Definition:Center (Abstract Algebra)/Ring

This page is about the center of a ring.

For the center of a Group, see Center of a Group.

For the center of a Circle, see Center of a Circle.

Definition
The center of a ring $\left({R, +, \circ}\right)$, denoted $Z \left({R}\right)$, is the subset of elements in $R$ that commute with every element in $R$.

Symbolically:
 * $Z \left({R}\right) = C_R \left({R}\right) = \left\{{x \in R: \forall s \in R: s \circ x = x \circ s}\right\}$

That is, the center of $R$ is the centralizer of $R$ itself.

It is clear that the center of a ring $\left({R, +, \circ}\right)$ can be defined as the center of the group $\left({R, \circ}\right)$.