I've been stuck on a problem, and I was wondering if anyone could help me out. The problem is:
Let $R$ be the $2 \times 2$ matrix ring over the reals $\mathbb{R}$ of the form $ \begin{bmatrix}a & b \\0 & c\end{bmatrix}, $ where $a, b, c \in \mathbb{R}$. Find an idempotent $e$ in $R$ such that $eRe$ is a field, but the right ideal $eR$ is not minimal.
I was thinking of using $e=\begin{bmatrix}0 & 1 \\0 & 1\end{bmatrix}$, which is idempotent. I also showed $eRe$ is a field, but I'm not sure how to show the right ideal $eR$ is not minimal.
If this $e$ doesn't work, I also tried $e=\begin{bmatrix}1 &0 \\0 & 0\end{bmatrix}$, but once again, I'm not sure how to show $eR$ is not minimal.
Any help would be greatly appreciated. Thanks!