Inverse of Unit in Centralizer of Ring is in Centralizer

Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $S$ be a subset of $R$.

Let $\map {C_R} S$ denote the centralizer of $S$ in $R$

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

Then:

$u \in \map {C_R} S \implies u^{-1} \in \map {C_R} S$

Proof

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

Let $u \in \map {C_R} S$.

Then from Commutation with Inverse in Monoid:

$u^{-1} \in \map {C_R} S$

$\blacksquare$

Sources

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