check the belt tension on the alternator first, if it is loose the belt will slip and cause this. pay attention to the pulleys also, if it is worn even a new belt will have a problem and may slip.
then get a vom and check the output of the alternator, it should be between 13.2 - 14.8v. If it is any lower then it's either a loose belt or a faulty alternator.
There are also some fuse links in the harness that run under the intake/spark plug area, check those if you do not find anything wrong elsewhere, especially if your vom readings for good alternator output are low or non-existent.
I would perform a load test on the battery too.

Not having any model information makes it harder but a general would be check the alternator first both idle and running. Also check the voltage cold and hot. Another thing i would check would be the fuse in the harness from the alternator to battery if it has one. If all that checks out then see if you can make the battery light come on while testing. If the battery light comes on but the battery and alternator voltages are correct then you have an issue in the wire harness and the ecm is seeing a voltage drop.