I wanted to ask about the hint given in Atiyah-Macdonald, Exercise 9.2. The question is as follows:
Let $A$ be a Dedekind domain. If f = $a_0 + a_1x+\cdots+ a_nx^{n}$ is a polynomial with coefficients in $A$, the content of $f$, $c(f)$, is the ideal $(a_0,...,a_n)$. Show that $c(fg)=c(f)c(g)$. (Hint: Localize at each maximal ideal.)
So if we are lucky, $A$ just so happen to be local; I proved the claim (assuming my proof is correct). So my guess is that the hint is trying to tell us to reduce to the case of local $A$. But then why not localize around each prime $P$ ? Why insist on localizing about $\mathfrak{m}$ of $A$ ?
Cheers