Consider a real quadratic field K. Let M be a complete Z-module in K.
I would like to see a proof that it can be multiplied by a totally positive number $x$ so that $$xM=\mathbb{Z} + w\mathbb{Z}$$ where $w \in K$ is such that $$0<\bar{w}<1<w.$$
Moreover, is there a specific method to find such a $w$?