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Let $D$ be a division ring, ${\mathcal V}$ a right or left vector
space over $D$, and ${\mathcal L(\mathcal V)}$ the ring of all right
(resp. left) linear transformations on $\mathcal V$. We characterize
certain one-sided ideals of the ring $\mathcal L(\mathcal V)$ in
terms of their rank-one idempotents. We use our result to
characterize a division ring $D$ in terms of the one-sided ideals of
$M_n(D)$. Some other consequences are presented.
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