$R$ is a commutative ring with $1$, prove that there exist epimorphism from $R[x]$ onto $R$.

Question

$R$ is a commutative ring with $1$, prove that there exist epimorphism from $R[x]$ onto $R$.

I maybe able to show that R[x] onto R is a homomorphism but I'm not sure how to show that it is onto and how will I use the fact that $R$ is a commutative ring with $1$ to prove the statement? Please can someone help.

Answer 1

Consider the map

$$ev_0: R[X] \to R: P \mapsto P(0)$$

Clearly this is a ring homomorphism and this is surjective since $ev_0(R) = R$.

In fact, evaluating in every other ring element works equally well.