Generalized Reverse Derivations and Commutativity of Prime Γ-Semirings

Abstract

We present the notion of generalized reverse derivation on Γ-semiring in this article and we prove that a prime Γ-semiring with center CΓ stratifying assumption (*) aαbβc = aβbαc, for all a, b, c S and α, Γ and I be a non-zero right ideal of S such that S admits a generalized reverse derivation f associated with a reverse derivation d satisfying d(CΓ) ≠ 0, if S satisfies any one of the properties: (i) f(aαb) + aαb Γ (ii) f([a, b]α) + [f(a), b]α Γ (iii) f([a, b]α) + [f(a), f(b)]α Γ (iv) [f(a), b]α + [a, f(b)]α Γ For of the all a, b I and α Γ, then S is commutative.