I came across the following problem
Let $k$ be a field and $R=k[t^2,t^3]/(t^4)$. The ring $k[t^2,t^3]$ is defined as the set $\{f(t^2,t^3)|f \in k[x,y]\}$ with the obvious ring structure.
- prove that the ring R is Artinian
- find ideals $A_0, \dots , A_n \unlhd R$ such that $$R=A_0 \supseteq \cdots \supseteq A_n$$ with each $A_i/A_{i+1}$ be a simple $R$ module.
for the first part, I know that in a short exact sequence of R-modules, the middle R-module is Artinian iff the two others are.
I thing that $R \simeq k[x,y,z]/(x-z^2, y-z^3, z^4)$ but I don't now how is this helpful.
for the second part I don't have a single clew how to proceed.
Any hints, useful remarks or theorems that may help are appreciated.
If we denote by $t^n$ the class of $t^n$ in $R$, the elements of $R$ can be written as $a_0+a_2t^2+a_3t^3+a_5t^5$, with $a_i\in{k}$, and a composition series is $0\subset(t^5)\subset(t^2)\subset(t^2,t^3)$.
Observe that $R$ is not a principal ideal ring; indeed the ideal $(t^2,t^3)$ is not principal.
Otherwise, the ring $k[x,y,z]/(x-z^2,y-z^3,z^4)\cong{k(z)/(z^4)}$ is a principal ideal ring; so it is not isomorphic to $R$.