Category:Centers (Abstract Algebra)
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This category contains results about center in the context of Abstract Algebra.
Definitions specific to this category can be found in Definitions/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:
- $\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$