Let $K$ a field, $n$ a non-negative natural number, and $f$ an irreducible polynomial, How to show that every decreasing ideals $A = K[X]/(f^n)$ sequence is stable.
I tried by saying that such sequence is non stable , for every $k,l;k<l$ we have $a_k \subset a_l$ , taking the detail of that the $n$ is fixed, but i couldn't reach any conclusion.
Thank you for your time ! Sorry for my english !
Big hint: If $d$ is the degree of $f$, then the quotient is $nd$-dimensional over $K$. Actually, all you care about is that it is finite dimensional over $K$, because each ideal is also a $K$ subspace.
Take it from here.