Differential Identities on Semigroup Ideals of Right Near-Rings

Abstract

In this paper, we investigate generalized derivations satisfying certain differential identities on semigroup ideals of right near-rings and discuss related results. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

A study of near-rings with generalized derivations

“…Recently, there has been a great deal of work concerning commutativity of prime and semiprime rings admitting suitably constrained derivations and generalized derivations (see [4][5][6][7]). In view of these results many authors have proved comparable results for nearrings (see [8][9][10][11]). In the present paper it is shown that near-rings with generalized derivations satisfying certain identities are commutative rings.…”

Structure of Near-Rings Satisfying Certain Polynomial Identities

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In this paper, we will introduce the concept of two-sided αgeneralized derivation in prime near-rings as it was outlined by the author N. Argac in [1]. Thereafter, we will generalize the same results proved by many authors (see [2], [4] and [5]) in the case of derivations, semiderivations and generalized derivations. Furthermore, we will give examples to demonstrate that the restrictions imposed on the hypothesis of various results are not superfluous.

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“…We begin with several lemmas, most of which are known. Those for which neither a proof nor a precise citation is given are to be found in [3], [4] and [6]. Unless it is stated otherwise, it will be assumed that N is a zero-symmetric near ring and d is a two-sided α-derivation of N associated with an endomorphism α.…”

“…The following lemma is very useful in the sequel and it is a special case of Theorem 1 and Theorem 3 of [4]. Lemma 5.…”

Two sided $\alpha $-derivations in 3-prime near-rings

“…In [13], Hongan generalizes some of their results by assuming that the commutativity condition is imposed on an ideal rather than on the wall near-ring. More recently, Bell et al [5,11] generalize several commutativity theorems for the 3-prime near-ring by treating the case of generalized derivations satisfying certain algebraic identities involving semigroup ideals. Some of our results, which deal with conditions on two sided α-derivations, extend earlier commutativity results involving similar conditions on derivations and semi-derivations.…”