Stabilizer of module in real quadratic number field

Question

Let $K$ be a real quadratic number field and $M \subseteq K$ be a $\mathbb Z$-module of rank 2. Why is there a non-trivial $\varepsilon \in K$ with $M = \varepsilon M$. With non-trivial I mean $\varepsilon \ne \pm 1$.

Answer 1

The set $\mathcal{O}$ of $\alpha\in K$ with $\alpha M\subseteq M$ is an order in $K$, that is a subring of the form $\Bbb Z[\beta]$ with $\beta\notin\Bbb Z$. An order in a real quadratic field has a unit $\varepsilon$ of infinite order (this is a power of the fundamental unit, which lies in the ring of integers of $K$.) Then $\varepsilon M=M$ and $\varepsilon\ne\pm1$.