Semi-Essentially Compressible Modules and Semi-Essentially Retractable Modules

: Let 𝑅 be a commutative ring with 1 and 𝑀 be a left unitary 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 . In this paper, the generalizations for the notions of compressible module and retractable module are given. An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed to be semi-essentially compressible if 𝑀 can be embedded in every of a non-zero semi-essential submodules. An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed a semi-essentially retractable module, if 𝐻𝑜𝑚 𝑅 (𝑀¸ 𝑁) ≠ 0 for every non-zero semi-essentially submodule 𝐾 of an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 . Some of their advantages characterizations and examples are given. We also study the relation between these classes and some other classes of modules.


1.Introduction:
Let R be a commutative ring with 1 and M be a left unitary − . A non-zero submodule is termed to be an essential submodule of if ∩ ≠ 0 for every non-zero submodule of [1]. In [2], the authors introduced and studied the notion of semi-essential ISSN: 0067-2904 submodules. Recall that a non-zero submodule of an −module is termed to be a semiessential submodule ( ≤ . ), if ∩ ≠ 0 for every non-zero prime submodule of . A submodule of is termed to be prime if is a proper and whenever ∈ for every ∈ , ∈ up to either ∈ or ∈ [ : ], where [ : ] = { ∈ : ⊆ } [3].
An − is termed to be compressible if can be embedded in every of it is a non-zero submodule. In this work, the notion of a semi-essentially compressible module as a generalization of compressible is introduced and studied. Further, some of their advantages characterizations and examples are given. Also, we view that below definite terms semi-essentially compressible, semi-uniform , essentially compressible and compressible modules are equivalent. In [4], the author defined a semi-essentially compressible as an − is a semi-essentially compressible module if for every essential submodule of , he found a monomorphism ∶ ⟶ ( ) for some set I. In this work, we give another definition for semi-essentially compressible, namely an −module is termed to be semi-essentially compressible module, if can be embedded in every of it is nonzero semi-essentially submodules. An is also called semi-essentially compressible, if one can find a monomorphism : ⟶ whenever is a non-zero semi-essentially submodule of . Moreover, we introduce the notion of a semi-essentially retractable module as a generalization of a retractable module . Finally, we study the relations between semiessentially compressible modules and some of the other as semi-essentially retractable.

Semi-Essentially Compressible Modules:
An − is termed to be a compressible , if can be embedded in every of it is a non-zero submodule of . Equivalently, is compressible if one can find a monomorphism : ⟶ whenever 0 ≠ ≤ [5]. A ring is termed compressible if the − is compressible [5]. That is can be embedded in any of it is non-zero ideal.

Definition (2.1): An −
is termed to be a semi-essentially compressible module if can be embedded in every nonzero semi-essential submodule. Equivalently, is a semiessentially compressible, if one can find a monomorphism : ⟶ whenever is a nonzero semi-essential submodule of . A ring is termed to be a semi-essentially compressible, if is a semi-essentially compressible as −module.

2.
− is not a semi-essentially compressible module, because there is no monomorphism from to , where ≤ . . 3. It is obvious that every compressible module is a semi-essentially compressible , but the converse is not true in general for example 6 − is semi-essentially compressible because 6 is the only semi-essential submodule of 6 , but it is 4. Every simple − is semi-essentially compressible module, however, the converse is not true, because − is a semi-essentially compressible but it is not simple. 5. 4 − is not semi-essentially compressible. Because 4 can not be embedded in 〈2 ̅ 〉 and 〈2 ̅ 〉 ≤ . 4 .

A homomorphic image of a semi-essentially compressible
needs not be semiessentially compressible in general. For example, − is a semi-essentially compressible module and 4 ≃ 4 is not a semi-essentially compressible module by (5).
7. If is a semi-simple module, then is a semi-essentially compressible module, since is the only semi-essential submodule of , see Example (2) (2) [2].
Recall that a non-zero − is termed to be semi-uniform, if every non-zero submodule of an − is semi-essential, see Definition(1) [2].

Proposition(2.3):
If is a semi-uniform − , then is a compressible if and only if is semi-essentially compressible .
Proof: Suppose that is semi-essentially compressible . Let be a non-zero submodule of because is semi-uniform, then is semi-essential submodule in . But is semi-essentially compressible , then can be embedded in for every 0 ≠ ≤ .

Proposition(2.4):
If is a semi-essentially compressible − in which all submodule of contains a non-zero semi-essential submodule of , then is a compressible .

Proposition(2.5):
A is a semi-essentially compressible module if and only if can be embedded in for every 0 ≠ ∈ ≤ . .

Corollary(2.6):
A semi-essentially compressible is compressible if every cyclic submodule of is semi-essential in .
Proof: The result is obviously obtained by Proposition 2.5.

Proposition(2.7):
A semi-essential submodule of a semi-essentially compressible module is also a semi-essentially compressible module.
. Because is semi-essentially compressible, so there exists a monomorphism : ⟶ and : ⟶ is the inclusion monomorphism ,then ∘ : ⟶ be a monomorphism. Therefore, can be embedded in .

Proposition(2.8):
Let be a fully prime − , then is a semi-essentially compressible module if and only if is an essentially compressible module, where an − is termed to be fully prime if every proper submodule of is a prime submodule [9].

Remark(2.9):
Let be any −submodule of containing a submodule of such that is a semi-essentially of , then is a semi-essentially submodule of .

Proposition(2.10):
Let be a semi-essentially compressible −module and be a submodule of such that contains in the inverse image of −monomorphism of , then is a semi-essentially compressible −module.

Proposition(2.11):
If 1 and 2 are isomorphic − , then 1 is a semi-essentially compressible if and only if 2 is a semi-essentially compressible.

Remark(2.13):
The direct sum of a semi-essentially compressible needs not to be semi-essentially compressible. Consider the following example let 4 ≃ 2 ⨁ 2 as Z-module . 2 is a semi-essentially compressible module, but 4 is not a semi-essentially compressible module.

Semi-Essentially Retractable Modules:
An − is termed to be a retractable if ( , ) ≠ 0for every non-zero submodule of . [8] A ring is termed to be retractable if the − is retractable. [8] Definition (3.1): An − is termed to be a semi-essentially retractable module, if ( , ) ≠ 0 for every nonzero semi-essentially submodule of .
An − is termed to be essentially retractable if (¸ ) ≠ 0 for every non-zero essential submodule of .
A ring is termed to be essentially retractable if the R-module R is essentially retractable. That is ( , ) ≠ 0 for every nonzero small ideal I of a ring [12] Remarks and Examples(3.2): 1.
4. Every semi-essentially compressible is a semi-essential retractable , but the converse is not true in general. For example, 4 − is a semi-essential retractable , however, it is not semi-essentially compressible . 5. Every semi-essentially retractable module is an essentially retractable 6. Every retractable is a semi-essentially retractable . 7. Every semi-simple − is a semi-essentially retractable because it is retractable. 8. Every compressible is a semi-essential retractable , however, the converse is not true for example 12 − is a semi-essential retractable module, then there exists a honomorphsim : 12 → (3 ̅ ) by ( ̅ ) = 3 ̅ ,which is not monomorphism.

Proposition(3.3):
If is a semi-uniform − , then is a retractable if and only if is semi-essentially retractable .
Proof: Suppose that is semi-essentially retractable . Let be a non-zero submodule of because is semi-uniform, then is semi-essential submodule in . But is semi-essentially retractable module. Thus, ( , ) ≠ 0. Therefore, is retractable. Conversely, it is obvious by Remarks and Examples(3.2) (5).

Proposition(3.4):
Let be a fully prime − , then is a semi-essentially retractable if and only if is an essentially retractable .

Proposition(3.5):
A semi-essential submodule of a semi-essentially retractable is also a semi-essentially retractable module.

Proposition(3.6):
If is a semi-essentially retractable − in which every submodule of contains a non-zero semi-essentially submodule of , then is a retractable .

Proposition(3.7)
: If 1 and 2 be isomorphic −modules, then 1 is a semi-essentially retractable module if and only if 2 is a semi-essentially retractable .