R[x] is an integral domain iff R is

Statement

Let $R$ be a commutative ring. Then the polynomial ring $R[x]$ is an integral domain if and only if $R$ is an integral domain.

Remarks

If $R$ is an integral domain and $f, g \neq 0$, then the leading coefficient of $f g$ is not zero, so $f g \neq 0$. Conversely, $R$ is a subring of $R[x]$, and a subring of an integral domain is an integral domain. By induction, the ring $R[x_1, \dots, x_n]$ of polynomials in several variables over an integral domain is an integral domain.

Explanations

L1 human

The product of two nonzero polynomials over the integers is never zero. The reason is that the leading coefficients multiply to a nonzero number. So the polynomial ring keeps the integral domain property of its coefficients.