One exercise from my textbook:

The set consisting of 0 and all zero divisors in a commutative ring with identity contains at least one prime ideal.

Having thought about this problem for days, I only come up with the solution to the simplest case in which there's no zero divisors in such a ring.

But what about the general case when zero divisors do exist in such a ring?

One more stupid question... does this set consisting 0 and zero divisors form a ring? Even an ideal?