Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1615–1622
We introduce a class of rings obtained as a generalization of rings with prime centers. A ring $R$ is called weakly prime center (or simply $WPC$) if $ab \in Z(R)$ ($R$) implies that $aRb$ is an ideal of $R$ where $Z(R)$ stands for the center of $R$. The structure and properties of these rings are studied and the relationships between prime center rings, strongly regular rings, and WPC rings are discussed, parallel with the relationship between the $WPC$ and commutativity.