[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](https://math.stackexchange.com/a/2502555/20748) 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](https://math.stackexchange.com/a/688140/20748) 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.



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


* P.M. Cohn, *Rings of zero divisors*, Proc. Amer. Math. Soc. **9**
(1958), 914-919;
* T. Porter, *Cohn's rings of zero divisors*, Arch.
Math. **43** (1984), 340-343.


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).


