Exercise 12, § 1

Let $A$ be a ring, $M$ an $A$-module. (a) Let $P \in A[X]$. Prove that if the kernel of the homothety of ratio $P$ in $M[X]$ is not zero, it contains a nonzero element of $M$ (let $Q$ be an element of $M[X]$ of minimal degree such that $PQ = 0$; if $P = a_0 X^p + \ldots + a_p$, $Q = m_0 X^q + \ldots + m_q$, observe that $a_0 Q = 0$, then by induction that $a_i Q = 0$ for all $i$). (b) Let $J$ be an ideal of finite type of $A$. Prove that the relation $\mathrm{prof}_A(J;M) > 0$ is equivalent to the existence of an $M[X]$-regular element in $JA[X]$. (c) Let $I$ and $J$ be ideals of finite type of $A$. Prove the equality $\mathrm{prof}_A(IJ;M) = \min(\mathrm{prof}_A(I;M), \mathrm{prof}_A(J;M))$ (reason by induction on $\mathrm{prof}_A(IJ;M)$, using $b$).