Ring of linear transformations modulo finite rank transformations

Question

Let $ K $ be a field and $ V $ be a vector space of countable dimension (infinite) over $ K $, and let $ L = L (V) $ be the vector space of $ K $-linear transformations on $ V $. Let $ I $ be the subset of $ U $ consisting of the transformations of finite rank. Clearly $ I $ is a bilateral ideal of $ L $.

a) Show that $L/I$ is simple ring.

b) Show that $L/I$ is not artinian. (left or right)

c) Show that $L/I$ is not noetherian. (left or right)

Answer 1

Hints:

1. Show $I$ is maximal. You can show that any transformation not of finite rank generates the whole ring.

2. Show $L$ is von Neumann regular, that is, for every element $a$, there is an $x$ such that $axa=a$.

3. Show $L/I$ is infinite dimensional over $k$. If $L/I$ were Artinian on either side, it would be semisimple and hence finite dimensional.

4. $L/I$ is also von Neumann regular, and such rings are Noetherian iff Artinian.