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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 with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds: (i) F (xy) ± xy ∈ Z(R) (ii) F ([x, y]) ± [F (x), y] ∈ Z(R) (iii) F ([x, y]) ± [F (x), F (y)] ∈ Z(R) (iv) F (x ο y) ± F (x) ο F (y) ∈ Z(R) (v) [F (x), y] ± [x, F (y)] ∈ Z(R) (vi) F (x) ο y ± x ο F (y) ∈ Z(R) for all x, y ∈ I, then R is commutative.

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 prime ring, I a nonzero ideal of R, and a ∈ R. Suppose that σ is a nontrivial automorphism of R such that a{(σ(x ∘ y))n − (x ∘ y)m} = 0 or a{(σ([x,y]))n − ([x,y])m} = 0 for all x,y ∈ I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative.

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

Let R be a prime ring with center Z(R), J a nonzero left ideal, \(\alpha \) an automorphism of R and R admits a generalized \((\alpha ,\alpha )\)-derivation F associated with a nonzero \({(}\alpha ,\alpha {)}\)-derivation d such that \(d(Z(R))\ne (0)\). In the present paper, we prove that if any one of the following holds: \(\textit{(i)}\) \(F([x,y])-\alpha ([x,y])\in Z(R)\) (ii) \(F([x,y])+\alpha ([x,y])\in Z(R)\) (iii) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) (iv) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) for all \(x,y\in J\), then R is commutative. Also some related results have been obtained.

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y])m = [x, y]n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)]m = [x, y]n for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

In this paper, we present some results concerning orthogonal generalized ()-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.

The concept of $\Gamma$-ring is a generalization of ring. Twoimportant 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, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u)ⁿ = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s₄; the standard identity in four variables. Нехай R — просте кільцє, L — нецентральний ідеал Лі та f — ненульова напівпохідна, асоційована з автоморфiзмом a таким, що f(u)ⁿ=0 для всіх uЄL, де n — фіксоване натуральне число. Якщо Char R>n+1 або Char R=0, то R задовольняє стандартну тотожність s₄ у чотирьох змінних.

WOS: 000338514500017, Let (r, *) be a 2-torsion free *-prime ring with involution * and center Z (R), U a nonzero square closed *-Lie ideal of R. An additive mapping F W R! R is called a generalized derivation if there exits a derivation d WR R such that F. xy/DF. x/yCxd. y/. In the present paper, we prove that U * Z. R/if any one of following conditions holds: 1) O F. u/; u _ D 0; 2) O d. u/; F. v/_ D 0; 3) d. u/oF. v/D 0; 4) O d. u/; F. v/ D * O u; v; 5) d. u/oF (v) D * uov; 6) d. u/F. v/* uv 2 Z. R/; for all u; v 2 U: Furthermore, an example is given to demonstrate that the *-primeness hypothesis is not superfluous.