Abstract
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. AMS Subject Classification: 16N60, 16W25, 16Y30
Highlights
A right near-ring is a set N with two operations + and . such that (N, +) is a group and (N, .) is a semigroup satisfying the right distributive law (x + y)z = x.z + y.z for all x, y, z ∈ N
A nonempty subset I of N is called stable by the additive law, if for any x, y ∈ I, x + y ∈ I An additive mapping σ : N → N is called an involution if σ(xy) = σ(y)σ(x) and σ2(x) = x for all x, y ∈ N
An additive mapping d : N −→ N is said to be a derivation if d(xy) = xd(y) + d(x)y for all x, y ∈ N
Summary
[6, Theorem 2] Let N be a prime near-ring and I be a nonzero semigroup ideal of N. Let N be a prime near-ring admitting a generalized derivation F associated with a nonzero derivation d. [3, Lemma 1.2(i), (ii)] Let N be a prime near-ring and z ∈ Z(N ) − {0}.
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