Let $f = \sum_{i=0}^n f_i x^i$ - an element of $R[x]$, where $R$ is a commutative ring. I want to prove that if $f$ is invertible if and only if $f_0$ is invertible and the rest of the coefficients $f_1, \ldots, f_n$ are nilpotent.
It's easy to prove that $f_0$ should be invertible. We consider $e = \sum_{i=0}^n e_i x^i$ such that $fe = 1$. Then we obtain following equations $$ f_0 e_0 = 1, \quad f_0 e_1 + f_1 e_0 = 0, \quad, \ldots, \quad \sum_{i=0}^n f_i e_{k-i} = 0. $$ Hence $f_0$ is invertible. But how to proof from there that $f_i^k = 0$ $(i \ge 1)$ for some $k$?
