The above post has multiple errors:
In known facts 1 the corrected statement should be:
An non-zero element is either unit or zero divisor.
In unknowns 1 and example 1:
The example is for commutative ring and should be mentioned that in such ring the number of zero divisors is always even. And even if
we have non-commultative ring with a as only zero divisor, we still cannot cancel, because ab=ac is always true for
Other than this, I think I solved it:
If a is zero divisor in R, then in R there is , s.t. ab=0. Then if c is another element of R, ac=ac-0=ac-ab=a(c-b), so for any c, ac=a(c-b). We can cancel the zero divisor only if c=c-b, or -b=0, or b=0. b is never zero by definition, so for any c, . We can never cancel zero divisor.