Definition:Center (Abstract Algebra)/Ring
This page is about the center of a ring.
For the center of a Group, see Center of a Group.
For the center of a Circle, see Center of a Circle.
Definition
The center of a ring $\left({R, +, \circ}\right)$, denoted $Z \left({R}\right)$, is the subset of elements in $R$ that commute with every element in $R$.
Symbolically:
- $Z \left({R}\right) = C_R \left({R}\right) = \left\{{x \in R: \forall s \in R: s \circ x = x \circ s}\right\}$
That is, the center of $R$ is the centralizer of $R$ in $R$ itself.
It is clear that the center of a ring $\left({R, +, \circ}\right)$ can be defined as the center of the group $\left({R, \circ}\right)$.
I think the group is not $R$, but $R^*$, to avoid finding an inverse for the zero under multiplication.
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Linguistic Note
The UK English spelling of this is centre.
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
