I know that if $V$ is a vector space over a field $k,$ then
- $\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$
- $\operatorname{End}(V)$ has exactly one non-trivial ideal if $\dim V=\aleph_0.$
Do we know how many ideals $\operatorname{End}(V)$ can have when $\dim V>\aleph_0?$ Can we describe them?