Intro I've been (not successfully) trying to educate myself a bit about singular/nonsingular modules and I came by a proposition on Wikipedia that I find really interesting.
For commutative rings, being nonsingular is equivalent to being a reduced ring.
Question Can anyone give me some hints on how to prove this?
My attempts I didn't really get far. I understand that the domain is non-singular. Because it has no non-zero zero divisors.
If I denote $Z$ the set of zero-divisors then clearly annihilators are subsets of $Z$. I even read a stronger statement in some article saying that that Annihilators in reduced rings are intersections of minimal primes (minimal primes are contained in $Z$ for reduced rings)
Also, it seems like a good direction to investigate when are ideals contained in $Z$ essential (which is a question I stumped upon also in different settings in past, so seems like an interesting problem on its own)