Over the last few decades lots of work has been done on commutativity of prime rings with derivations. It is natural to look for comparable results on near rings (see for example [6]). Historically speaking, the study of derivations of prime near-rings was initiated by H. E. Bell and G. Mason in 1987 [3]. An analogue of Posner’s result on prime near-rings was obtained by Beidar in [1]. Results concerning prime near-rings with derivations have since been generalized in several ways. Throughout the paper N is a zero-symmetric left near-ring with a multiplicative center Z. For x, y ∈ N, the symbol [x, y] will denote the commutator xy − yx, while the symbol (x, y) will denote the additive group commutator x + y − x − y. Our main aim in this paper is to give a correction for Theorem 3 on p.253 in [4].

The proof of Theorem 3 of [4], stated as Theorem 2.12 in this paper, is incorrect because of the assumption of both left and right distributivity. In Section 3, we first introduce Lemma 3.1, which is an extension of [2, Lemma 2.2], then give a correction for Theorem 2.12.

We first recall some of the background which will be used throughout this work.

Definition 2.1

A left near-ring is a set N together with two binary operations + and. such that (N, +) is a group (not necessarily abelian) and (N, .) is a semigroup satisfying the distributive law x(y + z) = xy + xz for all x, y, z ∈ N.

Definition 2.2

A near-ring N is said to be zero-symmetric if 0.x = 0 ∀a ∈ N.

Definition 2.3

A left near-ring N is called a prime near-ring if a, b ∈ N and aNb = 0 imply that a = 0 or b = 0.

Definition 2.4

A derivation on a near-ring N is defined to be an additive endomorphism satisfying the “product rule” d(xy) = xd(y) + d(x)y for all x, y ∈ N.

Definition 2.5

Let σ, τ be two endomorphisms of a near-ring N. An additive mapping d : N → N is called a (σ, τ )-derivation if d(xy) = σ(x)d(y) + d(x)τ (y) for all x, y ∈ N.

Definition 2.6

Let N be a near-ring, and d a derivation of N. An additive mapping f : N → N is said to be left generalized (σ, τ )-derivation of N associated with d if

f is said to be a right generalized (σ, τ )-derivation of N associated with d if

f is said to be a generalized (σ, τ )-derivation associated with d if it is both left and right generalized (σ, τ )-derivation associated with d.

We now recall some properties about near-rings with derivation and generalized (σ, τ )-derivation.

Lemma 2.7.([4, Lemma 1])

• (i) Let f be a right generalized (σ, τ )-derivation of near-ring N associated with d. Then f(xy) = σ(x)d(y) + f(x)τ (y) ∀x, y ∈ N.

• (ii) Let f be a left generalized (σ, τ )-derivation of near-ring N associated with d. Then f(xy) = σ(x)f(y) + d(x)τ (y) ∀x, y ∈ N.

Lemma 2.8.([4, Lemma 3])

Let N be a prime near-ring, f a generalized (σ, τ )-derivation of N associated with d, and a ∈ N.

• (i) If af(N) = 0, then a = 0.

• (ii) If f(N)τ (a) = 0, then a = 0.

Lemma 2.9.([5, Lemma 3])

Let N be a prime near-ring, d a non-zero (σ, τ )-derivation of N, and a ∈ N.

• (i) If d(N)σ(a) = 0, then a = 0.

• (ii) If ad(N) = 0, then a = 0.

Lemma 2.10.([4, Lemma 2])

• (i) Let f be a right generalized (σ, τ )-derivation of near-ring N associated with d. Then

  (f(x)τ(y)+σ(x)d(y))τ(z)=f(x)τ(y)τ(z)+σ(x)d(y)τ(z),,

  for all x, y ∈ N.

• (ii) Let f be a generalized (σ, τ )-derivation of near-ring N associated with d. Then

  (d(x)τ(y)+σ(x)f(y)τ(z))=d(x)τ(y)τ(z)+σ(x)f(y)τ(z),

  for all x, y ∈ N.

Theorem 2.11.([4, Theorem 2])

Let N be a prime near-ring with a non-zero generalized (σ, τ )-derivation f associated with d. If f(N) ⊂ Z then (N, +) is abelian. Moreover, if N is 2-torsion free then N is a commutative ring.

Theorem 2.12.([4, Theorem 3])

Let N be a prime near-ring with a non-zero generalized (σ, τ )-derivation f associated with non-zero (σ, τ )-derivation d such that τf = fτ, fσ = σf, σd = dσ, and τd = dτ. If [f(N), f(N)] = 0 then (N, +) is abelian. Moreover, if N is 2-torsion free, then N is commutative.

We first give the following auxiliary lemma.

Lemma 3.1

Let N be a prime near-ring and f be a generalized (σ, τ )-derivation associated with non-zero (σ, τ )-derivation d such that σf = fσ, τf = fτ and σd = dσ, τd = dτ. If d(f(N)) = {0}, then f(d(N)) = {0}.