Is restriction of a ring automorphism a subring automorphism

Question

$\phi:R\to R$ is a ring automorphism.

$S \subset R$ is a subring and $\phi(S)\subseteq S$

Is $\phi|_S$ necessarily an automorphism of $S$ ?

I can easily check that $\phi|_S:S\to S$ is a homomorphism and injective, but is it necessarily surjective?

Answer 1

No. Let $R$ be the polynomial ring in infinitely many variables $\{x_n\mid n\in\mathbb{Z}\}$, $S$ the polynomial subring with only the variables $\{x_n\mid n\in\mathbb{N}\}$, and the automorphism satisfying $\phi(x_n)=x_{n+1}$.