Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.

In particular, $\mathrm{id}_R$ is von Neumann regular if and only if every left $R$-module is flat, i. e. $R$ is von Neumann regular in the usual sense.

**Question:** Has this notion been studied? Does anyone know a reference?

The rings here are associative, unital and not necessarily commutative.