ac.commutative algebra - When is the set of zero divisors equal to the union of the minimal primes in a reduced ring?

The answer is: always, and argument is pretty simple:

Let $R$ be a reduced commutative unital ring.
If $ain R$ is a zero divisor, then $ab=0,$ for some $bneq 0.$ Hence
$bnot in 0 = text{nil}(R)= bigcap text{Spec}(R) =bigcap text{Specmin}(R),$ where $text{Specmin}(R)$ states for family of all minimal prime ideals of $R.$
So there exists a minimal prime ideal $mathfrak{p}triangleleft R$ s.t. $bnot in mathfrak{p}.$ But $mathfrak{p}$ is prime, and $abin mathfrak{p},$ thus $ain mathfrak{p}.$
Hence the set of zero divisors is contained in union of minimal prime ideals.

On the other hand it is well-known fact that minimal prime ideals in commutative unital rings consist of zero divisors, so the set of zero divisors in reduced commutative unital ring is exactly the union of minimal prime ideals.