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 $\displaystyle R$ which has zero divisors, what and when can we cancel, when trying to solve an equation?

Known facts:

1. An element is either zero or zero divisor.

2. If an element is unit, we can always cancel it.

Unknowns:

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 $\displaystyle ab\ne0$

Easy examples:

1. In $\displaystyle Z/14Z$ 2 and 7 are the only zero divisors. And we cannot cancel because $\displaystyle 2 * 0 \equiv 2 * 7$ but $\displaystyle 0 \not\equiv 7 $.

This example is trivial to rework into general proof.

2. In $\displaystyle Z/14Z$, $\displaystyle 2 * 8 \equiv 2 * 1 $ but $\displaystyle 8 \not \equiv 1$, 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 ?