Brauer's lemma states, that, given a ring $R$ and a minimal left ideal $I$ with $I^2 \neq 0$, there exists an idempotent $e \in R$ with $I = Re$ and $D = eRe$ is a division ring.
My question is, whether $e$ is a unique idempotent. If it is, how can I show this, if it is not, can you give a counter example?
Best Answer
No, $e$ is typically not unique. For instance, if $V$ is a vector space over some field and $W\subset V$ is a subspace of codimension 1, then the set $I\subseteq\operatorname{End}(V)$ of endomorphisms that vanish on $W$ is a minimal left ideal with $I^2\neq 0$. The idempotent $e$ can then be any projection whose kernel is $W$ (there is one such projection for every linear complement to $W$ in $V$).