule is a P-essential, so we are done. ( ) It follows similarly.

Proposition(2-14)
Let be an R-module and let and be P-essential submodules of such that , then is P-essential submodule of .

Proof
Let be a prime submodule of and let ( ) . This implies that ( ) . If , then we have a contradiction with the assumption, thus . This implies that is a submodule of [ 5 ]. Since is P-essential submodule of and, by our assumption, is a submodule of , then . But is P-essential submodule of , therefore , hence is P-essential submodule of .

3-P-Essential Submodules in Multiplication Modules
An R-module is called multiplication if every submodule of is of the form for some ideal of R [7] and an R-module is called faithfull if ( ) . In this section , we give a condition under which a submodule of is a faithful multiplication R-module that becomes P-essential.

Theorem(3-1)
Let be a faithful multiplication R-module and be a submodule of . Then is Pessential of if and only if is P-essential of .

Proof
Assume that is P-essential submodule of , let be a prime ideal of R and such that Since is a faithful multiplication R-module , then ( ) . Now, is a prime submodule of , and ( is P-essential submodule of ), implies that . Since is finitely generated faithful multiplication R-module , then . Therefore, is a P-essential . Conversely, let be a prime submodule of and be a submodule of such that . Since is multiplication , then there exists an ideal of R such that [8] . Hence ( ) . But is faithful , so . Since is a P-essential ideal of R, then , therefore , thus is a P-essential submodule of .

Theorem(3-2)
Let be a faithful multiplication R-module. Then is a P-essential submodule of if and only if , is a P-essential ideal of R for each .

Proposition(3-3)
Let be a finitely generated, faithful and multiplication R-module . If then for every ideals and of .

Proof
Let be a prime submodule of such that for some prime ideal of and , [8] and let be a submodule of such that . Since is a multiplication module , then for some ideal of . So , implies that ( ) . Since is a faithfull module, then . Since and is finitely generated, faithful and multiplication, so by [8], . Since is a P-essential ideal of , then and hence . That is, .

Proposition(3-4)
Let be a non-zero multiplication R-module with only one maximal submodule . If , then is an essential (hence P-essential) submodule of . Proof Let be a submodule of with . If , then , hence , which is a contradiction. Thus is a proper submodule of , and since is a non-zero multiplication module , so by [8] , is contained in some maximal submodule of . But has only one maximal submodule, which is . Thus implies that , that is is an essential (hence P-essentianl ) submodule of . Recall that a non-zero R-module is called fully essential if every non-zero semi-essential submodule of is an essential submodule of [5].

Definition(3-5):
A non-zero R-module is called fully P-essential if every non-zero Pessential submodule of is an essential submodule of . A ring is called fully P-essential if every non-zero P-essential ideal of is essential ideal of . Examples(3-6) 1as a Z-module is fully P-essential Z-module.

2-
as a Z-module is not fully P-essential , since the submodule ̅ of is P 2essential where ̅ , but not essential since ̅ ̅ ̅ but ̅ ̅ .

3-
Every fully essential is fully P-essential. The following theorem gives the hereditary of fully P-essential property between R-module and the ring .

Theorem(3-7)
Let be a non-zero faithfull and multiplication R-module , then is a fully P-essential module if and only if is a fully P-essential ring.
Proof ( ) Assume that is a fully P-essential module and let be a non-zero P-essential ideal of , then is a submodule of say . This implies that is a P-essential submodule of . Since and is faithful module, then . But is a fully P-essential module , thus is an essential submodule of . Since is a faithful and multiplication module, therefore is an essential ideal of R [8], that is is a fully P-essential ring. ( ) Suppose that is a fully P-essential ring and let . Since is a multiplication module, then for some P-essential ideal of . By assumption, is an essential ideal of R. But is faithful and multiplication module, then is an essential submodule of [8]. Thus is fully P-essential module.

4-P-Uniform Modules
Recall that a non-zero R-module is called uniform if every non-zero submodule of is essential [9]. Recall that a non-zero R-module is called semi-uniform if every non-zero submodule of is semi-essential [10]. In this section , we give a P-uniform module concept as a generalization of the uniform module concept. We also generalize some properties of uniform modules to P-uniform modules.

Definition(4-1)
A non-zero R-module is called P-uniform if every non-zero submodule of is P-essential . A ring is called P-uniform if is a P-uniform R-module. Remarks(4-2) 1-Each uniform R-module is P-uniform, but the converse is not true in general . For example, as a Z-module is P-uniform but not uniform since ̅ ̅ ̅ while ̅ ̅ see remarks and examples(2,2),(2).

2-
Each simple R-module is P-uniform. But the converse is not true in general. For example, is a P-uniform Z-module where ̅ , but not simple Z-module. 3-as a Z-module is not P-uniform , where ̅ is prime submodule of , ̅ ̅ ̅ and ̅ ̅ .

4-
We can note that a semi-uniform R-module needs not to be P-uniform, as shown in the following example: The Z-module is semi-uniform [3], but not P 1 -uniform and not P 2 -uniform, where ̅ ̅ , since ̅̅̅̅ ̅̅̅̅ ̅ , but ̅̅̅̅ ̅ as in the following table:

Proposition(4-3)
Let be an R-module , then is uniform if and only if is P-uniform and fully Pessential. Proof:-( ) It is clear. ( ) Let be a non-zero submodule of . since is P-uniform module, then . But is fully essential module , then , implies that is uniform module.

Theorem(4-4)
Let be a faithful multiplication R-module , then is a P-uniform R-module if and only if is a P-uniform ring.

Proof
Suppose that is P-uniform and let be a non-zero ideal of . Thus is P-essential submodule of . By theorem (3-1), is a P-essential ideal of . Conversely, assume that is P-uniform and is a submodule of . Since is multiplication, then there exists an ideal of such that . But is P-uniform, so is P-essential . Thus is P-essential by theorem .

Proposition(4-5)
Let and be two R-modules and let be an epimorphism. Then: 1-If is P-uniform R-module , then is also P-uniform R-module.

2-If
is P-uniform R-module for each prime submodule of , then is ( )uniform R-module. Proof 1-Let be a non-zero submodule of , then ( ) is a non-zero submodule of . Since is P-uniform R-module, thus ( ) is a P-essential submodule of . By remark(2-9), we get ( ( )) is a P-essential submodule of . Therefore, is P-uniform Rmodule. 2-Let be a non-zero submodule of , then ( ) is a non-zero submodule of . Since is P-uniform R-module, then ( ) is a P-essential submodule of . By proposition(2-8), we get ( ( )) is a ( )essential submodule of . Therefore, is ( )uniform R-module.

Proposition(4-6)
Let be R-module, where and are R-modules. If is P-uniform, then and are P-uniform modules Proof Let be non-zero submodule of , so . But is a P-uniform, then is a Pessential submodule of . Thus, is a P-essential submodule of . Therefore, is Puniform R-module. In a similar way, we can proof that is a P-uniform R-module.