I've been studying composition series. I've been struggling with problems related to
finding composition series and composition factors.
I feel like there have been very limited examples in a lot of the material I have
encountered. In saying this could someone help me compute (and explain how you do it) the following composition series
and composition factors for the $k[x]$-module $k[x]/\langle x^3-x^2\rangle$ where $k$ is a field.
Note that $x^3-x^2 = x^2(x-1)$, and that $x^2$ and $x-1$ are coprime in $k[x]$, because $1 = x^2 -(x+1)(x-1)$. Hence, we can apply the Chinese Remainder Theorem and get $$ A:= k[x]/(x^3-x^2) \cong k[x]/(x^2) \times k[x]/(x-1). $$ Writing $k(\lambda) := k[x]/(x-\lambda)$ as a $k[x]$-module (where $\lambda\in k$), it is clear that $k[x]/(x^2)$ has composition factors $k(0), k(0)$, and $k[x]/(x-1) = k(1)$. Hence, $A$ has composition factors $k(0), k(0), k(1)$.
If you want to find an explicit composition series, then note that $\overline{x}^2 \in A$ corresponds to $(0,1)$ on the right hand side of the above isomorphism. Hence, for $M_1 := (\overline{x}^2) \cong k(1)$, we have $A/M_1 \cong k[x]/(x^2)$. Next, $M_2 := (\overline{x}^2, \overline{x})$ satisfies $M_2/M_1 \cong (x)/(x^2) \cong k(0)$ and $A/M_2 \cong (k[x]/(x^2))/(\overline{x}) \cong k(0)$. Therefore, one composition series is given by $$ (\overline{x}^2) \subseteq (\overline{x}^2, \overline{x}) \subseteq A. $$