Battery Light always on

There is a key piece of information missing-- The model year of your Miata! Different model years & NA vs. NB use different schemes and hardware to control the Alternator. In the old days, from your complaint, it would be assumed that your regulator is suspect, or there is a wiring problem. Some years use an alternator that is controlled by of all things, the instrument panel module. Others have a separate regulator. There are some variations that make the alternator year or part year specific. I don't even want to get into differences between US, European, Japaness, and other market area variations. It's also quite possible that a blocking diode is bad, and the real problem is to find it. Finally, some Mazda models will turn on the alternator light if a brake light bulb is bad or have a poor connection. It's best to have a wiring diagram in hand when chasing this sort of problem, unless you enjoy being frustrated.