Search

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 \(R\) 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 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, 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 of characteristic different from \(2\) and \(3\), \(L\) a non-central Lie ideal of \(R\), \((d,\sigma )\) a nonzero skew derivation of \(R\), \(n\) a fixed positive integer. If \([d(x),x]^{n}=0\) for all \(x\in L\), then \(R\) satisfies \(s_{4}\).

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) oF(y) = 0 for all x, y ∈ I. (ii) [d(x), F(y)] = 0 for all x, y ∈ I. (iii) d(x) o F(y) = x o 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 = beta on I.