Is there any characterization for the left primitive rings which are left duo ? Clearly division rings satisfy this property. 



A question on primitive rings
The class of rings in the question is exactly the class of Division rings. Here is another proof based on the characterization of primitive rings. 


**Proof.** Since you have asked about primitive rings I can assume that you are familiar with the **Jacobson's Density Theorem**. Therefore $R$ can be regarded as a dense subring of linear transformations of a vector space $V$ over a division ring $D$. It is enough to show that $\mbox{dim}\_D V = 1$ (since then $R$ would be equal to $D$) . Suppose not and consider two linearly independent vectors $u,v \in V$. By density there are $f,h \in R$ such that $f(u) = u , f(v) = 0$ and $h(u) = u , h(v) = u$. Now Take $I$ to be the left ideal generated by $f$ in $R$. We show that $fh \not \in I$ (which is a contradiction since by assumptions any left ideal of $R$ is two-sided): 


If $fh \in I$ then there exists $g \in R$ such that $fh = gf$. Hence $u = fh(v) = gf(v) = g(0) = 0$ a contradiction ! 


