General aviation instructor pilot reported a near miss with another aircraft that was transiting the non-towered airport traffic pattern without communicating during a training flight. The instructor took control and maneuvered to avoid the other aircraft before re-entering the traffic pattern.

Narrative

I was flying in the traffic pattern with a student in Aircraft X. I had just extended upwind to make room for an aircraft that was entering the pattern for landing. As we rolled out of our (right) crosswind turn; I caught sight of another aircraft dangerously close in front of me (the first aircraft was turning base at the time and not involved). I immediately pulled power to idle and executed a left turn to avoid the traffic before re-entering the pattern. I checked ADS-B data; and the other aircraft was Aircraft Y flying from ZZZ to ZZZ1. The other pilot was at pattern altitude; not on frequency or making radio calls; flying straight to our airport before he saw us and also turned. If pilots want to fly directly over the airport; they should be communicating. If they do not want to communicate; they should avoid the airport. Honestly; I think whoever the other pilot was needs to be retrained on those kinds of procedures and communication. Additionally; I think it would be helpful if public airports had a small protected zone around them where air traffic is not allowed unless they are on the airport's CTAF frequency. I would not have to be large; perhaps a 2nm radius circle up to 1;500 or 2;000 AGL. I know this would be a big change to the national airspace system; but I think it would improve safety.