# Definition:Center (Abstract Algebra)/Ring

*This page is about Center in the context of Ring Theory. For other uses, see Center.*

## Definition

The **center of a ring** $\struct {R, +, \circ}$, denoted $\map Z R$, is the subset of elements in $R$ that commute under $\circ$ with every element in $R$.

Symbolically:

- $\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$

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

It is clear that the **center of a ring** $\struct {R, +, \circ}$ can be defined as the center of the semigroup $\struct {R, \circ}$.

## Also see

## Historical Note

The notation $\map Z S$, conventionally used for the **center** of a structure $\struct {S, \circ}$, derives from the German **Zentrum**, meaning **center**.

## Linguistic Note

The British English spelling of **center** is **centre**.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling **center**, but it is appreciated that there may be lapses.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains