A (unital) ring $R$ with the property that every element other than the identity $1\_R$ is a (two-sided) 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 is:



> 
> Does there exist any non-commutative $\mathcal O$-ring? If so, can you provide a reference where this is discussed?
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I've tried to track the citations of Cohn's paper, but couldn't find an answer to my question. (See also [here](https://mathoverflow.net/questions/395683/) and [there](https://mathoverflow.net/questions/284684/).)



Is there any non-commutative ring such that every element other than the identity is a zero divisor?
[Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]


The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper


* "Rings with a unique regular element", pp. 78-87 in B.J. Gardner (ed.), *Rings, modules and radicals* (Proc. Conf., Hobart/Aust. 1987), Pitman Res. Notes Math. Ser. **204**, Longman Sci. Tech., Harlow, 1989,


where Henriksen writes:



> 
> We do not know if there is a ring with a unique regular ring [*sic*] that fails to be commutative.
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In Henriksen's paper, a ring need not be unital; and a regular element is nothing else than a cancellative element of the multiplicative semigroup of the ring (loc. cit., Definition 2.1).


The question is marked as open by David Feldman in a [2012 post](https://mathoverflow.net/a/100432/16537) from the "Not especially famous, long-open problems which anyone can understand" [big list](https://mathoverflow.net/questions/100265/) (see also the comments under the same post), where Feldman writes:



> 
> Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?
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> 
> 


