Question

in a ring prove that the cancellation law of multiplication is true if and only if the ring has non zero divisors (i.e if and only if xy=0 -> x=0 or y=0)

Answer 1

got any answer to this problem to work around with

Answer 2

You probably want to attempt a proof by contradiction here. For the forward direction, assume a ring \(R\) has a cancellation law and that there are zero divisors. If \(x,y\in R\) are zero divisors, then $$x\cdot y=0$$with both \(x\ne 0,y\ne 0\). However: $$0=x\cdot 0$$for any \(x\in R\). So we have this: $$xy=x\cdot 0.$$How can you use the cancellation law here to get a contradiction?

Answer 3

For the converse, a direct proof works. Assuming that we have no zero divisors in our ring, we want to show that if \(xy=xz\) and \(x\ne 0\), then \(y=z\). If \(xy=xz\), then: $$xy-xz=0\Longrightarrow x(y-z)=0.$$How can we use the fact that our ring has no zero divisors to conclude that \(y=z\)?