1. The problem statement, all variables and given/known data Suppose that a commutative ring R, with a unit, has no zero divisors. Does that necessarily imply that every nonzero element of R is invertible? 2. Relevant equations 3. The attempt at a solution We have to show that there exists some b in R such that ab = e. Having no zero divisors implies that if b[itex]\neq[/itex]0 then ab[itex]\neq[/itex]0. To show that every nonzero element of R is not invertible we must find a case where ab = c for some c in R and c [itex]\neq[/itex] e. The question seems easy but I can't wrap my head around how to write it down.