Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} R&R\\ xR&R \end{pmatrix}$.
Question: what are the left ideals of the ring $A$?
We have $A=M_1\oplus M_2$ for the two projective left $A$-modules $M_1=\begin{pmatrix} R\\ xR \end{pmatrix}$ and $M_2=\begin{pmatrix} R\\ R \end{pmatrix}$.
So for example we can take a left $A$-submodule $N_1$ of $M_1$, say $N_1=\begin{pmatrix} I\\ xJ \end{pmatrix}$ for two ideals $I,J\subset R$. Then $AN_1\subset N_1$ gives us $xJ\subset I\subset J$, doing the same with $M_2$ we get left ideals of the form $N=\begin{pmatrix} I&K\\ xJ&L \end{pmatrix}$ with four ideals $I,J,K,L\subset R$ satisfying $xJ\subset I\subset J$ and $xK\subset L\subset K$.
Are these all left ideals (I doubt that)? Or can you write down more? I tried to use the knowledge of the form of left idelas in $M_2(R)$, but that does not help much, because we dont have Morita equivalence between $A$ and $R$ in this case because of the $xR$ factor in $A$.
Another idea of mine was to look at the map $"x=0"$: $f: \begin{pmatrix} R&R\\ xR&R \end{pmatrix} \rightarrow \begin{pmatrix} k[[y]]&k[[y]]\\ 0&k[[y]] \end{pmatrix}$. Then look at left ideals in this ring of upper trinagular matrices and study their preimages, maybe this gives new ideals?
Any help or idea is very welcome.