(I fear that I’m missing something obvious here, but I’ll dare to ask anyway.)

A division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let an anti-division ring be a ring where every element other than the (multiplicative) identity is a zero divisor. Is there any classification whatsoever of anti-division rings?

There are many simple things one can actually say about these rings: They are closed under direct products; they all have characteristic $2$; every boolean ring is a member of the family; any such ring is, in fact, a subdirect product of domains (see Jose Brox’s answer to a related question on SME); the only nilpotent element in such a ring is zero; if a ring in the family is semilocal, then it’s a direct product of copies of the field $F_2$ with two elements (see znc’s answer from the same thread of Jose Brox’s); etc.

Question.Is it perhaps the case that a more precise characterization (say, than the one provided by Jose Brox for rings whose non-units are all zero divisors) is indeed possible?

If so, it’s clear to me that this must be very well known (at least in some large circles) and I would much appreciate it if you could offer a reference.