If I had this "in front of me" I could probably figure it
This could easily be a regulator, and I THINK it could be a bad diode in the alternator as well.
Is this a "relay" regulator or a solid state replacement?
First thing I'd do (if it's relay/ mechanical) is turn start the engine, and give the regulator a "whack" and see if the light goes out.
Check that all regulator wiring is intact.
Then pull the regulator connector off -- I believe the lamp should go out. If it does not, you have a wiring issue.
Replace the regulator.