The problem is to prove that, in a commutative ring with identity, the set of ideals in which every element is a zero divisor has a maximal element with respect the order of inclusion, and that every maximal element is prime. But I´m thinking* that considering the set of all zero-divisors, this set as I see it´s an ideal, and thus must be the only maximal element.
An ideal is defined as a subset J of the ring R , such that for every $ x\in R$ we have $ xJ \subset J $
I think that I´m wrong, only because it´s rare.
An extra question but related, there exist rings with element x, such that x is not a zero divisor nor a unit?