Let $R$ be a ring (here, rings are always associative, unital, and non-zero). An element $r \in R$ is quasi-regular if $1_R + r$ is a unit (so in particular, $0_R$ is quasi-regular); and $R$ 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 quasi-regular element is zero?

It is perhaps worth noting that, if the only quasi-regular element of $R$ is $0_R$, then $R$ has characteristic $2$ (and the Jacobson radical of $R$ is of course trivial).

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.

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). An element $r \in R$ is quasi-regular if $1_R + r$ is a unit (so in particular, $0_R$ is quasi-regular); and $R$ 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 regular element is zero?

It is perhaps worth noting that, if the only quasi-regular element of $R$ is $0_R$, then $R$ has characteristic $2$ (and the Jacobson radical of $R$ is of course trivial).

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). An element $r \in R$ is quasi-regular if $1_R + r$ is a unit (so in particular, $0_R$ is quasi-regular); and $R$ 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 quasi-regular element is zero?

It is perhaps worth noting that, if the only quasi-regular element of $R$ is $0_R$, then $R$ has characteristic $2$ (and the Jacobson radical of $R$ is of course trivial).