Determine $Soc(R_R)$ for $R =\left(\begin{smallmatrix} K & K \\ 0 & K \end{smallmatrix}\right)$

Question

Let me continue the study I began in a previous post on the ring of upper triangular matrices:

We consider $K$ a field and the ring $R = \begin{pmatrix} K & K \\ 0 & K \end{pmatrix}$. Recall that $e_{11},e_{22}$ are non-central idempotents. Prove that:

3. $e_{12}R$ is simple. $e_{12}R \cong e_{22}R?$
4. Let $a \in K \setminus \{0\}$ then $(e_{12}+ae_{22})R$ has dimension $1$ over $K$. $(e_{12}+ae_{22})R \cong e_{22}R?$
5. If the previous two points are true, determine $Soc(R_R)$ (the socle).

My approach

3. To show that $e_{12}R$ is simple I use the same argument as in point 2 (of the previous post). For the isomorphism, it seems that the mapping $\begin{pmatrix} 0 & 0 \\ 0 & k \end{pmatrix} \mapsto \begin{pmatrix} 0 & k \\ 0 & 0 \end{pmatrix}$ is a module isomorphism (bijective, preserves sums and scalars) between $e_{22}R$ and $e_{12}R$.

4. It seems that the dimension here is referring to the fact that matrices in $(e_{12}+ae_{22})R$ are written $\begin{pmatrix} 0 & 1 \\ 0 & a \end{pmatrix}K$. For the isomorphism, it seems that the mapping $\begin{pmatrix} 0 & 1 \\ 0 & a \end{pmatrix}k \mapsto \begin{pmatrix} 0 & 0 \\ 0 & k \end{pmatrix}$ is a module isomorphism (bijective, preserves sums and scalars) between $(e_{12}+ae_{22})R$ and $e_{22}R$.

5. We have that $R = \begin{pmatrix} K & K \\ 0 & 0 \end{pmatrix} \oplus \begin{pmatrix} 0 & 0 \\ 0 & K \end{pmatrix}$ and then using proposition 9.19 of Rings and Categories of Modules, by Anderson and Fuller, we would have that $Soc(R) = Soc(\begin{pmatrix} K & K \\ 0 & 0 \end{pmatrix}) \oplus Soc(\begin{pmatrix} 0 & 0 \\ 0 & K \end{pmatrix})$ but the second submodule is simple and for the first, we can reason by imposing that scalar multiplication is closed that the only simple submodule is $\begin{pmatrix} 0 & K \\ 0 & 0 \end{pmatrix}$. Therefore, $Soc(R) = \begin{pmatrix} 0 & K \\ 0 & K \end{pmatrix}$.

How does 3.4. help me to find $Soc(R_R)$?

Problems with linked questions

Question 1 is more general than my question so there are things to clarify:

1. In this context minimal ideals coincide with the notion of simple submodules.

Answer 1

Yes, one way to see 3 is that if you take the unit $u=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ in $R$, then $x\to ux$ is a right $R$ linear map from $e_{22}R\to e_{12}R$. And what's more, it is its own inverse map. So yes, they are isomorphic.

For 4, you are close, but I think you should be using $u=\begin{bmatrix}0&a\\1&0\end{bmatrix}$ for which you can compute the inverse also. The map you gave is not an invertible mapping (and I suspect may be a typo.)

This will be helpful for computing the socle, yes, and I will also refer to the links I gave above for a ground-up computation of the socle:

Socle of the well-known triangular matrix ring

The simple modules of upper triangular matrix algebras.