Exercise 2, § 4

Let A be a local ring, m its maximal ideal, $k = A/m$ its residue field and $f : A \to k$ the canonical homomorphism. Let $P \in A[X]$ be a monic polynomial of degree n. We write $B = A[X]/P.A[X]$ and denote by $x$ the canonical image of X in B.

(a) Let Q, $Q'$ be two strongly relatively prime monic polynomials in $A[X]$ such that $P = QQ'$. Show that the ring B is the direct sum of the ideals $B.Q(x)$ and $B.Q'(x)$.

(b) Conversely, let $B = b \oplus b'$ be a decomposition of B as a direct sum of two ideals. Show that there exist polynomials Q, $Q'$ in $A[X]$ satisfying the hypotheses of (a) and such that $b = B.Q(x), b' = B.Q'(x)$. (Show first that $b/mB$ and $b'/mB$ are generated by the images in $B/mB$ of monic polynomials $Q_0 \in \tilde{f}(b), Q'_0 \in \tilde{f}(b')$) such that

Let $r = \deg(Q_0), s = \deg(Q'_0)$. Show that b is a free A-module with basis $Q_0(x), xQ_0(x), \ldots, x^{s-1}Q_0(x)$ (apply Chapter II, § 3, no. 2, Proposition 5); we may then write $x^s Q_0(x) = a_0 Q_0(x) + a_1 x Q_0(x) + \cdots + a_{s-1} x^{s-1} Q_0(x)$, where $a_i \in A$ ($0 \leq i \leq s-1$); show that, if we write then $\tilde{f}(P) = \tilde{f}(Q_0) \tilde{f}(Q')$ and $Q' \in \tilde{f}(b')$. Similarly define Q starting with $Q'$ and $Q_0$ and show that Q and $Q'$ solve the problem, using Proposition 2 of no. 1.)