Do automorphisms of quotient fields preserve the underlying ring?

Question

Suppose $R$ is an integral domain. Let $\gamma: \mathrm{Frac}(R) \rightarrow \mathrm{Frac}(R)$ be an automorphism of the quotient field. Is it true that $\gamma(R) = R?$ I don't think that this is true in general, but I cannot think of any examples. Are there examples for which this is not true?

Answer 1

No, a field automorphism $\gamma: \text{Frac}(R)\to\text{Frac}(R)$ need not map $R$ to $R$. Take $R = \mathbb{Q}[x]$ and $\gamma: \mathbb{Q}(x) \to \mathbb{Q}(x)$ defined by preserving $\mathbb{Q}$ and by $\gamma(x) = 1/x$.