Ring over which nonzero polynomials define nonzero functions

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 $$f(a) \ne 0$$.
 * The natural map from $$R[x]$$ to the ring of functions from $$R$$ to itself is injective.