Volume 5, Issue 5 (May 2018), Pages: 79-81

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 Original Research Paper

 Title: The commutativity of prime rings with homoderivations

 Author(s): E. F. Alharfie ^1, *, N. M. Muthana ²

 Affiliation(s):

 ¹Department of Mathematics, Tabuk University, Tabuk, Saudi Arabia
 ²Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

 https://doi.org/10.21833/ijaas.2018.05.010

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 Abstract:

Let R be a ring with center Z(R), and I be a nonzero left ideal. An additive mapping h:R→R is called a homoderivation on R if for all h(xy)=h(x)h(y)+h(x)y+xh(y)for all x.y ∈R. In this paper, we prove the commutativity of R if any of the following conditions is satisfied for all   x.y ∈R: (i)  xh(y)±xy∈Z(R).(ii)     xh(y)±yx∈Z(R).(iii)     xh(y)±[x.y]∈Z(R)     (iv)[x.y]∈Z(R)(v)[h(x)y]±xy Z(R)and (vi)    [h(x).y]±yx∈Z(R). This result is in the sprite of the well-known theorem of the commutativity of prime and semiprime rings with derivations satisfying certain polynomial constraints. Also, we prove that the commutativity of prime ring on R, if R admits a nonzero homoderivation h such that h([x.y])=±[x.y] for all x.y in a nonzero left ideal. 

 © 2018 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

 Keywords: Prime ring, Homoderivation, Commutativity theorems

 Article History: Received 4 November 2017, Received in revised form 19 February 2018, Accepted 12 March 2018

 Digital Object Identifier: 

 https://doi.org/10.21833/ijaas.2018.05.010

 Citation:

 Alharfie EF and Muthana NM (2018). The commutativity of prime rings with homoderivations. International Journal of Advanced and Applied Sciences, 5(5): 79-81

 Permanent Link:

 http://www.science-gate.com/IJAAS/2018/V5I5/Alharfie.html

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