Let $\mathcal{O}_K$ be the ring of integers of some number field $K$.
It happens that $\mathcal{O}_K$ might not have unique factorization, but...
- We can form the multiplicative group of ideals of $\mathcal{O}_K$
- It has unique factorization
- This construction doesn't seem to be a ring
- Each ideal can be put into the form $(\alpha,\beta)$ with both $\alpha,\beta \in \mathcal{O}_K$
I think the ideal $(\alpha,\beta)$ represents the gcd of $\alpha$ and $\beta$ (analogous to field of fractions) so why can't we build a new ring out of the algebraic integers which has gcd closed and unique factorization?