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The concept of $\Gamma$-ring is a generalization of ring. Two important classes of $\Gamma$-rings are prime and semi-prime $\Gamma$-rings. In this paper, we consider the concept of derivations on prime and semi-prime $\Gamma$-rings and we study some of their properties.

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.

Let R be a 6-torsion free semiprime *-ring, \tau an endomorphism of R, \sigam an epimorphism of R and f : R ! R an additive mapping. In this paper we proved the following result: f is a generalized Jordan ( \sigma,\tau) *-¡derivation if and only if f is a generalized Jordan triple ( \sigma,\tau) *-derivation.

Let R be a 2-torsion-free prime ring with center Z(R), F a generalized derivation associated with a nonzero derivation d, L a Lie ideal of R. If (d(u)l1F(u)l2d(u)l3F(u)l4…F(u)lk)n=0 for all u ∈ L, where l1, l2, … , lk are fixed non-negative integers not all zero, and n is fixed positive integer, then L ⊆ Z(R). We also examine the case when R is a semiprime ring. Finally, we apply the above result to Banach algebras.

Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i)d(x)◦F(y) = 0 for all x, y ∈ I. (ii) [d(x),F(y)] = 0 for all x,y ∈ I. (iii) d(x)◦F(y) = x◦y for all x, y ∈ I. (iv) d(x)F(y) − xy ∈ Z(R) for all x, y ∈ I. We also discuss (alpha,beta)-derivations of sigma-prime rings and prove that if G is an (alpha,beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = on I.