Ring over which nonzero polynomials define nonzero functions

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

Suppose $R$ is a commutative unital ring. We say that $R$ is a ring over which nonzero polynomials define nonzero functions if it satisfies the following equivalent conditions:

• For any nonzero polynomial $f(x)\in R[x]$, there exists $a\in R$ such that $f(a)\neq 0$.
• The natural map from $R[x]$ to the ring of functions from $R$ to itself is injective.