Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$ e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0. $$
I am trying to find the composition series for the regular $A$-module (as vector space over itself with action given by the multiplication).
It is easy to find that $Z(A)$ is one-dimensional and that all the nonzero idempotents of $A$ are $1,e+ps+qt$ and $1-e-ps-qt$ for $p,q \in k$. I was trying to explore the decomposition $$ A=Ae \oplus A(1-e), $$ but this gave me nothing. Can you please suggest how to find the composition series (my goal is actually to find all the simple $A$-modules up to isomorphism, so I decided that this is an equivalent problem).