Fuel injector

With key on should have power at all injectors
ECM pulse ground on all injectors to turn them on and off
if electrically the injector forced to be open all the time I would suspect bad ECM
check with a noid light if light on all the time suspect bad ECM if not suspect bad injector
good luck

The injector staying on all the time would indicate that the trigger wire for that injector is shorted to ground. This could be happening in the harness or could be due to a fault within the PCM itself. You could try locating that wire at the PCM and cutting it for testing and see if that shuts the injector off. If so the PCM is bad, if not locate the shorted wire in the harness. When that wire gets spliced back together, it need to be carefully soldered and sealed up with shrink tubing.