I'm reading the following paper: http://math0.bnu.edu.cn/~huwei/paper/Holm-Hu-1.pdf
On page 795 and 796 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon $A_n:=k[x,y]/\langle x^2,xy^{n+1},y^{n+2}\rangle$.
All the $A_n$ modules defined there should be indecomposable.
Question: Is there an easy way to prove this?
Of course, one could show that the endomorphism ring is local, but that seems to end in a lot of extensive computations.
Can we use anyhow that $A_n$ is local and commutative, or the fact that $x$ and $y$ are nilpotent?
Thanks for the help.