Does the belt that charges the battery and runs the power stering have anything to do with the A/C as well?

A/C stopped working a while back. Was told I needed a new compressor and some other parts to fix it. Meanwhile my car which usually runs very quiet and smooth began running rough and made a terrible scraping noise when I started it but this dissipated after a few moments. I recently started my car and it gave me a message that the charging system needs servicing. I checked under the hood and saw that the belt on the power steering side was loose. I pulled on it and it came off. It was broken in half. Now the car sounds perfect but my battery is not being charge and I have no power steering. Obviously this belt does more than that. I assume it is the alternator belt too. Can someone explain all of this to me. Thanks in advance!