Category:Definitions/Centers (Abstract Algebra)

This category contains definitions related to center in the context of Abstract Algebra.
Related results can be found in Category:Centers (Abstract Algebra).

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:

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