When is the torsion submodule of a $k[x,y]$-module a summand?

Question

Let $A=k[x,y]$ and $M$ be a finitely generated graded $A$-module. I want to know if the torsion submodule $T$ of $M$ is a direct summand.

Apparently, Kaplansky, Irving: A characterization of Prufer rings shows that if every finitely generated $A$-module contains its torsion submodule as a direct summand, then $A$ is a Prüfer domain. If I've got it right, then $k[x,y]$ is no Prüfer domain. So there must be a finitely generated module $M$ whose torsion submodule is no summand.

Which?

Answer 1

I would suggest that you try the following $M=Ae_1\oplus Ae_2/(xye_1-x^2e_2, y^2e_1-xye_2)$.