center (rings)CenterOfARing2002-06-07 03:53:182002-06-09 12:18:22Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereIf $A$ is a ring, the center of $A$, sometimes denoted $\operatorname{Z}(A)$, is the set of all elements in $A$ that commute with all other elements of $A$. That is,
$$\operatorname{Z}(A) = \{ a \in A \mid ax = xa \text{} \forall x \in A \}$$
Note that $0 \in \operatorname{Z}(A)$ so the center is non-empty. If we assume that $A$ is a ring with a multiplicative unity $1$, then $1$ is in the center as well. The center of $A$ is also a subring of $A$.