The voltage regulator is internal to the engine's processor.
The processor works with the alternator to provide proper vehicle voltage.
The battery light is probably coming on due to lower amperage and voltage because of a faulty alternator. Check here first.
The battery light can only come on due to low amperage or voltage, or a faulty wiring connector/wiring that leads to the instrument cluster from the alternator electrical connector.
Anything else is not a valid cause.
Testing the charging system, you should have a peak amperage of probably 50-80 amps at 2000 RPM, and peak voltage of about 13.9 - 14.4 at 2000 RPM. If both figures are not in those ranges, you have an alternator issue.