The ring $\mathrm{End}_D(V) $ is simple if $V$ is finite dimensional.

Question

Theorem: Let $V$ be an $n$-dimensional vector space over a division ring $D$. Then the rings $\mathrm{End}_D(V)$ and $ M_n(D^{\mathrm{o}})$ are isomorphic.

Remark: If $D$ is a division ring, then $M_n(D)$ is a simple ring for every $n \in\mathbb N$. We infer that the ring $\mathrm{End}_D(V)$ is simple if $V$ is finite dimensional.

From above theorem and remark we infer that the ring $\mathrm{End}_D(V) $ is simple if $V$ is finite dimensional.

Is the converse actually true?

Can we prove directly that $\mathrm{End}_D(V)$ is simple? (by definition)

Answer 1

No. The full ring of linear transformations of an infinite dimensional vector space is never simple.

It always has, at least, a nontrivial ideal made up of linear transformations with finite dimensional image.