Mind bending started kicking in now. To solve equations such as (x+2)(x-2) = 0, we know x could be either 2 or -2, premised on the universal default assumption that only zero product another part leads to zero. But it is not necessary true in abstract algebra!

In the integer module 12 ring, to solve an equation x^2 + 5^x + 6 = 0, we found there are 4 solutions 1, 6, 9, 10 more than only two 10 and 9.

In this case, two get a product of two elements to be zero, neither need to be zero:

There are various numbers to be zero divisors, meaning 2 can divide 0 by 6, like in our elementary school math, we know 2 can divide 6 by 3. What if the ring is Integers mod 11 (Z/11Z)? it has no “zero divisors” because 11 is prime.

Here the mathematicians coined another new notion called “Integral Domain” to denote a commutative Ring with 1 as multipliable identity and NO zero divisors. If it’s not specifically commutative, it’s called “domain”.

In math, cancellation property are frequently used. However if it is not an integral domain, the cancellation may not be working. Only Integral domain allows one to use cancellation property safely.