Semiderivations Satisfying Certain Algebraic Identities on Prime Near-Rings

Abstract

In this paper it is shown that zero symmetric prime left near-rings satisfying certain identities involving semiderivations are commutative rings.

Structure of Near-Rings Satisfying Certain Polynomial Identities

“…The main purpose of this paragraph is to generalize several results du to some authors (see [2], [4] and [5]) concerning the study of commutativity of prime right near-ring satisfying certain differential identities. For more details see the following results.…”

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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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On Semiderivations of Near-Rings

“…( [4]) Let N be a 2-torsion free prime near-ring and let f be a nonzero semiderivation associated with a surjective function g : Proof. By hypothesis, we have…”

Some commutativity criteria for 3-prime near-rings

“…An additive mapping f : N → N is said to be a generalized derivation on N if there exists a derivation δ : N → N such that f (xy) = f (x)y + xδ(y) for all x, y ∈ N . An additive mapping d : N → N is called semiderivation if there is a function g : N → N such that d(xy) = xd(y) + d(x)g(y) = g(x)d(y) + d(x)y and d(g(x)) = g(d(x)) for all x, y ∈ N , or equivalently, as noted in [5], that d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and d(g(x)) = g(d(x)) for all x, y ∈ N . Obviously, any derivation is a semiderivation, but the converse is not true in general (see [5]).…”