Center of Ring is Commutative Subring

Theorem

The center $\map Z R$ of a ring $R$ is a commutative subring of $R$.

Proof

Follows directly from the definition of center and Centralizer of Ring Subset is Subring.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Theorem $21.5$: Corollary
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $2$