# Definition:Center (Abstract Algebra)

*This page is about centers in the context of Abstract Algebra. For other uses, see Center.*

## Definition

### Semigroup

The **center of a semigroup** $\struct {S, \circ}$, denoted $\map Z S$, is the subset of elements in $S$ that commute with every element in $G$.

Symbolically:

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

### Group

The **center of a group** $G$, denoted $\map Z G$, is the subset of elements in $G$ that commute with every element in $G$.

Symbolically:

- $\map Z G = \map {C_G} G = \set {g \in G: g x = x g, \forall x \in G}$

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

### Ring

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:

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

## Also see

- Definition:Centralizer
- Results about
**centers**(in the context of abstract algebra)**can be found****here**.

## 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.