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 is (weakly) nil clean provided that every element in is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let be abelian, and let . We prove that is nil clean if and only if is Boolean and is nil. Furthermore, we prove that is weakly nil clean if and only if is periodic; is , or where is a Boolean ring, and that is weakly nil clean if and only if is nil clean for all .
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ; where is a Boolean ring; local ring with nil Jacobson radical; or ; or the ring of a Morita context with zero pairings where the underlying rings are or .