# Category:Definitions/Centers (Abstract Algebra)

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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:

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

## Pages in category "Definitions/Centers (Abstract Algebra)"

The following 7 pages are in this category, out of 7 total.