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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!

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up vote 3 down vote accepted

Your last $e$ works: $$ e=\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$ Then $eRe$ consists of the matrices $\begin{bmatrix}a&0\\0&0\end{bmatrix}$, so it is a field.

We have $$ eR=\{\begin{bmatrix}a&b\\0&0\end{bmatrix}:\ a,b\in R\} $$ is a right ideal, and it contains the right ideal $$ J=\{\begin{bmatrix}0&b\\0&0\end{bmatrix}:\ b\in R\}. $$ As $J\subsetneq eR$, $eR$ cannot be minimal as a right ideal.

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Thank you! I knew it was simple, but I Wasn't seeing it. – Maria Nov 30 '12 at 15:49

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