Problem 1.24, Introduction to representation theory, Etingof Let $k$ be a field and $n$ and $N$ be two nonnegative integers. Let $A = k[x_1, \ldots, x_n]$, and let $I \neq A$ be any ideal in $A$ containing all homogeneous polynomials of degree $\geq N$. Show that $A/I$ is an indecomposable representation of $A$. 
 A: Okay, I might have solved this finally. I think this may be a possible solution, but twice before this I thought I had a solution that subsequently turned out to be completely bogus.
If $A/I = A_1 \oplus A_2$, then either $A_1$ or $A_2$ must contain an element with a nonzero constant term. Without loss of generality, take this to be $A_1$, and let this element be $a$.  Furthermore, without loss of generality, take the constant term to be 1.
By multiplying each homogeneous polynomial of degree $N-1$ by $a$ (which kills off all the non-constant terms), we see that $A_1$, an invariant subspace, must in fact contain all homogeneous polynomials of degree $N-1$.
Then we successively multiply each homogeneous polynomial of degree $N-k$ by $a$, for $k = 2, 3, \ldots$. This kills off all the terms of degree at least $k$, and all the terms of degree $1, 2, \ldots k-1$ go to homogeneous polynomials of degree $N-k+1, \ldots, N-1$ which are in $A_1$, so we may subtract them off. This leave the contribution from the constant term, the original homogeneous polynomial. So we can successively show that all homogeneous polynomials of degree $N-k$ are in $A_1$, and conclude that $A_1 = A/I$.
Therefore $A/I$ is indecomposable.
A: 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 $$0=(\overline{u}+\lambda\overline{v})^N=\sum_{j=0}^N \binom{N}{j} \overline{u}^j(\lambda\overline{v})^{N-j}=\overline{u}^N+(\lambda\overline{v})^{N}=\overline{u}+\lambda^N\overline{v},$$
which is obviously contradictory.
A: I guess you had the right idea but I didn't get all the details of what you wrote. (especially, is a a polynomial of degree 1?)
That essential idea is that $A/I$ has a unit, which is a cyclic vector. Then, in full generality, a representation with a cyclic vector is indecomposable.
(I also discovered that great lesson by etingof and I'm try to do the exercices!!)
