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In this paper, we define a set including of all f(a) with a is an element of R generalized derivations of R and is denoted by f(R). It is proved that (i) the mapping g : L (R) -> f(R) given by g (a) = f(-a) for all a is an element of R is a Lie epimorphism with kernel N-sigma,N-tau; (ii) if R is a semiprime ring and sigma is an epimorphism of R, the mapping h : f(R) -> I (R) given by h(f(a)) = i(sigma)(-a) is a Lie epimorphism with kernel 1 (f(R)); (iii) if f(R) is a prime Lie ring and A, B are Lie ideals of R, then [f(A), f(B)] = (0) implies that either f(A) = (0) or f(B) = (0).

Let R be a *-prime ring with characteristic not 2, sigma,tau : R -> R be two automorphisms, U be a nonzero *-(sigma, tau)-Lie ideal of R such that tau commutes with *, and a,b be in R. (i) If a is an element of S*(R) and [U, a] - 0, then a is an element of Z (R) or U subset of Z (R) : (ii) If a is an element of S* ( R) and [U,a](sigma),(tau) subset of C-sigma,C-tau, then a is an element of Z (R) or U subset of Z (R). (iii) If U not subset of Z (R) and U not subset of C-sigma,C-tau, then there exists a nonzero *-ideal M of R such that [R, M](sigma, tau) subset of U but [R, M](sigma,tau) not subset of C-sigma,C-tau . (iv) Let U not subset of Z (R) and U not subset of C-sigma,C-tau . If aUb = a*U b = 0, then a = 0 or b = 0

Let R be a sigma-prime ring with characteristic not 2, Z (R) be the center of R, I be a nonzero sigma-ideal of R, alpha, beta : R -> R be two automorphisms, d be a nonzero (alpha, beta)-derivation of R and h be a nonzero derivation of R : In the present paper, it is shown that (i) If d (I) subset of C-alpha,C-beta and beta commutes with sigma then R is commutative. (ii) Let alpha and beta commute with sigma. If a is an element of I boolean AND S-sigma (R) and [d(I), a](alpha,beta) subset of C-alpha,C-beta then a is an element of Z(R). (iii) Let alpha, beta and h commute with sigma. If dh (I) subset of C-alpha,C- beta and h(I) subset of I then R is commutative.

Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivation such that δD = Dδ and ΓD = DΓ(i)If D(N)⊂ Z; or [D(N);D(N)] = 0 or [D(N);D(N)]σ, γ= 0; then (N; +)is abelian. (ii) If N is 2-torsion free, d1 is a (α, γ)-derivation and d2 is a derivation on N such that d1d2(N) = 0, then d1 = 0 or d2 = 0.

WOS: 000448214700015, Let R be a (*)-prime ring with characteristic not 2, U be a nonzero (*)- (sigma, tau)-Lie ideal of R and d be a nonzero derivation of R. Suppose sigma, tau be two automorphisms of R such that sigma d = d sigma, tau d = d tau and * commutes with sigma, tau, d. In the present paper it is shown that if d(2)(U) = (0), then U subset of Z., Scientific Research Project Fund of Cumhuriyet University [F-514], This work is supported by the Scientific Research Project Fund of Cumhuriyet University under the project number F-514.