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 $O\_K$?



Multiplicative monoid of ring modulo units
For general (commutative) rings $R$, we can construct $M = R \setminus \{0\}$ and $\sim$ as you did. It's not too hard to see (as GH mentioned) that in this general situation, $a \sim b$ if and only if they generate the same ideal, so $M/\sim$ will be isomorphic to the monoid of nonzero principal ideals of $R$.


Again, as GH mentioned, the fact that $\mathbb{Z}[\phi]$ is a PID means that this is free of infinite rank, generated by the primes. For general number rings $\mathcal{O}\_K$, it should have finite index in some free monoid (that of the nonzero integral ideals) by finiteness of the class number. Being inside a free monoid should tell you it's torsion free and that no non-identity elements have inverses, but I don't know if we can get anything interesting from this, or if we can get anything extra from the finite index.


