A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring". These rings were first studied by P.M. Cohn (though only in the commutative setting) in

• Rings of zero divisors, Proc. Amer. Math. Soc. 9 (1958), 914-919.

Moreover, every right (or left) artinian $\mathcal O$-ring is, in fact, a boolean ring (and hence commutative), see

• H.G. Moore, S.J. Pierce, and A. Yaqub, Commutativity in rings of zero divisors*, Amer. Math. Monthly 75 (1968), 392

Thence the question:

Does there exist any non-commutative $\mathcal O$-ring? If so, can you provide a reference where this is discussed?

I've tried to track the citations of Cohn's paper, but couldn't find an answer to my question. (See also here and there.)