Ring over which every nonzero polynomial has finitely many roots

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

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.

Relation with other properties

Stronger properties

• Finite ring
• Integral domain

Weaker properties

• Ring over which every monic polynomial has finitely many roots
• 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