Let  be a commutative ring with 1 and  be left unitary  . In this papers we introduced and studied concept P-small compressible  (An     is said to be P-small compressible if  can be embedded in every of it is nonzero P-small submodule of . Equivalently,  is P-small compressible if there exists a monomorphism  , ,     is said to be P-small retractable if  , for every non-zero P-small submodule of . Equivalently,  is P-small retractable if there exists a homomorphism  whenever  as a generalization of compressible  and retractable  respectively and give some of their advantages characterizations and examples.

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .     In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of their adv

Let R be a commutative ring with identity and M be a unitary R- module. We shall say that M is a primary multiplication module if every primary submodule of M is a multiplication submodule of M. Some of the properties of this concept will be investigated. The main results of this paper are, for modules M and N, we have M N and HomR (M, N) are primary multiplications R-modules under certain assumptions.

The main goal of this paper is to introduce and study a new concept named d^*-supplemented which can be considered as a generalization of W- supplemented modules and d-hollow module. Also, we introduce a d^*-supplement submodule. Many relationships of d^*-supplemented modules are studied. Especially, we give characterizations of d^*-supplemented modules and relationship between this kind of modules and other kind modules for example every d-hollow (d-local) module is d^*-supplemented and by an example we show that the converse is not true.

Inhibition of T cell derived cytokine production could help suppress osteoclast differentiation in inflammatory skeletal disorders. Bisphosphonates are typically prescribed to prevent inflammatory bone loss but are not tolerated by all patients and are associated with an increased risk of osteonecrosis of the jaw. In light of this other anti-resorptives such as phytoestrogens are being considered. However the effect of phytoestrogens on T cell-induced osteoclast formation is unclear. The effect of genistein and coumestrol on activated T cell-induced osteoclastogenesis and cytokine production was therefore examined. Concentrations of genistein and coumestrol (10−7 M) previously shown to directly inhibit osteoclast formation also suppressed

    In this paper, we introduce the concept of s.p-semisimple module. Let S be a semiradical property, we say that a module M is s.p - semisimple if for every submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S. we prove that a module M is s.p - semisimple module if and only if for every submodule A of M, there exists a direct summand B of M such that A = B + C and C has S. Also, we prove that for a module M is s.p - semisimple if and only if for every submodule A of M, there exists an idempotent e ∊ End(M) such that e(M) ≤ A and (1- e)(A) has S. 

Throughout this paper, T is a ring with identity and F is a unitary left module over T. This paper study the relation between semihollow-lifting modules and semiprojective covers. proposition 5 shows that If T is semihollow-lifting, then every semilocal T-module has semiprojective cover. Also, give a condition under which a quotient of a semihollow-lifting module having a semiprojective cover. proposition 2 shows that if K is a projective module. K is semihollow-lifting if and only if For every submodule A of K with K/( A) is hollow, then K/( A) has a semiprojective cover.

Let be a commutative ring with unity and let be a non-zero unitary module. In
this work we present a -small projective module concept as a generalization of small
projective. Also we generalize some properties of small epimorphism to δ-small
epimorphism. We also introduce the notation of δ-small hereditary modules and δ-small
projective covers.