S-Essentially Compressible Modulesand S-Essentially Retractable Modules

Let 𝑅 be a commutative ring with 1 and 𝑀 be a left unitary R -module. In this paper, we give a generalization for the notions of compressible (retractable) Modules. We study s-essentially compressible (s-essentially retractable). We give some of their advantages, properties, characterizations and examples. We also study the relation between s-essentially compressible (s-essentially


Introduction
Let R be a commutative ring with 1 and M be left unitary R-module. A non-zero submodule is termed an essential submodule of , if ∩ ≠ 0 for every non-zero submodule of , [2]. A submodule of an −module is termed a small submodule ( ≪ ), if + = for every submodule of implies = , [2]. An R-module is termed a compressible if can be embedded in every non-zero submodule. In [3] . An R-module is called s-essentially retractable if ( , ) ≠ 0 for each non-zero s-essential submodules of . In this paper we introduce and study the notion of s-essentially compressible module and s-essentially retractable module as a generalization of compressible module and retractable module, respectively. Also, we give some of their properties, examples and some of their advantages characterizations and examples. We noticed that small-essentially compressible, s-uniform and compressible modules are equivalent. and ⊴ . But is not an s-essential in . Example 2.6: Assume that = , = 12 , = (3 ̅ ) and = (4 ̅ ), then we have ⊴ 12 . Since ≪ 12 and ∩ = 0 with ≠ 0.

3.S-Essentially compressible modules:
In this section, we introduce the concept of s-essentially compressible module as a generalization of compressible module, and some of basic properties, examples and characterized of this concept has been given.

Definition 3.1: An R-module
is said to be an s-essentially compressible if can be embedded in every non-zero s-essential submodule of . Equivalently, is an s-essentially compressible if there exists a monomorphism : ⟶ whenever 0 ≠ ⊴ . A ring is called an s-essentially compressible if is an s-essentially compressible as R-module.

Remarks and Examples 3.2:
(1) Each compressible is an s-essentially compressible, but the converse is not true.
(2) The set as Z-module is an s-essentially compressible (since is compressible).
(3) Every simple R-module is an s-essentially compressible, but the converse is not true in general since as Z-module is an s-essentially compressible. However, it is not simple. (4) The set as Z-module is not an s-essentially compressible (since ⊴ ,  (5) The set 6 as R-module is not an s-essentially compressible (since there is no a monomorphism : 6 ⟶ (3 ̅ ) and (3 ̅ ) ⊴ 6 ). (6) A homomorphic image of an s-essentially compressible module needs not to be an sessentially compressible in general, for example as Z-module is an s-essentially compressible and 6 ⁄ ≃ 6 is not an s-essentially compressible. This is can be shown in Remarks and Examples 3.2 (5). (7) By Remarks and Examples 3.2 (5), we can obtain that every semisimple R-module needs not to be an s-essentially compressible. Since is an s-uniform and an s-essentially compressible module. Hence, can be embedded in . Therefore, is a compressible.

Remark 3.4:
Every s-essentially compressible is essentially compressible. However, the converse is not true.
Proof: Suppose is an s-essentially compressible module, let ≤ , then ⊴ . Since is an s-essentially compressible module, we have : ⟶ is a monomorphism. Thus, is an essentially compressible. The converse is not true because 6 as Z-module is an essentially compressible. However, by Remarks and Examples 3.2 (5), we obtain 6 is not an s-essentially compressible.

Proposition 3.5:
Every non-zero submodule of an s-essentially compressible R-module contains a non-zero s-essential submodule of is also an s-essentially compressible.

Proof: Let ≤
and 0 ≠ ⊴ ≤ . But, is an s-essentially compressible, so there exists a monomorphism : ⟶ and : ⟶ which is the inclusion monomorphism. Therefore, is an s-essentially compressible module.

Remark 3.6:
The direct sum of an s-essentially compressible is not necessarily be an sessentially compressible. For example, let 6 = 3 ⨁ 2 as Z-module 3 and 2 are s-essentially compressible modules, but by Remarks and Examples 3.2 (5) we obtain 6 is not an sessentially compressible module.
Proof: Let 0 ≠ ⊴ . Then by Proposition 2.5 we have = 1 ⨁ 2 for some 0 ≠ 1 ⊴ 1 ≤ and 0 ≠ 2 ⊴ 2 ≤ , hence, by Proposition 2.7(3) we have 1 and 2 are s-essential submodules in 1 and 2 , respectively . But, 1 and 2 are s-essentially compressible modules, so there exists a monomorphisms : 1 ⟶ 1 and : 2 ⟶ 2 . Define : ⟶ by ( , ) = ( ( ), ( )), it is clear that is a monomorphism. Therefore, is an s-essentially compressible. Recall that an R-module is called an s-essential prime if ( ) = ( ) for each non-zero s-essential submodule of [1] Lemma 3.8: Let be an s-essential prime R-module then ( ) is a prime ideal of for every non-zero s-essential submodule of in which every submodule is an s-essential submodule of .

Definition 3.10 An − module
is called an s-essentially uniform if every non-zero sessential submodule of is an essential submodule in .

Remark 3.11:
Every uniform is an s-essentially uniform. However, the converse needs not to be true in general for example 6 as Z-module is an s-essentially uniform module, but it is not uniform.

Remark 3.12:
If is an s-essentially compressible R-module. Then is isomorphic to a submodule s-essential for all 0 ≠ ∈ .

Remark 3.13:
If is an s-essentially compressible R-module. Then, is isomorphic to Rmodule ⁄ for some s-essential prime ideal of and an ideal of contains property.
Proof: Let 0 ≠ ∈ , then ⊴ . Then, by Proposition 3.10 we have is an sessentially prime , so there exists a monomorphism ∶ ⟶ . Hence, M is isomorphic to a submodule of . On the other hand, as ≃ ⁄ ( ) and by Lemma 2.10 we have ( ) is a prime ideal and hence an s-essential prime ideal of R. Put ( ) = , then ≃ ∕ where is an ideal of contains properly. Proposition 3.14: Let be an integral domain, then every finitely generated s-uniform − module is a compressible.

4.S-Essentially retractable modules:
In this section, we introduce the concept of an s-essentially retractable module as a generalization of retractable module and, some of its basic properties, examples and characterizations of this concept has been given.

Definition 4.1: An R-module
is said to be an s-essentially retractable if ( , ) ≠ 0 for every non-zero s-essential submodules of . Equivalently, is an s-essentially retractable if there exists a non-zero homomorphism : ⟶ , 0 ≠ ⊴ . A ring is called an s-essentially retractable if is an s-essentially retractable as an R-module.
Recall that an R-module is said to be an essentially retractable if ( , ) ≠ 0 for every essential submodule of . A ring is said to be an essentially retractable if the R-module R is an essentially retractable. That is ( , ) ≠ 0 for every non-zero essential ideal I of a ring , [6].

Remarks and Examples 4.2:
1. It's clear that every retractable module is a s-essentially retractable module. 2. Every s-essentially retractable module is essentially retractable module.
Proof: Let 0 ≠ ≤ , then ⊴ . Since is an s-essentially retractable module, then there exists 0 ≠ : ⟶ . Thus is an essentially retractable. 3. The set as Z-module is a s-essentially retractable module, because is a retractable. 4. The set 6 as Z-module is an s-essentially retractable module. 5. It's obvious that every s-essentially compressible module is an s-essentially retractable module, but the converse is not true in general for example 6 as Z-module is an s-essentially retractable module, but not an s-essentially compressible, which can be shown by Examples 4.2 and 3.2. 6. The set as Z-module is not an s-essentially retractable module, since ( , ) = 0 and ⊴ . 7. Every compressible module is an s-essentially retractable module, but the converse is not true for example 6 as Z-module is an s-essential retractable module, but not compressible. 8. Every simple R-module is an s-essentially retractable module but not conversely, because as Z-module is an s-essentially retractable module but not simple. 9. Every semisimple R-module is an s-essentially retractable because it is retractable.

Proposition 4.3:
Let be an s-uniform R-module , then is an s-essentially retractable module if and only if is retractable module .
Proof: ⟸) It is clear, by Remark 4.2 (1). ⟹)Suppose that is an s-essentially retractable and let 0 ≠ ≤ , then ⊴ (since is an s-uniform), and since is an s-essentially retractable module. Therefore, is retractable. Hence, there exists a non-zero homomorphism : ⟶ for some 0 ≠ ≤ .

Proposition 4.4:
Every submodule of an s-essentially retractable contains a non-zero sessential submodule of is also an s-essentially retractable .