s-Compressible and s-Prime Modules

Let R be a ring with identity and Ą a left R -module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R -module A is s-compressible if for any nonzero submodule B of A there exists a small f in Hom R ( A , B ). An R -module A is s-prime if for any submodule B of A , ann R ( B ) A is small in A . These concepts and related concepts are studied in as well as many results consist properties and characterizations are obtained.


Introduction
Compressible module was introduced by Zelmanowitz [1] simultaneous with introducing the concept of weakly primitive ring in the way of generalizing the Jacobson density theorem. He also introduced critically compressible module. In [2], the author studied those concepts in details. A left R-module is compressible if it can be embedded in any of its nonzero submodule [1]. A compressible module A is critically compressible if it cannot be embedding in any factor A/B, where B is a nonzero submodule of A . In[1], Zelmanowitz defined a ring to be weakly primitive if it possesses a faithful critically compressible module. In [3]- [6], authors have been extensively studied compressible, critically compressible and prime modules. By using small submodules one direction of generalizations of compressible and prime modules e appeared in [7]- [9]. A small compressible module is defined as a module that can be embedded in its small submodules, as well as small prime module is defined as a

ISSN: 0067-2904
Alhossaini Iraqi Journal of Science, 2022, Vol. 63, No. 3, pp: 1200-1207 2122 module A in which ann R B=ann R A for each small submodule B of A. Note that a module A is prime, if ann R B=ann R A for each nonzero submodule B of A [7].
Throughout this work, we use the notion of small submodule . Different generalizations are given. We recall that, an R-homomorphism in Hom(A, B) is said to be small if its kernel is small in A [10]. In the new generalization the zero kernel will be replaced by small kernel. An R-module A is said to be s-compressible if for each nonzero submodule B of A there exists a small element f in Hom(A, B), that is kerf is small in A. Note that this definition is also appeared in [11] with different abbreviation, sk-compressible. An s-compressible module A is critically s-compressible if Hom(A, A/ B) has no small element for any non-small submodule B of A. A module A is s-prime if (ann R B) A is small in A for any nonzero submodule B of A. These concepts are studied, and their relationships among them and with other related concepts are discussed. Some properties and characterizations are obtained. Firstly, it is shown that s-compressible with small compressible modules are independent, as well as the s-prime and small prime modules are also independent. The class of compressible modules contains both classes of s-compressible and small compressible modules. As well as the class of prime modules contains both classes of s-prime and small prime modules.
Throughout this article some definitions and notations are given. A module is a left unitary module over a ring R with identity. A submodule B of a module A will be abbreviated by B ≤ A. A submodule B of a module A is said to be small in A(abbreviated by B << A) if it is proper and its sum with any other proper submodule of A is again proper, "in other word if B + C= A, where C≤ A, then C= A [10]. A is said to be hollow if all its proper submodules are small.  [12]. An R-module A is retractable if Hom R (A, B)≠0 for any nonzero submodule B of A [13].
In Section 2 s-compressible and critically s-compressible modules are introduced and investigated. The notion s-compressible is appeared in [11]. It is abbreviated by skcompressible. In this work this notion is studied in details and more results are given. Section 3 devotes to introduce s-prime module and study the relationships between the present notions and old related notions.

s-Compressible and Critically s-Compressible Modules
Any compressible module is s-compressible, however the converse is not true. Remark (2.3): Any simple module is s-compressible. Example (2.4): Consider the -module n , if n=mp k where p is a prime which is not dividing m, thus if s n is a small submodule of n , then s=pt for some t. Note that, in a R -module A, the submodule Ra is small in A if and only if a belongs to all maximal submodules of A [10]. Now, if f: n → p k n is a -homomorphism such that ker f = s n small in n , then |ker f|= n/s, so that | n /ker f|=s=pt , while | p k n |= m, this gives a contradiction with the fact that n /ker f is isomorphic to a submodule of p k n . Therefore, there is no small homomorphism from n into p k n , that is, n is not s-compressible if n=mp k and p is a prime which is not dividing m.
On the other hand n=p k , the -module n is hollow, all its proper submodules are small. it is easy to see that it is s-compressible. Therefore the -module n is s-compressible if and only if n= p k where p is prime. We note that the two notions small compressible and s-compressible are independent. For example 6 is small compressible -module which is not s-compressible, while 4 is scompressible that is not small compressible -module [8]. Both of two -modules are not compressible. The two classes of small and s-compressible modules contain the class of compressible modules. Remark 2.5: It is clear that any s-compressible module is retractable. However the converse is not true to see that 6 as a -module is retractable but not s-compressible. Next proposition gives However, a condition can be added to a retractable module to get scompressible module, see the following. Proposition 2.6: Any hollow retractable module is s-compressible. Proof: Assume that A is hollow retractable module, and B is a nonzero submodule of A, then there exists 0≠f Hom(A, B) such that kerf is a proper submodule of A, hence small in A. Therefore A is s-compressible. This proposition can be applied to example 2.4 so that is s-compressible. We note that the -module is s-compressible but not hollow, and this proves that the converse of proposition 2.6 is not true. □ It is well known that a nonzero submodule of a compressible module is compressible. In the following this property will be discussed under certain condition for s-compressibility.
Recall that, an R-module A is said to be fully stable, if for each submodule B of A and for each f Hom (B, A), it follows f(B) [12], In fact A is fully stable if and only if Hom(B, A) End(B) for each submodule B of A ,and more details about fully stable modules can be found in [12]. For completeness a proof will be given.
□ It is known that any small submodule of a module is contained in its Jacobson radical,while a submodule that contained in the Jacobson radical of the module is small if it is finitely generated [10]. Proposition 2.13: A finitely generated direct summand of a fully stable s-compressible module is s-compressible. Proof : Assume that A = A 1 A 2 is an s-compressible module and B is a submodule of A 1 , then B is a submodule of A, by assumption there exists f Hom(A, B) with kerf<< A. Let g=f , then kerg= A 1 ∩ kerf. It is known that =J(A 1 )(by full stability) so that kerg J(A 1 ) and kerg<< A 1 . Therefore A 1 is s-compressible. □ Remark 2.14: The converse of Proposition 2.13 is not true to see that let 6 = 〈 〉 〈 〉 is fully stable [11] and both 〈 〉 and 〈 〉 are s-compressible ,however 6

) A and A 2 = (ann R A 1 ) A, then A = (ann R A 2 ) A (ann R A 1 ) A. But by Proposition 2.3 (ann R A 1 ) A and (ann R A 2 )
A are both small in A, which is a contradiction. Therefore A is indecomposable. □ An R-module A is said to be duo if any submodule of A is full invariant, that is, for each f End(A) and for each B≤ A, f(B) " [14].,and it is said to be torsion free if rm≠0 whenever 0≠r R and 0≠m A, or equivalently 0≠m A and rm=0 implies r=0 Next theorems give a characterization of Duo modules, we will start with the following lemma. Lemma 2.19: "An R-module A is duo if and only if for each f End(A) and for each m A there exists r R such that f(m)=rm" [14]. Theorem 2.20: Let A be a duo torsion free R-module. Then A is compressible if and only if it is retractable. Proof: ( ) It is clear so that it is omitted . ( ) Assume that A is a duo torsion free R-module and retractable, let 0≠B≤ A, then there exists 0≠f Hom (A, B), it can be considered that f End(A). By Lemma2.19, for each m A there exists r R such that f(m)=rm. So kerf= { m A| rm=0 for some r R}, as A is torsion free and 0≠f, it follows kerf=0, that is A embed in B. Therefore A is compressible. □ A compressible module is said to be critically compressible if it cannot be embedded in any of its proper factors [2]. This notion was generalized in [7] using small submodule this way gives that a small compressible module A is called small critically compressible if A cannot be embedded in any proper quotient module A/ B with 0≠ B≪ A".
Another generalization will be given by using small submodule. , is not small critically compressible module for k>1.While 6 is small critically compressible -module but not critically s-compressible. (iii) The -module , also is critically s-compressible. (iv) Any critically compressible module is critically s-compressible. But the converse is not true.
(v) Any simple module is critically s-compressible.
By partial endomorphism of a module A it means an element of Hom (B, A) where B is a submodule of A. Proposition 2.24: If A is a critically s-compressible module, then any nonzero partial endomorphism of A has kernel small in A. Proof: Assume that A is a critically s-compressible module and 0≠f Hom(B, A), where B≤ A, suppose that kerf is not small in A. Then Imf ≠0 and there exists 0≠g Hom(A, Imf) such that kerg<< A since A is s-compressible. On the other hand Imf N/ kerf≤ A /kerf, let h: Imf→ B/ kerf be an isomorphism and i: B/ kerf→/kerf be the inclusion map. Then ihg Hom(A, A /kerf) and ker ihg= kerg<< A. This contradicts the assumption that A is critically s-compressible. To prove the converse of Proposition 2.24, we need a condition this is given in next proposition.

s-Prime Modules
Prime modules are defined and investigated in the literatures see [3][4] [15]. An R-module A is said to be prime if for any nonzero submodule B of A, ann R B = ann R A. This notion is generalized in [7] using the concept of small submodules in this way an R-module A is a small prime module if ann R A = ann R B for each non-zero small submodule B of A . We also use small submodules to give a different generalization for prime module, its properties, and characterizations as well as relations with s-compressible is also studied. Definition 3.1: A nonzero R-module A is called s-prime if for any nonzero submodule B of A, (ann R B) A << A. Remark 3.2: (i) The two notions small prime, and s-prime are independent. For example the -module 4 is an s-prime but not small prime (can be easily checked), while the -module 24 is small prime , this is also shown in [7], However it is not s-prime since ( ( 〈 ̅ 〉 ) 24 =3 ( 24 )= 〈 ̅ 〉 not small in 24 . (ii) We have seen that any torsion free Ɽ -module is s-prime. (iii) It is clear that any prime module is s-prime. However the converse is not true, for example the -module 8 is not prime since (4 8 )= 2 , while ( 8 )=8. But 8 is sprime -module (can be easily checked). ( )Assume that A is a multiplication retractable R-module and 0≠ Ɓ≤ A. Then, there exists 0≠ f Hom (A, B), since A is retractable, that is Imf ≠0. As A is s-prime, it follows (ann R Imf) A << A. Now, ann R (Imf)={r R |rf(m)=0, m A }={r R |f(rm)=0, m A}= {r R |rm kerf m A }= [kerf: A]. Hence (ann R Imf) A = [kerf: A] A =kerf, since A is multiplication. Therefore, we have kerf<< A, and A is s-compressible. □ In [16] author proved that a faithful multiplication R-module is retractable according to this result Theorem 3.6 can be rewritten as following. Corollary 3.7: Let A be a faithful multiplication R-module, then A is s-compressible if and only if it is s-prime.
Recall that a ring R is called left duo if any left ideal is two sided ideal [14]. Proposition 3.8: Let R be a left duo ring. A nonzero R-module A is s-prime if and only if for each 0≠ x A and for each ideal I of R, Ix=0 implies I A << A. Proof: ( ) Assume that 0≠x A and Ix=0, then IRx= RIx=0 where 0≠Rx<< A and by assumption I A << A.