I've proven that $R=\text{End}_k(k^2)$ where $k$ is a field, is simple. But out of this, I think that this should imply that $\text{End}_k(k^2)$ is a division ring, which is obviously false, but what is wrong in my reasoning.
If $R$ is simple, then $AR= R $ for any $A \in R$. So let's take an non-invertible matrix $A$, so then $AB \neq I$ for any $B$. But.... $I \in R$, so also $I \in AR$, so $I= A \cdot B$ for some $B$. But that is impossible !
For non-commutative rings (such as the endomorphism ring you are considering) simple means no non-trivial two-sided ideals. On the other hand, non-commutative simple rings can have non-trivial left or right ideals, and this is what you have just observed.