Center of a ring

Let $R$ be a ring . The center of $R$ is

It is a subring of $R$, and in fact a commutative ring .

The center measures how far $R$ is from being commutative: $R$ is commutative iff $Z(R)=R$. The center also controls constructions like the opposite ring and scalar actions on modules and representations.

Examples:

• If $k$ is a field and $n\ge 1$, then $Z(M_n(k))$ consists of scalar matrices and is isomorphic to $k$.
• The center of the quaternion division ring $\mathbb{H}$ is $\mathbb{R}$.
• In $M_2(k)$, the matrix $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ is not central since it does not commute with all matrices.