I'm trying to figure out a solution to the question:
If a commutative ring $R$ has a nontrivial proper ideal $I$ that contains no nontrivial zero divisor of $R$, is $R$ an integral domain?
I haven't been able to find a counterexample, and it is simple to answer this in the affirmative if $I$ is prime. However, I have found it very difficult to make progress in the general case. I considered the contrapositive, and obviously in a ring that is not an integral domain there is an ideal that contains zero divisors, but I can't figure out how to show that every ideal contains zero divisors.
Does anyone have any suggestions for a proof? Or a counterexample? Thanks so much, and I apologize that I don't have more progress to show.