Kyungpook Mathematical Journal 2020; 60(2): 415-421
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Correction to "On prime near-rings with generalized (σ,Ƭ)- derivations, Kyungpook Math. J., 45(2005), 249-254"
Hassan J. Al Hwaeer*, Gbrel Albkwre, Neşet Deniz Turgay
Department of Mathematics and Computer applications, Al-Nahrain University,Iraq
e-mail : email@example.com
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
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Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey
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Received: June 12, 2019; Revised: September 24, 2019; Accepted: November 18, 2019
In the proof of Theorem 3 on p.253 in [
Keywords: prime near-rings, derivation, generalized (σ,Ƭ,)-derivation
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 ). Historically speaking, the study of derivations of prime near-rings was initiated by H. E. Bell and G. Mason in 1987 . An analogue of Posner’s result on prime near-rings was obtained by Beidar in . Results concerning prime near-rings with derivations have since been generalized in several ways. Throughout the paper
The proof of Theorem 3 of , 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.
A left near-ring
We now recall some properties about near-rings with derivation and generalized (
Lemma 2.7.([4, Lemma 1])
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.
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])
If af( N) = 0, then a= 0.
If f( N) τ( a) = 0, then a= 0.
Lemma 2.9.([5, Lemma 3])
If d( N) σ( a) = 0, then a= 0.
If ad( N) = 0, then a= 0.
Lemma 2.10.([4, Lemma 2])
Let f be a right generalized( σ, τ) -derivation of near-ring N associated with d. Then , for all x, y∈ N.
Let f be a generalized( σ, τ) -derivation of near-ring N associated with d. Then 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.
By using (
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
By Lemma 2.8, we get
and this gives
and this gives
Now, assume that
and this gives
If we take
So we have
Assume, by way of contradiction, that
So, we have
We claim that
By Lemma 3.1 we get
and from this we obtain
Rewrite the equation above and use Lemma 2.10, to find
By using our claim and the
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