Definition:Center (Abstract Algebra)/Ring

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This page is about the center of a ring. For other uses, see Definition:Center.

Definition

The center of a ring $\struct {R, +, \circ}$, denoted $\map Z R$, is the subset of elements in $R$ that commute with every element in $R$.

Symbolically:

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


That is, the center of $R$ is the centralizer of $R$ in $R$ itself.


It is clear that the center of a ring $\struct {R, +, \circ}$ can be defined as the center of the group $\struct {R, \circ}$.



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.


Sources