Let R be a commutative ring. it is known that for ab to be a zero divisor of a ring, it means that, (ab)x=0 for some x in the ring R such that \(\displaystyle{x}\ne{0}\)

Now, by associativity and commutatibity property of a ring, we have,

a(bx)=0

And

b(ax)=0

If, \(\displaystyle{0}\ne{b}{x}\in{R}\), then we can say by definition of zezro divisor that a is zero divisor. Also, on the other hand, if we have bx=0, then as the value of \(\displaystyle{x}\ne{0}\), by our assumption, get again by using the definition of zero divisor that, b is a zero divisor

Hence, it can be a concluded that either a or b is a zero-divisor

Now, by associativity and commutatibity property of a ring, we have,

a(bx)=0

And

b(ax)=0

If, \(\displaystyle{0}\ne{b}{x}\in{R}\), then we can say by definition of zezro divisor that a is zero divisor. Also, on the other hand, if we have bx=0, then as the value of \(\displaystyle{x}\ne{0}\), by our assumption, get again by using the definition of zero divisor that, b is a zero divisor

Hence, it can be a concluded that either a or b is a zero-divisor