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...

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let $R$ be a noncommutative prime ring of characteristic different from two and let $S$ and $T$ be left centralizers on $R$. Suppose that $\left[S\right(x),T(x\left)\right]S\left(x\right)+S\left(x\right)\left[S\right(x),T(x\left)\right]=0$ is fulfilled for all $x\in R$. If $S\ne 0$$(T...$

The main result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping. Suppose that $T\left(xyx\right)=xT\left(y\right)x$ holds for all $x,y\in R$. In this case $T$ is a centralizer.

Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵\left(𝒢\right)$ be the center of $𝒢$. Suppose that $𝔮:𝒢\times 𝒢\to 𝒢$ is an $ℛ$-bilinear mapping and that ${𝔗}_{𝔮}:𝒢\to 𝒢$ is a trace of $𝔮$. The aim of this article is to describe the form of ${𝔗}_{𝔮}$ satisfying the centralizing condition $[{𝔗}_{𝔮}\left(x\right),x]\in 𝒵\left(𝒢\right)$ (and commuting condition $[{𝔗}_{𝔮}\left(x\right),x]=0$) for all $x\in 𝒢$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) ${𝔗}_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.