Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Define $R \subset \mathcal{O}$ to be the set of all $\mathbb{Z}$-linear combinations of units. Since the product of two units is a unit, the set $R$ is a ring.

**Question** : Under what circumstances do we have $R = \mathcal{O}$?

Of course, this holds for $\mathcal{O} = \mathbb{Z}$. It is also clearly false for imaginary quadratic number rings aside from the Gaussian integers. But it is true for all the real quadratic number rings I have played with. Maybe it holds whenever $\mathcal{O}$ has infinitely many units? This might be too optimistic...

In case the above has a wild or overly complicated answer, the following question might be easier.

**Question** : Under what circumstances is $R$ finite-index in $\mathcal{O}$ as an abelian group?