I am trying to solve the following problem, so some help would be appreciated.
Prove that the ring $R=k[t^2,t^3]/(t^4)$ is Artinian and find a composition series for the ring.
Where $k$ is a field and 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.
So for the first part I noticed that, $R= \mathrm{span}_k([1],[t],[t^2],[t^3])$, considering $R$ as a $k$ vector space.
An ideal $\mathfrak{a} \unlhd R$ is a $k$ vector subspace, considering considering $R$ as a $k$ vector space. So a descending chain of ideals,is a descending chain of vector subspaces which it has to stabilize since R is finite dimensional. This I think proves that $R$ is Artinian.
For the second part, I don't now how to proceed. I thought that somehow I would manipulate a chain of vector subspaces into a composition series of ideals, but I don't see a reasonable way to do so.
On second thought, I tried to manipulate the ring $$\frac{k[x,y,z]}{(x-z^2,y-z^3,z^4)} \simeq R$$ but it seems even more complicated.
Putting the above problem in a broader context I would like to introduce my question:
Let the ring $R$ be a $k$ algebra, such that $R$ is finite dimensional $k$ vector space and thus a Artinian ring. Is there a method to find a composition series of ideals for the ring $R$?