Battery light on dashboard will not go off.

Is it dim or bright when still on? The battery dash light in our 14 yr old truck is always on dimly since the truck was new. It goes bright at start then drops to dim when truck starts. It went bright when we did have a real battery problem one time. On second alternator and still like that. It must be some "leakage" thing as mentioned above.
Mex went through the four wire plug in the alternator a couple years ago with me on this too.

So IMO the OP needs to worry if the light stays bright but not if it acts properly but just stays dim instead of going right out.