Exercise 11, § 9

Let A be a commutative ring and B the ring A[X].

a) First, assume that A has no nonzero nilpotent elements. Let $f=\sum_ia_iX^i$ and $g=\sum_ib_iX^i$ be two elements of B. Prove that we have $a_pb_q= 0$ for $p+q >$ deg($f g$). b) Let $\mathfrak{N}$ be the nilradical of A (Exercise 6). Prove that a polynomial $f\in B$ is invertible in B if and only if its constant term is invertible in A and its other coefficients belong to $\mathfrak{N}$ (apply a) to the image of $f$ in $(A/\mathfrak{N})[X])$. c) Deduce that the radical of B is the ideal $\mathfrak{N}[X]$ and that it is equal to the nilradical of B.