Exercise 2, § 1

Let $A$ be a valuation ring of height 1, non-noetherian, $m$ its maximal ideal, $n$ a principal ideal of $A$, distinct from $(0)$ and from $A$. Exercise prove that one has $\mathrm{prof}_A(n; A) = 1$ and $\mathrm{prof}_A(m; A) \geq 2$, although one has $V(m) = V(n)$. (Observe that $\mathrm{Ext}_A^1(A/m, A/n)$ is isomorphic to $\mathrm{Hom}_A(A/m, A)$.)