# Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor?

[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.

• The Jacobson radical must be zero as well since it x is in the radical 1-x is a unit Jun 18, 2021 at 20:02

Sorry for answering my own question, but it turned out that what I'm calling "anti-division rings" in the OP were already studied by P.M. Cohn under the name of "$$0$$-rings" (though Cohn's work on this stuff is seemingly restricted to the commutative setting), see
In particular, Cohn's paper (Sect. 3) contains a couple of structural results for commutative $$0$$-rings, and it seems unlikely that one can do any better (without throwing in further conditions on the ring).
• See also mathoverflow.net/a/395778/16537, where a reference is provided for the fact that every right (or left) artinian $0$-ring is boolean (and hence commutative). Jun 20, 2021 at 8:54