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Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$.

Question. Is there a non-commutative, left duo ring whose only unit is the identity?

It is perhaps worth noting that, if the only unit of $R$ is the identity, then $R$ has characteristic $2$ and the Jacobson radical of $R$ is trivial.

Edit. In a previous version of this question, I was asking for the existence of a left duo ring whose only quasi-regular element is zero: I hadn't realized that this is only possible if the group of units of the ring is trivial (whence the new formulation), which makes the question vaguely reminiscent of an open problem by M. Henrikson.

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  • $\begingroup$ Only a weak attempt to keep alive algebra on mathoverflow. Henriksen's problem and yours can be weakened using algebras where all inverible elements are scalars (algebras over the field with two elements give the original problem). Cohn's method for the commutative case works in this setting (extending to an algebra where each non-zero divisor is a scalar). Skew polinomial rings don't give duo examples for the weakened version of your problem, but skew power series rings might (only over a field with nonidentity auto/endo/morphism). $\endgroup$
    – NameNo
    Commented Jan 26, 2022 at 9:18
  • $\begingroup$ @NameNo Are you implicitly referring to [G. Marks, Duo rings and Ore extensions, J. Algebra 280 (2004) 463–471] when you write that skew polynomial rings "don't give duo examples for the weakened version" of my problem"? $\endgroup$
    – Salvo Tringali
    Commented Jan 26, 2022 at 19:32
  • $\begingroup$ Yes. Also: On the structure of skew polynomial rings, Gil Alon and Elad Paran. Structure of weakly one-sided duo Ore extensions doi.org/10.1007/s12044-020-00600-9 $\endgroup$
    – NameNo
    Commented Jan 26, 2022 at 19:52

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