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).