units in a polynomial ring

1. Proposition

Let \(A\) be a commutative ring, \(A[X]\) the polynomial ring and \(f = \sum_{i=0}^{n} \alpha_i X^i \in A[X]\). TFAE:

\(\alpha_0 \in A^{\times}\) and \(\alpha_1,...,\alpha_n \in \mathcal{N}_A\) are nilpotent
\(f \in A[X]^{\times}\) is a unit

Suppose there exists an \(f^{-1} = \sum_{i=0}^{n} b_i X^i\) Then

\begin{align*} f^n \cdot f^{-1} = \end{align*}