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.

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 $n\ge 2$.