A non-zero module M is called hollow, if every proper submodule of M is small. In this work we introduce a generalization of this type of modules; we call it prime hollow modules. Some main properties of this kind of modules are investigated and the relation between these modules with hollow modules and some other modules are studied, such as semihollow, amply supplemented and lifting modules.

       Th  goal of the pr s nt p p r is to obt in some differ tial sub rdin tion an  sup r dination the rems for univalent functions related b  differential operator  Also, we discussed some sandwich-type results.

Let M be an R-module, where R be a commutative;ring with identity. In this paper, we defined a new kind of submodules, namely; ET-coessential and ET-Coclosed submodules of M. Let T be a submodule of M. Let K  H  M, K  is called  ET-Coessential of H in M (K⊆_ET.ce H), if     . A submodule H is called ET- coclosed in M of H has no proper coessential submodule in M, we denote by  (K⊆_ET.cc H) , that is, K⊆_ET.ce H implies that   K = H. In our work, we introduce;some properties of ET-coessential and ET-coclosed submodules of M.

The goal of this research is to introduce the concepts of Large-coessential submodule and Large-coclosed submodule, for which some properties are also considered. Let M  be an R-module and K, N are submodules of M such that , then K is said to be Large-coessential submodule, if . A submodule N of M is called Large-coclosed submodule, if K is Large-coessential submodule of N in M, for some submodule K of N, implies that  .

The goal of this research is to introduce the concepts of Large-coessential submodule and Large-coclosed submodule, for which some properties are also considered. Let M  be an R-module and K, N are submodules of M such that , then K is said to be Large-coessential submodule, if . A submodule N of M is called Large-coclosed submodule, if K is Large-coessential submodule of N in M, for some submodule K of N, implies that  .

The main goal of this paper is introducing and studying a new concept, which is named H-essential submodules, and we use it to construct another concept called Homessential modules. Several fundamental properties of these concepts are investigated, and other characterizations for each one of them is given. Moreover, many relationships of Homessential modules with other related concepts are studied such as Quasi-Dedekind, Uniform, Prime and Extending modules.

In this paper, we will generalized some results related to centralizer concept on
prime and semiprime Γ-rings of characteristic different from 2 .These results
relating to some results concerning left centralizer on Γ-rings.

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

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A non-zero submodule N of M is called essential if N L for each non-zero submodule L of M. And a non-zero submodule K of M is called semi-essential if K P for each non-zero prime submodule P of M. In this paper we investigate a class of submodules that lies between essential submodules and semi-essential submodules, we call these class of submodules weak essential submodules.