Problem: find indecomposable injective modules and indecomposable projective modules over $k[x]/ \langle x^n \rangle. (n\geq 2)$.
From this post, I learned that any indecomposable module over $k[x]/ \langle x^n \rangle$ is of the form $k[x]/ \langle x^i \rangle, 1\leq i \leq n$. Hence the problem reduces to: pick out the injective modules and projective modules from $k[x]/ \langle x^i \rangle, 1\leq i \leq n$.
I could determine that $k[x]/ \langle x^n \rangle$ is projective, since it's a free module. And from this post $k[x]/ \langle x^n \rangle$ is also an injective module.
What about the rest? Any help is appreciated.
Best Answer
First, every finitely generated indecomposable projective module of a finite-dimensional algebra $A$ is a summand of the regular module $A\in\mathrm{mod}A$. Think about what this means in your case.
Second, if $I$ is an injective module and $f:I\to M$ an injective morphism then, by the definition of an injective module, there exists a morphism $g:M\to I$ such that $gf = \mathrm{id}_I$. You can apply this to the inclusions $M_i\to M_{i+1},\: [1]\mapsto [x]\;$ (where $M_i:=k[x]/\langle x^i\rangle$) to prove non-injectivity of many of the modules $M_i$. Then you should be able to sort out the rest of it for yourself.