Definition:Center (Abstract Algebra)/Ring
This page is about Center in the context of Ring Theory. For other uses, see Center.
Definition
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:
- $\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$
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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 semigroup $\struct {R, \circ}$.
Also see
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.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains