Inverse of Central Unit of Ring is in Center
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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