3
$\begingroup$

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!

$\endgroup$
3
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.