Well it is simple.

If you used (schools) materials and school time (on or off time) - yes the school should get a cut.

Keywords being = school materials and or (made) for the school to use.

I remember a case where the UH of Hawaii sued a student over patents to the tune of 11 million bucks. This was bacause they could prove a student did creat a patent under school time (lab work) and with his professor education <- no school materials were used.

The court decided that the lab work = using the schools equipment was indeed schools (material) and thus sided with the school.