C172 pilot reported alternator failure inflight resulting in a return to the departure airport.

Narrative

Conducting a training flight with a pre-solo Student when we encountered an alternator failure while in cruise flight. This was determined by an aural annunciation of the low voltage annunciation and a visual indication of an electrical discharge on the main battery as indicated on the MFD. We immediately headed for ZZZ airport and informed them of a potential electrical failure and eventual loss of comms due to low battery voltage. I elected to shed electrical load by turning off all exterior lights and turn off the MFD to extend remaining battery life as much as possible. We made it back on the Ground at ZZZ and taxied back to Parking without further incident or complete electrical failure. No light guns were used as we were able to retain radio communication for the rest of the flight. After post flight assessment of the aircraft there were no external indications of an alternator or electrical failure. The alternator belt was appropriately tensioned and showed no signs of wear. This failure could have been a direct failure of the alternator or the solenoid may have been intermittently failing. This is going to be determined by a Licensed Mechanic and will be remedied soon.