Let M is a Г-ring. In this paper the concept of orthogonal symmetric higher bi-derivations on semiprime Г-ring is presented and studied and the relations of two symmetric higher bi-derivations on Г-ring are introduced.

      This paper investigates the concept (α, β) derivation on semiring and extend a few results of this map on prime semiring. We establish the commutativity of prime semiring and investigate when (α, β) derivation becomes zero.

We introduced the nomenclature of orthogonal G -m-derivations and orthogonal generalized G -m-derivations in semi-prime G -near-rings and provide a few essentials and enough provision for generalized G -n-derivations in semi-prime G -near-rings by orthogonal.

 The current paper studied the concept of right n-derivation satisfying certified conditions on semigroup ideals of near-rings and some related properties. Interesting results have been reached, the most prominent of which are the following: Let M be a 3-prime left near-ring and A_1,A_2,…,A_n are nonzero semigroup ideals of M, if d is a right n-derivation of M satisfies on of the following conditions,
d(u_1,u_2,…,(u_j,v_j ),…,u_n )=0 ∀ 〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u;
d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=0 ∀u_1,v_1 〖ϵA〗_1,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u ;
d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=(u_

Let R be a 2-torision free prime ring and ?, ?? Aut(R). Furthermore, G: R×R?R is a symmetric generalized (?, ?)-Biderivation associated with a nonzero (?, ?)-Biderivation D. In this paper some certain identities are presented satisfying by the traces of G and D on an ideal of R which forces R to be commutative

In this paper, we proved that if R is a prime ring, U be a nonzero Lie ideal of R , d be a nonzero (?,?)-derivation of R. Then if Ua?Z(R) (or aU?Z(R)) for a?R, then either or U is commutative Also, we assumed that Uis a ring to prove that: (i) If Ua?Z(R) (or aU?Z(R)) for a?R, then either a=0 or U is commutative. (ii) If ad(U)=0 (or d(U)a=0) for a?R, then either a=0 or U is commutative. (iii) If d is a homomorphism on U such that ad(U) ?Z(R)(or d(U)a?Z(R), then a=0 or U is commutative.

We present the concept of maps Γ- periodi2 on Γ -near-ring S. Our main goal is to research and explore the presence and mapping traits such as h Γ –hom anti-Γ –hom, Γ –α-derivations of Γ -periodi2 on Γ- near-rings.

"In this article, "we introduce the concept of a WE-Prime submodule", as a stronger form of a weakly prime submodule". "And as a "generalization of WE-Prime submodule", we introduce the concept of WE-Semi-Prime submodule, which is also a stronger form of a weakly semi-prime submodule". "Various basic properties of these two concepts are discussed. Furthermore, the relationships between "WE-Prime submodules and weakly prime submodules" and studied". "On the other hand the relation between "WE-Prime submodules and WE-Semi-Prime submodules" are consider". "Also" the relation of "WE-Sime-Prime submodules and weakly semi-prime submodules" are explained. Behind that, some characterizations of these concepts are investigated".

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.