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 determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J\left(M_2\left(R\right)\right)$, or $I_2-A\in J\left(M_2\left(R\right)\right)$, or $A$ is similar to $\left(\begin{array}{cc}0& \lambda \\ 1& \mu \end{array}\right)$, where $\lambda \in J\left(R\right)$, $\mu \in 1+J\left(R\right)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J\left(R\right)$ and a root in $1+J\left(R\right)$. We further prove that $f\left(x\right)\in R\left[\right[x\left]\right]$ is strongly $J$-clean if $f\left(0\right)\in R$ be optimally $J$-clean.

A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in \mathbb{N}$. We prove that $M_n\left(R\right)$ is nil clean if and only if $R/J\left(R\right)$ is Boolean and $M_n\left(J\left(R\right)\right)$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J\left(R\right)$ is ${\mathbb{Z}}_3$, $B$ or ${\mathbb{Z}}_3\oplus B$ where $B$ is a Boolean ring, and that $M_n\left(R\right)$ is weakly nil clean if and only if $M_n\left(R\right)$ is nil clean for all $n\ge 2$.

Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in D^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$