Inverse of Central Unit of Ring is in Center

Theorem

Let $R$ be a ring.

Let $\map Z R$ denote the center of $R$.

Let $u \in R$ be a unit of $R$.

Then:

$u \in \map Z R \implies u^{-1} \in \map Z R$

Proof

Follows directly from the definition of center and Inverse of Unit in Centralizer of Ring is in Centralizer.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Theorem $21.5$: Corollary