In abstract algebra those would be rings with zero divisors. But all my early algebra education was in integral domains (ring without zero divisors like the integers) or fields which are even nicer.

I was talking mainly about the simple case presented above.

x*y can be 0 (mod 6) but I don't think it takes knowing abstract algebra and a deep knowledge of modulo and axioms to figure that out. I hope math teachers that don't know abstract algebra know that!