Semi-Small Compressible Modules and Semi-Small Retractable Modules

Let 𝑅 be a commutative ring with 1 and 𝑀 be left unitary 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 . In this paper we introduced and studied concept of semi-small compressible module (a 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is said to be semi-small compressible module if 𝑀 can be embedded in every nonzero semi-small submodule of 𝑀 . Equivalently, 𝑀 is semi-small compressible module if there exists a monomorphism : 𝑀 ⟶ 𝑁 , 0 ≠ 𝑁 ≪ 𝑠𝑒𝑚 𝑀 , 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is said to be semi-small retractable module if 𝐻𝑜𝑚(𝑀, 𝐾) ≠ 0 , for every non-zero semi-small sub module 𝐾 in 𝑀 . Equivalently, 𝑀 is semi-small retractable if there exists a homomorphism 𝑓: 𝑀 ⟶ 𝑁 whenever 0 ≠ 𝑁 ≪ 𝑠𝑒𝑚 𝑀 . In this paper we introduce and study the concept of semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 and semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 s as a generalization of compressible 𝑚𝑜𝑑𝑢𝑙𝑒 and retractable 𝑚𝑜𝑑𝑢𝑙𝑒 respectively and give some of their advantages characterizations and examples.


Introduction
Let R be a commutative ring with 1 and M be a left unitary  − .Authors that introduced and studied the concept of small sub modules where a proper sub module  of an  −module  is termed a small sub module ( ≪ ), if  +  ≠  for every sub module  of  [1].A proper sub module  of  is said to be primary if whenever  ∈  ,  ∈  with  ∈  implies either  ∈  or   ∈ [: ] for some positive integer  , where [: ] = { ∈ :  ⊆ } [2] .In [3] Mijbas and K. Abdullah introduced and studied the concept of semi-small sub modules , where a sub module  of an  −   is termed semi-small sub module  ≪   if  +  ≠  for any primary sub module  of .An  −   is termed compressible if  can be embedded in every non-zero sub module in , [4].An  −   is said to be semi-small compressible if  can be embedded in every non-zero semi-small sub module of.Equivalently,  is semi-small compressible if there exists a monomorphism :  ⟶  whenever0 ≠  ≪  .
Under which condition we introduce and study the concept of semi-small compressible as a generalization of compressible module, and we give some properties, characterization and examples.In addition, we see that under condition semi-small compressible, small compressible and compressible are equivalent.An  −   is said to be semi-small retractable module if (, ) ≠ 0 , for every non-zero semi-small sub module of , some of their advantages characterizations and examples are given.We also study the relation between semi-small compressible, semi-small retractable module and some of classes of modules.
(2) (0) is the only semi-small sub module If  is a semi-simple .
(4) Each small sub module is semi-small.However, conversely is true or not in general.

Semi-Small Compressible Modules
In this section, we introduce the concept of semi-small compressible  as a generalization of compressible .Give some of it is basic properties, examples and characterizations of this concept.

Definition (3.1):
An  −   is said to be semi-small compressible if  can be embedded in every non-zero semi-small sub module of .Equivalently,  is semi-small compressible if there exists a monomorphism :  ⟶  whenever 0 ≠  ≪  .

Remarks and Examples (3.2):
1.It is obvious that every compressible module is semi-small compressible , but the converse is not true.2.  6 as Z-module is not semi-small compressible since (0 ̅ ) is the only semi-small sub module, see, [3]. 3.    −  is semi-small compressible module, because it is compressible , see [4]. 4. If   −  is semi-simple, then  is not semi-small compressible module (Because (0) is the only semi-small sub module in ). 5. Every simple  −  is semi-small compressible module but not conversely, because    −  is a semi-small compressible  but not simple.6.  12   −  is not semi-small compressible.(Because  12 cannot be embedded in 〈6 ̅ 〉 and 〈6 ̅ 〉 ≪   12 ).In addition    −  is not semi-small compressible module, since   (, ) = 0 , where  ≪  .(Since every finitely generated sub module of  is semi-small sub module in .

7.
A homomorphic image of a semi-small compressible  need not be semi-small compressible in general for example    −  is a semi-small compressible module and  12 ≃  12 is not semi-small compressible module see (5).

Remark (3.5):
The direct sum of semi-small compressible  need not be semi-small compressible.Consider the following example, let 6 =  3 ⨁ 2 as  − . 3 ,  3 are semismall compressible modules, but  6 is not semi-small compressible module see remarks and examples (3.2) point ( 2).An  −   is said to be small compressible if  can be embedded in every nonzero small sub module of .Equivalently,  is small compressible if there exists a monomorphism :  ⟶  whenever 0 ≠  ≪  [4].

Conclusion
In this work, the class of compressible and retractable modules have been generalized to new concepts called semi-small compressible and semi-small retractable modules.Several characteristics of this type of modules have been studied.Sufficient conditions under which these modules with compressible and retractable are discussed.In addition, we see relations between semi-small compressible modules and other related modules as semi-small retractable module semi-small quasi-Dedekind, semi-small monoform.