It is straightforward to show, for example, that the set of zero divisors of a (commutative unital) reduced Noetherian ring is precisely the union of its minimal primes. When else can we say that the set of zero divisors is equal to the union of the minimal primes? Are there other useful cases where this is true? Is there a structure theory for such rings? I'm primarily looking for conditions that do not assume that the ring is Noetherian.

Edit: I messed up this question and received a correct answer for the wrong assumptions, so I have accepted the answer. The reducedness assumption was for the example, not for the general question.

The correct question is in there, but I accidentally changed the title, so I apologize for the confusion.