Ring over which every monic 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 monic polynomial has finitely many roots if every 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.

Relation with other properties

Stronger properties

• Ring over which every nonzero polynomial has finitely many roots
• Finite ring
• Integral domain