We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi R\to R$ defined by $\psi \left(x\right)=T\left(x\right)x-xT\left(x\right)$ for all $x\in R$ is a free action. We also show that for a generalized $(\alpha ,\beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha ,\beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha +d.$ Furthermore, we prove that for a centralizer $f$ and...

Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F\left(u\right){[F\left(u\right),u]}^n=0$ for all $u\in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: (1) ...

Let $R$ be a prime ring and $I$ a nonzero ideal of $R.$ The purpose of this paper is to classify generalized derivations of $R$ satisfying some algebraic identities with power values on $I.$ More precisely, we consider two generalized derivations $F$ and $H$ of $R$ satisfying one of the following identities: (1) ...

Generalized radical rings (braces) were introduced for the study of set-theoretical solutions of the quantum Yang-Baxter equation. We discuss their relationship to groups of I-type, virtual knot theory, and quantum groups.

Let $R$ be a prime ring with center $Z\left(R\right)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d\left(Z\right(R\left)\right)\ne 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F\left(xy\right)\pm xy\in Z\left(R\right)$, (ii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),y]\in Z\left(R\right)$, (iii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),F(y\left)\right]\in Z\left(R\right)$, (iv) $F(x\circ y)\pm F\left(x\right)\circ F\left(y\right)\in Z\left(R\right)$, (v) $\left[F\right(x),y]\pm [x,F(y\left)\right]\in Z\left(R\right)$, (vi) $F\left(x\right)\circ y\pm x\circ F\left(y\right)\in Z\left(R\right)$ for all $x,y\in I$, then $R$ is commutative.