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Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

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) *-iderivation 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 ) 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 prime ring with Utumi quotient ring U and m, n fixed positive integers. If R admits a generalized derivation F associated with a derivation d such that (F([x, y]))m = [x, y]n for all x, y ∈ R, then R is commutative or d = 0. Moreover, we also examine the cases R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a noncommutative version of the Singer–Wermer theorem.

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.

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, . Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and for all x, , then either R is commutative or n = 1, , R contains a non-zero central ideal and for all .

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.

Let R be a semiprime ring with center Z(R) and µ be a surjective homomorphism. In this paper, we prove that T is a µ-centralizer if one of the following holds: (i) T(x)µ(y) = µ(x)T(y) for all x;y 2 R, where T is a mapping. (ii) [T(x);µ(x)] = 0 for all x 2 R, where T is a left µ-centralizer. (iii) 2T(xyx) = T(x)µ(y)µ(x) + µ(x)µ(y)T(x) for all x;y 2 R, where T is an additive mapping, µ(Z(R)) = Z(R) and R is 2-torsion free. 1