Let $R$ be a commutative semiring, and consider the property that for all $x \in R$, the set of divisors of $x$ only has finitely many elements.
What are some conditions that will guarantee such a property in a ring (and more generally, in a semiring)? This seems to be a kind of ``discreteness'' condition, for instance, $\mathbb{Z}$ satisfies it, whereas $\mathbb{Q}$ does not -- polynomial rings built over $\mathbb{Z}$ and their quotients should also satisfy this property, for instance.
One of my thoughts is that perhaps this is related to the discreteness property $p + q = 1 \implies$ either $p$ or $q$ is equal to $0$, or something similar, but I haven't seen anything like this in the literature. Is there a name for these sorts of semirings? Do they have a nice classification theorem?