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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.

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  • $\begingroup$ Morita equivalence does not "see" left or right ideals, so it was probably a dead end anyway. $\endgroup$ – rschwieb Jun 8 '15 at 20:34
  • $\begingroup$ Yes you are right. Maybe it helps, if one looks at ideals in the ring of upper triangular matrices to find different ideals? I aded some lines about this. $\endgroup$ – Bernie Jun 9 '15 at 10:24
  • $\begingroup$ maybe. It is an interesting question :) It is not "far" from Morita theory. Have you ever heard of a "Morita context ring"? $\endgroup$ – rschwieb Jun 9 '15 at 12:48

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