not necessarily inherited from the quotient of a ring by a non-zero divisor. As there is an example of hypersurface ring (a quotient of a regular domain modulo a non-zero element) which is $F$-pure, but not $F$-regular, the converse of the above question does not hold.

Let $R$ be an $F$-coherent local ring. Then is the Hilbert-Kunz multiplicity of $R$ rational?

This question is in connection with the fact that the Hilbert-Kunz multiplicity of a local ring having finite $F$-representation type is rational. It is also interesting to ask:

Let $R \subseteq S$ be a pure extension such that $S$ is $F$-coherent. Then is $R$ also $F$-coherent?

As our examples of $F$-coherent rings always came from the regular rings, it is natural to ask:

Is there an example of an $F$-coherent ring whose perfect closure does not coincide with the one which comes from a regular ring?

Suppose $R$ is a Noetherian ring of characteristic $p>0$. Then can one find any practical characterization other than the definition that $R$ is $F$-coherent?

Let us make some observations about the arithmetic deformation of $F$-coherent rings over ${\operatorname{Spec}}\mathbb{Z}$ in the one-dimension case. First, we consider $\mathbb{Z}[t^{2},t^{3}] \simeq \mathbb{Z}[x,y]/(x^{3}-y^{2})$. Then in characteristic $p>0$, the 