I have the feeling that this question will be another manifestation of grand-scale stupidity, but anyways ...
I'm trying to put it all together. In a ring which has zero divisors, what and when can we cancel, when trying to solve an equation?
1. An element is either zero or zero divisor.
2. If an element is unit, we can always cancel it.
1. If an element is the only zero divisor in the ring, can we cancel it from equations like ab=ac ?
2. How about when we are sure
1. In 2 and 7 are the only zero divisors. And we cannot cancel because but .
This example is trivial to rework into general proof.
2. In , but , so we cannot cancel the zero divisor even when we know that the multiple of it and the unknown is not zero. Same for 2*9 and 2*2, 2*10 and 2*3, 2*11 and 2*4, 2*12 and 2*5, 2*13, and 2*6.
How to rework this example into proof. Alternatively this is like asking the question on which exact conditions we can cancel zero divisor.
1. Never ?
2. On some conditions ?