For this ring homomorphism, show that $\ker f$ is a principal ideal and find an element that generates it

Question

Let $R$ be a commutative ring with identity and let $f : R[x] → R$ be a ring homomorphism defined by $f(r_0 + r_1x + · · · + r_nx^n) = r_0 + r_1 + · · · + r_n$. Prove that $\ker f$ is a principal ideal of $R[x]$ and find an element of $R[x]$ which generates $\ker f$.

I know $\ker f = \{r\in R:f(r) = 0\}$ and I know that $0 \in R = 0 \in R[x]$. Also here $f = 0$ iff $\sum_{i=0}^n r_i = 0$, but I have no idea how to proceed.

Answer 1

Let $p(x)=r_0 + r_1x + · · · + r_nx^n\in\operatorname{ker}(f)$. Then $p(1)= r_0 + r_1 + · · · + r_n=0$. Or $(x-1)\mid p(x)$. And any polynomial which can be written as $(x-1)g(x)$, belongs to $\operatorname{ker}(f)$. Hence $\operatorname{ker}(f)=\langle x-1\rangle$, the ideal generated by $x-1$.