It is true in both fields and ringswith identitythat "if ab= 0 for all b in A then a= 0". Just take a times the identity, i. ai= a= 0. To find a counter-example, you would need to look at rings without identity.

In a field, it is true that if ab= 0 forsomenon-zero b (not necessarily "all" b) then a= 0. If b is non-zero, it has an inverse, so .