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...$

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]).

Let $m\ge 0,r\ge 0,s\ge 0,q\ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y\in R$ there exist polynomials $f\left(X\right)\in X^{2}ZZ\left[X\right],g\left(X\right),h\left(X\right)\in XZZ\left[X\right]$ such that $\{1-g\left(y{x}^m\right)\}[x,x^{r}y-x^{s}f\left(y{x}^m\right)x^q]\{1-h\left(y{x}^m\right)\}=0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.