Proper integrally closed subring has infinite index

Statement

Suppose $S$ is a commutative unital ring and $R$ is a proper integrally closed subring of $S$. Then, $R$ has infinite index in $S$: in other words, the quotient group $S/R$ is infinite.

Proof

Given: A ring $S$, a proper integrally closed subring $R$ of $S$.

To prove: $R$ has infinite index in $S$.

Proof: Suppose $R$ has finite index, say $r$, in $S$. Let $x$ be an element in $S\setminus R$. Consider the elements $x,x^2,x^3,\dots ,$. Since there are only $r$ cosets of $R$ in $S$, there must exist $m>n\ge 1$ such that $x^m,x^n$ are in the same coset. Let $a=x^m-x^n$. Then, $a\in R$, so $x$ satisfies the monic polynomial in $R[x]$:

$x^m-x^n-a=0$.