$Ra=Rb$ if and only if $aR=bR$

On which classes of (non commutative) rings we have the following property: $aR=bR$ if and only if $Ra=Rb$ ?

While I googling around I found the notion of "Duo Ring" in which $aR=Ra$ for every $a\in R$. This is stronger that what I am looking for. Even for this, I don't know any example of duo ring.

Solution 1:

One place to start would be the third question here, in which the property you are considering is referred to only as (*). Unfortunately, while skimming through articles that cited this paper, I didn't see anything else of particular relevance.

Nonetheless, the paper above proves:

Theorem. Let $R$ be a noetherian integrally closed duo domain. Then $aRbR = bRaR$ for all $a,b \in R$ and ideal multiplication is commutative.