# An indecomposable module

Suppose $A=k[x_1,\dots,x_n]$ and let $I$ be a proper ideal containing all monomials of degree $\geq N$. Then $A/I$ is an indecomposable $A$-module.

1. Suppose $A/I=U\oplus V$ and write $\overline{1}=\overline{u}+\overline{w}$ with $\overline{u}\in U$, $\overline{v}\in V$. It is easy to see that $\overline u,\overline v$ are orthogonal idempotents. If we suppose $\overline{u}$ and $\overline{v}$ are non-zero, it follows that both $u$ and $v$ have non-zero constant term, since otherwise $\overline{u}^N=\overline{v}^N=0$, contradicting their idempotence.
Let $\lambda\in k^\times$ be such that $u+\lambda v$ has zero constant term. Then
$0=(\overline{u}+\lambda\overline{v})^N=\sum_{j=0}^N \binom{N}{j} \overline{u}^j(\lambda\overline{v})^{N-j}=\overline{u}^N+(\lambda\overline{v})^{N}=\overline{u}+\lambda^N\overline{v},$