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Let $\mathcal X$ be a real or complex locally convex vector space
and \( {\mathcal L}_c(\mathcal X)\) denote the ring (in fact the
algebra) of continuous linear operators on $\mathcal X$. In this
note, we characterize certain one-sided ideals of the ring \(
{\mathcal L}_c(\mathcal X)\) in terms of their rank-one idempotents.
We use our main result to show that a one-sided ideal of the ring
of continuous linear operators on a real or complex locally convex
space is triangularizable if and only if the one-sided ideal is
generated by a rank-one idempotent if and only if ${\rm rank}(AB-BA)
\leq 1$ for all $A, B$ in the one-sided ideal. Also, a description
of irreducible one-sided ideals of the ring \( {\mathcal
L}_c(\mathcal X)\) in terms of their images or coimages will be
given. (The counterparts of some of these results hold true for
one-sided ideals of the ring of all right (resp. left) linear
transformations on a right (resp. left) vector space over a general
division ring.)
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