Equivalence of a integer domain

Question

If $R$ is a commutative ring with $1$.

Suppose that for all polynomial $P(X)\in{R[X]\setminus{R}}$ has at most $n$ roots, with $n=grad\ (f)$ then $R$ is an integer domain.

Any suggestion, please.

Answer 1

Suppose to the contrary that the ring is not an integral domain. Then there are ring elements $a$, $b$, neither equal to $0$, such that $ab=0$.

Then the polynomial $ax$ has at least $2$ roots, but has degree $1$.