The canonical example of a ring is in fact $\Bbb{Z}$. That's what ring theory abstracts, the integers...
Primes are just $ab \in (p) \implies a \in (p)$ or $ b \in (p)$. If true, then $p$ is prime and vise versa.
which is actually exactly what it looks like
These $n\Bbb{Z}$ are precisely the ideals of the ring $\Bbb{Z}$.
They are subgroups that absorb elements multiplicatively
I think if anything, elegance will come from abstract algebra, not analysis, but most results in NT are done in analysis, kind of annoying!
Usually both fields though
