center (rings)

If $A$ is a ring, the center of $A$, sometimes denoted $\mathrm{Z}(A)$, is the set of all elements in $A$ that commute with all other elements of $A$. That is,

$$\mathrm{Z}(A)=\{a\in A\mid ax=xa\forall x\in A\}$$

Note that $0\in \mathrm{Z}(A)$ 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 $A$.