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 $$\left({R, \circ}\right)$$.