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In this paper, we obtain several results on generalized derivations and partial generalized automorphisms in prime rings. Also, some examples are given to show that the restrictions imposed on the hypothesis of the various results cannot be omitted.

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C the center of U, F a non-zero generalized derivation of R and L a non-commutative Lie ideal of R. Suppose that there exists $$0\ne a\in R$$ such that $$a(u^s[F(u),u]u^t)^n=0$$ for all $$u \in L$$ , where $$s\ge 0, t\ge 0, n\ge 1$$ are fixed integers. Then either $$F(x)=\alpha x$$ for all $$x\in R$$ with $$\alpha \in C$$ or R satisfies $$s_4(x_1,\ldots ,x_4)$$ , the standard identity in four variables, and $$F(x)=bx+xb+\alpha x$$ for all $$x\in R$$ , for some $$b\in U$$ and $$\alpha \in C$$ .

Let \(R\) be a prime ring of characteristic different from \(2\) and \(k\ge 1\) fixed positive integer. If \(R\) admits a generalized derivation \(G\) associated with a deviation \(d\), here we study the following cases: (1) \(G([r_1,r_2]_k)\circ ([r_1,r_2]_k)=0\) (2) \(G([r_1,r_2]_k)\circ G([r_3,r_4]_k)=[r_1,r_2]_k\circ [r_3,r_4]_k\) (3) \(G([r_1,r_2])\circ _k G([r_3,r_4])=0\) (4) \(G([r_1,r_2])\circ _k G([r_3,r_4])=[r_1,r_2]\circ _k [r_3,r_4]\). We obtain a description of the structure of \(R\) and information on the form of \(G\) in terms of the commutativity of \(R\) and the multiplication by a specific element from the extended centroid of \(R\).

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 ) l 1 F ( u ) l 2 d ( u ) l 3 F ( u ) l 4 … F ( u ) l k ) 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 *-prime ring with involution *, L ≠ 0 be a square closed *-Lie ideal of R and α, β automorphisms of R commuting with *. An additive mapping F: R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)-derivation d such that F(xy) = F(x)α(y) + β(x)d(y) holds for all \({x, y \in R}\). In the present paper, we shall show that \({L\subseteq Z(R)}\) such that R is a *-prime ring admits a generalized (α, β)-derivation satisfying several conditions, but associated with an (α, β)-derivation commuting with *.