In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be very $J^{\#}$-clean in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen...

The purpose of this paper is to prove the following result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping, such that $2T\left(x^2\right)=T\left(x\right)x+xT\left(x\right)$ holds for all $x\in R$. In this case $T$ is left and right centralizer.

Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z\left(U\right)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $\left[F\right(u),u]F\left(u\right)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F\left(x\right)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F\left(x\right)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove that, if $F$...