REM (n.1)

also R.E.M., rem, unit for measuring ionizing radiation, 1947, acronym of roentgen equivalent man.

REM (n.2)

also R.E.M., rem, 1957, initialism (acronym) for rapid eye movement.

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Definitions of REM from WordNet

rem (n.)
a recurring sleep state during which dreaming occurs; a state of rapidly shifting eye movements during sleep;
Synonyms: paradoxical sleep / rapid eye movement sleep / REM sleep / rapid eye movement
rem (n.)
(roentgen equivalent man) the dosage of ionizing radiation that will cause the same amount of injury to human tissue as 1 roentgen of X-rays;