I shall prove some properties on a polynomial ring $R[x]$ over a commutative ring $R$, and there are two with which I struggle:
For some $f=a_0+a_1x+\cdots+a_nx^n\in R[x]$,
$f$ is invertible in $R[x] \Leftrightarrow a_0$ is invertible, and $a_1,...,a_n$ are nilpotent.
$fg$ is primitive $\Leftrightarrow f$ and $g$ are primitive, where primitive means: $(a_0,...,a_n)=(1)$.
The first one is quite intuitive and $a_0$ must be invertible since the first coefficient of the unit-polynomial shall be $1$, but I do not follow the nilpotency of the other coefficients.
Can anyone help?