## On rings with prime centers.(English)Zbl 0814.16029

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.

### MSC:

 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras) 16N60 Prime and semiprime associative rings 16P20 Artinian rings and modules (associative rings and algebras)

### Keywords:

prime center; semiprime center
Full Text: