Abstract
Let R be a ring with center Z(R), and.. be a nonzero left ideal. An additive mapping h : R -> R is called a homoderivation on R if h(xy) = h(x)h(y) + h(x)y + xh(y) for all x.y is an element of R In this paper, we prove the commutativity of R if any of the following conditions is satisfied for all x.y is an element of R (i) xh(y) +/- xy is an element of Z(R) (ii) xy(y) +/- yx is an element of Z(R). (iii) xh(y) +/- [x.y] is an element of Z(R) (iv) [x.y] is an element of Z(R)(v)[h((x)y)] +/- xy Z(R) and (vi) [h(x).y] +/- yx is an element of 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. (C) 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/).