# von Neumann regular ring homomorphisms

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.

• Out of curiosity, does the flatness of $f^\ast M$ imply anything about the flatness of $M$? I have no mastery of flatness, especially under this transfer. – rschwieb Oct 11 '18 at 15:11
• @rschwieb: It means that tensoring with $M$ preserves exactness for exact sequences which are induced up from $R$. But not any exact sequence is of this form. I don't know if anything stronger is true. – nikola karabatic Oct 12 '18 at 8:18
• Flatness of $f^*M$ does not tell you anything about $M$ in general, as can be seen by taking $R$ to be absolutely flat (e.g., a field). – Fred Rohrer Oct 12 '18 at 10:56