Exercise 3, § 1

Let $A$ be the local ring $k[[X, Y]]$ and $M$ the direct sum of the $A$-modules $A/\alpha$, where $\alpha$ runs through the set of the nonzero principal ideals of $A$. Now prove that one has $\mathrm{prof}_A(M) = 1$, but that there does not exist an $M$-regular element in $m_A$.