Article
Kyungpook Mathematical Journal 2020; 60(2): 415421
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Correction to "On prime nearrings with generalized (σ,Ƭ) derivations, Kyungpook Math. J., 45(2005), 249254"
Hassan J. Al Hwaeer*, Gbrel Albkwre, Neşet Deniz Turgay
Department of Mathematics and Computer applications, AlNahrain University,Iraq
email : hjh@sc.nahrainuniv.edu.iq
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
email : gmalbkwre@mix.wvu.edu
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey
email : neset.turgay@emu.edu.tr
Received: June 12, 2019; Revised: September 24, 2019; Accepted: November 18, 2019
In the proof of Theorem 3 on p.253 in [
Keywords: prime nearrings, derivation, generalized (σ,Ƭ,)derivation
1. Introduction
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 nearrings was initiated by H. E. Bell and G. Mason in 1987 [3]. An analogue of Posner’s result on prime nearrings was obtained by Beidar in [1]. Results concerning prime nearrings with derivations have since been generalized in several ways. Throughout the paper
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.
2. Preliminaries
We first recall some of the background which will be used throughout this work.
Definition 2.1
A
Definition 2.2
A nearring
Definition 2.3
A left nearring
Definition 2.4
A
Definition 2.5
Let
Definition 2.6
Let
We now recall some properties about nearrings with derivation and generalized (
Lemma 2.7.([4, Lemma 1])

(i)
Let f be a right generalized (σ, τ )derivation of nearring 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 nearring N associated with d .Then f (xy ) =σ (x )f (y ) +d (x )τ (y ) ∀x, y ∈N .
Lemma 2.8.([4, Lemma 3])

(i)
If af (N ) = 0,then a = 0. 
(ii)
If f (N )τ (a ) = 0,then a = 0.
Lemma 2.9.([5, Lemma 3])

(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 nearring N associated with d. Then $$(f(x)\tau (y)+\sigma (x)d(y))\tau (z)=f(x)\tau (y)\tau (z)+\sigma (x)d(y)\tau (z),$$ ,for all x, y ∈N . 
(ii)
Let f be a generalized (σ, τ )derivation of nearring N associated with d. Then $$(d(x)\tau (y)+\sigma (x)f(y)\tau (z))=d(x)\tau (y)\tau (z)+\sigma (x)f(y)\tau (z),$$ for all x, y ∈N .
Theorem 2.11.([4, Theorem 2])
Theorem 2.12.([4, Theorem 3])
3. Corrected Proof of Theorem 2.12
We first give the following auxiliary lemma.
Lemma 3.1
Assume that
Applying
Substituting
Replacing
Hence (
Replacing
Taking
By using (
Replacing
∀
The preceding lemma will now be used to establish the correction.
We start with the same argument given in the proof of [4, Theorem 3]. If
So we get
Substituting
Since
By Lemma 2.8, we get
and this gives
Replacing
and this gives
Since
Now, assume that
Using
and this gives
If we take
So we have
Hence
If
Assume, by way of contradiction, that
So, we have
Replacing
We claim that
Also
By Lemma 3.1 we get
Substituting (
Now, substituting
and from this we obtain
Rewrite the equation above and use Lemma 2.10, to find
By using our claim and the
Since
References
 KI. Beidar, Y. Fong, and XK. Wang.
Posner and Herstein theorems for derivations of 3prime nearrings .,.  H. E. Bell, On prime nearrings with generalized derivation, In. J. Math. Math. Sci., (2008), Art. ID 490316, 5 pp. Comm. algebra, 24(1996), 15811589.
 HE. Bell, and G. Mason.
On derivations in nearrings and nearfields . NorthHolland Mathematics Studies.,137 (1987), 3135.  Ö. Gölbaşı.
On prime nearrings with generalized (σ, τ )derivations . Kyungpook Math J.,45 (2005), 249254.  Ö. Gölbaşı, and N. Aydın.
Results on prime nearring with (σ, τ )derivation . Math J Okayama Univ.,46 (2004), 17.  XK. Wang.
Derivations in prime nearrings . Proc Amer Math Soc.,121 (1994), 361366.