On rings with prime centers (Q1338417)

A ring \(R\) with center \(C\) is said to have prime center (resp. semiprime center) if \(ab \in C\) implies \(a \in C\) or \(b \in C\) (resp. \(x^ n \in C\) implies \(x \in C\)). After giving some basic results on these rings, the authors investigate sufficient conditions for such rings to be commutative; and they give an example of a ring with prime center which is not commutative. A sample result is the following: If \(R\) with 1 has prime center and for each \(x \in R\) there exists a monic polynomial \(f(t)\) with integer coefficients such that \(f(x) \in C\), then \(R\) is commutative.