# Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.

I know that if $V$ is a vector space over a field $k,$ then

1. $\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$
2. $\operatorname{End}(V)$ 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?

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If $\text{dim}(V)$ is infinite, then for each infinite cardinal $\kappa\leq\text{dim}(V)$ we have the ideal of endomorphisms whose ranges have dimension $<\kappa$.
Yes: if $I$ is an ideal and contains an endomorphism whose range is dimension $\lambda$, then it contains all endomorphisms with range of dimension $\lambda$ (just compose with an appropriate isomorphism between the ranges). –  Arturo Magidin Feb 16 '12 at 16:58