In Etingof's notes entitled "Introduction to Representation Theory," he asks the reader to produce an example of an indecomposable module which is not cyclic (Problem 1.25(c)). The exercise even comes with a hint: if $A = k[x,y]/(x^2,xy,y^2)$, then he suggests $A^*$ with the standard $A$-action. But this seems unnecessarily complicated: why not simply use $(x,y) \subset k[x,y]$? Am I correct in thinking that this is an indecomposable ideal? Why would Etingof suggest the first example instead of this one?
Perhaps he just wants to use the complicated example later on, too? It might also be some weird difference between modules and representations.
What you say looks right to me. Any nonzero ideal of $k[x,y]$ will intersect any other nonzero ideal nontrivially since the ring is a commutative domain, and it's pretty clear $(x,y)$ isn't a cyclic module...