Near – Rings with Generalized Right -Derivations

We define a new concept, called " generalized right -derivation", in near-ring and obtain new essential results in this field. Moreover we improve this paper with examples that show that the assumptions used are necessary.


Introduction
A near-ring is defined to be a "set with two binary operations and such that (i) is a group that is not necessarily abelian: (ii) is a semi group; (iii) ". The elements products, such as and in will be which is specified by . is zero-symmetric whenever , for each ( yields from left distributivity). The centre of multiplicative of will be represented by . For each [ ] symbolizes the commutator and is the additive commutator, while will denote the well-known Jordan product. is referred to as prime nearring in case of { } which infers that or . "A non-empty subset U of is named as semigroup left ideal, resp. semigroup right ideal in case of (resp. ). But, if U represents both semigroup right and left ideal, then it will be termed as semigroup ideal". For more about near-ring theory and its applications, we make reference to Pilz [1]. In [2], X. Wang defined the derivation as an additive mapping from into itself which satisfies for each . Later, the derivation concepts generalization have been achieved through various means according to different authors. "Ashraf and Siddeeque well-defined the concepts of derivations, generalized derivations, and derivation in near ring [3 -6]". Also, various properties of such derivations were examined. In 2015, Abdul Rehman and Enaam defined a new concept, called " right n-derivation", in near-ring and obtained new essential results for researchers in this field [7].

ISSN: 0067-2904
Adhab Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2334-2342 5332 "An additive mapping from into itself is said to be right derivation of if for each and additive (i.e. additive in each argument) mapping ⏟ is said to be right derivation of if the following equations hold for each ": " " " " " " Motivated by the previous studies, we define here the concepts of generalized right derivation and generalized right n-derivation in near-ring . After that, we will give new essential results in this field and generalize some results presented in [7]. Finally, we improve this paper with examples that show that the assumptions used are necessary. Note that we will use the abbreviation C.R to refer to the commutative ring. Definition 1.1. Let be a right derivation of . An additive mapping from into itself is said to be generalized right derivation of connected with if , for each . Definition 1.2. If is a right -derivation of and ⏟ is an additive mapping on , then will be called " generalized right derivation of connected with " if the following equations hold for each . " ) = + " " ( ) = + " " ( ) = + " Example 1.3. If be a near-ring and zero symmetric then it is obvious that {( ) } is a near-ring with the addition and multiplication of matrices.
Let and , : ⏟ defined by: Simply can check that is a generalized right derivation connected with right derivation of and is a nonzero generalized right derivation connected with right derivation of .

2.
Preliminaries The next lemmas are fundamental to develop the proofs of our work. Lemma 2.1 [8]. "Let be a near-ring. If there is an element of such that , then ( ) is abelian".

Lemma 2.2 [9] "Let be a prime near-ring. If
{ } and is an element of such that or , then ". Lemma 2.3 [9]. "Let be a prime near-ring and contains a nonzero semigroup left ideal or nonzero semigroup left ideal, then is a C.R".  (1) and (2) (7) and (8) . Using (9) in previous equation yields ( )( ) . Using Lemma 2.8, we conclude that ( x 1 ') = 0 for each , ϵ N. Hence, is an abelain group. Therefore, for each and, using Lemma 2.5, we finally obtain that is a C.R.