Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

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...$

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 $n\ge 2$.

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.

Let $R$ be an associative ring with identity $1$ and $J\left(R\right)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[x,[x,y^m]]=0$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (ii) is replaced by (ii)’ $[{\left(yx\right)}^m{x}^m-x^m{\left(xy\right)}^m,x]=0$, for all $x,y\in R\setminus N\left(R\right)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).