TITLE: Nilpotent/invertible polynomial over commutative ring.
QUESTION [6 upvotes]: Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial over a commutative ring $R$. Prove that
(a) $p$ is unit in $R[x]$ iff $a_0$ is unit and $a_1,a_2,\ldots,a_n$ are nilpotent in $R$.
(b) $p$ is nilpotent in $R[x]$ iff $a_0,a_1,\ldots,a_n$ are nilpotent in $R$.

I have got no lead in part (a) and in part (b) I have proved the if part.
for the only if part can we use the fact that a polynomial is vanish everywhere iff all its coefficients are $0$?

REPLY [2 votes]: For the "only-if"-part of b) consider the first coefficent and show that it is nilpotent. Then use the fact that the nilpotent elements form an ideal to derive b) inductively.