[wz1000]

> Does this really require knowing abstract algebra? Seems obvious to anyone doing any sort of multiplication that if the output is 0 then one of variables/functions has to be 0

No, this is only true when you are doing multiplication in an integral domain.

https://en.wikipedia.org/wiki/Integral_domain

There are many rings where this is not true, like Z/nZ where n is not prime.

    2*2 = 0 (mod 4)
    2 /= 0  (mod 4)

[posterboy]

mod should be part of the signature of the ring, so you are not looking at pure multiplication. Then the domain would be still integral. Why wouldn't it?

[opportune]

An integral domain is any commutative ring with no zero divisors (a,b are zero divisors if ab = 0 but a != 0, b != 0). In this case 2*2 = 0, so 2 is a zero divisor, so Z/4z = Z_4 is not an integral domain.