Narrow Results

Let $R$ be a noncommutative prime ring, with $\mathrm{\text{char}}(R)\ne 2,3$, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid.

Suppose that $F$ and $G$ are generalized skew derivations of $R$, associated with the same automorphism $\alpha$ of R, and $m\ge 1$ a fixed integer. If

Let $R$ be a noncommutative prime ring, with $char(R)=p$, and suppose that either $p=0$ or $p>0$ and $p\nmid (2^n-2)$. Let $Q_r$ be the right Martindale quotient ring of $R$, $C$ its extended centroid, $F$ a generalized skew derivation of $R$, $m\ge 1$ and $n\ge 2$ fixed integers. If

Let $R$ be a prime ring with the extended centroid $C$, $L$ a noncommutative Lie ideal of $R$ and $g$ a nonzero $b$-generalized derivation of $R$. For $x,y\in R$, let $[x,y]=xy-yx$. We prove that if $[[\cdots [[g(x^{n_0}),x^{n_1}],x^{n_2}],\dots ],x^{n_k}]=0$ for all $x\in L$, where $n_0,n_1,\dots ,n_k$ are fixed positive integers, then there exists $\lambda \in C$ such that $g(x)=\lambda x$ for all $x\in R$ except when $R\subseteq M_2(F)$, the $2\times 2$ matrix ring over a field $F$. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc.118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra42 (2014), 139–152.]

Let R be a non-commutative prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring and C its extended centroid. Suppose that L is a non-central Lie ideal of R, F a nonzero b-generalized skew derivation of R. If

• (a)there exists $\lambda \in C$ such that $F(x)=\lambda x$, for all $x\in R$;
• (b)$R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over C, and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$, for all $x\in R$.

Let $R$ be a noncommutative prime ring of characteristic different from $2$, with right Martindale quotient ring $Q_r$ and extended centroid $C$. Let $m,n\ge 1$ be fixed integers and $F$ a nonzero generalized skew derivation of $R$. In this paper, we investigate the set

Let R be a prime ring of characteristic different from 2 and 3, Q_r be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))ⁿ = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Q_r such that F(x) = xc and G(x) = cx; (b) R satisfies s₄ and there exist a, b, c ∈ Q_r such that F(x) = ax + xc, G(x) = cx + xb and (a − b)² = 0.

Let $R$ be a ring of characteristic different from $2$, $m,n,s\ge 1$ fixed positive integers, $L$ a noncentral Lie ideal of $R$ and $F:R\to R$ a nonzero generalized skew derivation of $R$.

We prove the following results:

• ^(a)If $R$ is prime and there exists $0\ne a\in R$ such that
  $$a{(F{(x)}^{m}F{(y)}^n-y^n{x}^m)}^s=0\forall x,y\in L$$
  then $R\subseteq M_2(K)$, the $2\times 2$ matrix ring over a field $K$.
• ^(b)If $R$ is semiprime and
  $${(F{(x)}^{m}F{(y)}^n-y^n{x}^m)}^s=0\forall x,y\in L$$
  then either $L$ centralizes a nonzero ideal of $R$ or $[s_4(x_1,\dots ,x_4),x_5]$ is a polynomial identity for $R$.