# Definition:Center (Abstract Algebra)

## 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:

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

## 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.