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

linear-algebra ring-theory ideals cardinals

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?


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 $<\kappa$.

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