Center of a ring - Imported Knowlpedia Content

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

Z(R)=\{\,z\in R:\forall r\in R,\; zr=rz\,\}.

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

Remarks

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.