Ring over which every monic polynomial has finitely many roots

Definition
Let $$R$$ be a commutative unital ring. We say that $$R$$ is a ring over which every monic polynomial has finitely many roots if every fact about::monic polynomial over $$R$$ has finitely many roots, i.e., for any monic polynomial:

$$f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_0$$,

the set:

$$\{ a \in R \mid f(a) = 0\}$$

is finite.

Stronger properties

 * Weaker than::Ring over which every nonzero polynomial has finitely many roots
 * Weaker than::Finite ring
 * Weaker than::Integral domain