Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form $k[x^p,y^p,f]$, where $f\in k[x,y]\setminus k[x^p,y^p]$.

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 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) ...

A ring $R$ is called a right $\mathrm{PS}$-ring if its socle, $\mathrm{Soc}\left(R_R\right)$, is projective. Nicholson and Watters have shown that if $R$ is a right $\mathrm{PS}$-ring, then so are the polynomial ring $R\left[x\right]$ and power series ring $R\left[\right[x\left]\right]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R\left[[x^{-1};\alpha ,\delta ]\right]$ and the skew polynomial ring $R[x;\alpha ,\delta ]$, where $R$ is an associative ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Our results extend and unify many existing...

In this note, for a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha$-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if $T_n\left(R\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz if and only if $R\left[x\right]/\left(x^n\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz ring for any positive integer $n$.

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $$2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right)$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $$\begin{array}{c}2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒢\left(x^{n-1}\right),\\ 2𝒢\left(x^n\right)=𝒢\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right),\end{array}$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. ...

Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a{[\delta \left(x^n\right),x^n]}_k=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.

An element in a ring $R$ is called CJ if it is of the form $c+j$, where $c$ belongs to the center and $j$ is an element from the Jacobson radical. A ring $R$ is called CJ if each element of $R$ is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show...

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) ...