I'm trying to prove that if a polynomial over a commutative ring is nilpotent, then all its coefficients are nilpotent.
Let $f= \sum_{i=0}^n a_i x^i$.
I'm using induction on the degree, and then by subtracting $f-\sum_{i=0}^{n-1} a_i x^i = a_n x^n$, we get that $a_n x^n$ is nilpotent.
I'm having a hard time showing this implies that $a_n$ is nilpotent, and the only thing i can think of is showing that $x$ is a unit in a larger ring of the formal polynomials of the form $\sum_{i=-m}^n a_i x^i$.
Am i missing some super-simple proof?