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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,2012-05-04
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    Thanks Matt, I believe you're right. He says "representation" when he means "finite-dimensional representation" elsewhere in these notes.2012-05-04

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