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.



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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.



Proving indecomposability of special modules
Hello. 


I believe that the endmorphism ring may not be hard to compute in this case.


Let $R = k[x,y]$ where $k$ is a field, and let $I = (x^2, xy^n, y^{n-1})$. Then $End\_R(R/I) = Hom\_R(R/I,R/I) \cong Hom\_{R/I} (R/I,R/I) \cong R/I$. But $R/I$ is a local ring since $\sqrt{I} = (x,y)$ which is a maximal ideal in $R$.


You may also want to take a look at Section 2 of <http://arxiv.org/abs/1212.2912> by Yi Zhang. It discusses indecomposibility of a module in terms of a presentation matrix of the module.


