# indecomposable module which is not cyclic

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?

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Dear Justin, He probably suggests the example you mention because it is finite dimensional over the ground field $k$, and so closer to the world of group rings and Lie algebras than your example, which is infinite dimensional over $k$. Regards, – Matt E May 4 '12 at 0:17
Thanks Matt, I believe you're right. He says "representation" when he means "finite-dimensional representation" elsewhere in these notes. – Justin Campbell May 4 '12 at 0:28

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...