A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$. It turns out that a ring $R$ is right Rickart iff every principal right ideal of $R$ is projective as a right $R$-module. (Prop. 7.48 in Lam - "Lectures on Modules and Rings".)
Recall that a ring $R$ is a right non-singular ring if the left annihilator of each essential right ideal of $R$ is zero.
Question: For a ring $R$, being right Rickart implies being right non-singular (loc.cit., p.262). The converse (right non-singular $\Rightarrow$ right Rickart), however, does not hold in general. What would be (a candidate for) the mildest possible assumption that one can impose on $R$ and get that the converse holds?
