Exercise 21, § 2

Let $A$ bearing, $F$ a monic polynomial of $A[X]$, $B$ the ring $A[X]/F.A[X]$, $m$ a maximal ideal of $A$, $k = A/m$ the quotient field and $\overline{F}$ the canonical image of $F$ in $k[X]$.

(a) Let $\overline{F} = \prod_{i=1}^n f_i^{e_i}$, where the $f_i$ are distinct irreducible polynomials of $k[X]$. Let $F_i \in A[X]$ be a monic polynomial such that its canonical image in $k[X]$ is $f_i$. Let $\mathfrak{M}_i$ denote the ideal $mB + F_i.B$ of $B$; show that the $\mathfrak{M}_i$ are distinct and are the only maximal ideals of $B$ containing $mB$ (note that $B/mB$ is an algebra of finite rank over $A/m = k$ generated by the roots of $\overline{F}$).

(b) For all $i$ let us write $\Omega_i = mB + F_i^{e_i}.B$. Show that the $\Omega_i$ are $\mathfrak{M}_i$-primary in $B$ and that $mB = \Omega_1 \cap \cdots \cap \Omega_n = \Omega_1 \Omega_2 \cdots \Omega_n$.

(c) For B to be a local ring, it is necessary and sufficient that A be a local ring and that $\overline{F}$ be a power of an irreducible polynomial of $k[X]$.