Prove that if every principal left ideal of a ring R is generated by an idempotent, the ring R is regular.
2026-03-26 07:33:13.1774510393
Principal Ideal of a regular ring
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For an arbitrary $x$, if $Rx=Re$ for an idempotent $e$, you get $rx=e$ and $x=se$.
Two trivial computations later, you verify that one of the elements written here already shows $x$ is regular. The set of candidates here is so small you should have no trouble finding it.