Units with no finite prime-union complement ($\mathbb{Z}\setminus {\mathbb{Z}}^{\times }={\bigcup }_p(p)$)

In $\mathbb{Z}$, the saturated multiplicative subset $\{-1,1\}$ has complement equal to the union of $(p)$ over all positive primes. No finite subcollection of prime ideals covers this complement: a prime outside the finite list is a nonunit not in any listed ideal.