Ring over which every nonzero polynomial has finitely many roots

Definition
Let $$R$$ be a commutative unital ring. We say that $$R$$ is a ring over which every nonzero polynomial has finitely many roots if for every nonzero $$f(x) \in R[x]$$, the set:

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

is a finite set.

Stronger properties

 * Weaker than::Finite ring
 * Weaker than::Integral domain

Weaker properties

 * Stronger than::Ring over which every monic polynomial has finitely many roots
 * Stronger than::Ring with at most n nth roots of unity

Related properties

 * Ring over which nonzero polynomials define nonzero functions: Any infinite ring over which every nonzero polynomial has finitely many roots has the property that nonzero polynomials define nonzero functions.

Analogues in other algebraic structures
Here are some analogous properties in groups:


 * Groupprops:Group with at most n elements of order dividing n