[I fear that I'm missing something obvious here, but I'll dare to ask anyway.]

As we all know, 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 any element other than the identity is a (two-sided) zero divisor.

There are many simple things one can actually say about these objects: Their class is closed under direct products; they all have characteristic $2$; every boolean ring is a member of the family; any anti-division ring is, in fact, a subdirect product of domains (see Jose Brox's answer to a related question on MSE); the only nilpotent element in such a ring is zero and, more generally, the Jacobson radical is trivial (as noted by Benjamin Steinberg in the comments below); if a ring in the family is semilocal, then it's a direct product of copies of the field with two elements (see znc's answer from the same thread on MSE); 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.