Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio. We define the equivalence relationship $x\sim y$ iff $x =uy$ for some unit.

Is there a good description of the quotient $M/\sim$? My initial thought was to do $a+b \phi \sim a+b$ since $\phi \sim 1$ but this is just plain wrong.

Is this also known for other integral rings $O_K$?