---
tag: 0B5Z1EC66J
kind: statement
role: theorem
name: R[x] is an integral domain iff R is
msc: [13F20]
uses: [0Q9EZS90GY, 09S3PFBEPM, 0T233AZ1GE]
sources:
  - url: https://en.wikipedia.org/wiki/Polynomial_ring
    locator: Properties that pass from R to R[X]
---

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is an [[integral domain|09S3PFBEPM]] 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|0CES91M79A]] 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.