What was found
WHILE CRUISING AT 38,000 FEET IN AND OUT OF CLOUDS NO PRECIPITATION OR TURBULENCE. RIGHT ENGINE FIRE INDICATION PUSH BUTTON ILLUMINATED AND "FIRE" ANNUNCIATOR IN THE N1 INDICATION CAME ON. THE FLIGHT CREW PERFORMED MEMORY ITEMS OF: THROTTLE TO IDLE AND WAIT FOR THE LIGHTS TO EXTINGUISH - THEY DID NOT. THE CREW MOVED THE THROTTLE TO CUT OFF AND THE LIGHTS EXTINGUISHED. COMMUNICATED TO CENTER THEY WANTED TO START A DECENT AND CHANGED THEIR ARRIVAL AIRPORT FROM 27K TO KLEX FOR GROUND AND SAFETY SUPPORT REASONS. THE ARM BUTTON WAS PRESSED THAT TURNED FUEL AND HYDRAULICS OFF AND ARMED THE SQUIB IF THEY NEEDED TO BLOW THE BOTTLE. FIRE BOTTLE WAS NOT BLOWN AS THE FIRE INDICATION WAS NO LONGER PRESENT. RIGHT ENGINE REMAINED SHUT DOWN FOR THE REMAINDER OF THE FLIGHT. THE CREW RAN THE "ONE ENGINE INOPERATIVE" APPROACH CHECK LIST IN ITS ENTIRETY. LANDED AT KLEX AND WAS ALERTED THAT BOTH ENGINES SEEMED NORMAL. FIRE CAME TO THE AIRCRAFT TO PERFORM THERMAL SCANNING AND A VISUAL INSPECTION WITH NO INDICATIONS OF A CURRENT OR HISTORY OF FIRE TAKING PLACE. WE BELIEVE THIS TO BE A FALSE FIRE INDICATION.