V \subseteq K$ and $P=R \cap tV$, where $K$ is the field of fractions of $R$. If $R \subseteq S$ is an integral extension domain, then the valuation $v$ attached to $V$ extends to $S$ with $\mathbb{Z}$ or $\mathbb{Q}$ as its value group. In particular, for some non-zero element $a \in R$, we get $v(a^{1/r})=\frac{1}{r} \cdot v(a)$. In what follows, we shall say that $(V,tV)$ as above is $\textit{attached}$ to the pair $(S,P)$ for the brevity of notation.

A question which naturally arises is, of course, to ask how $F$-coherent rings are related with those rings that are studied in tight closure theory. We occasionally use some basic facts on coherent modules, for which we refer to [@Glaz].

\[coherent-4\] Let $R$ be a reduced Noetherian ring of characteristic $p>0$ with module-finite normalization. If $R$ is $F$-coherent, then $R^{\infty}$ is a normal ring. In particular, for any such an $F$-coherent ring, the normalization $R\ \to \overline{R}$ is purely inseparable.

Let $R \to \overline{R}$ be the normalization map. Since this map is module-finite, there exists a non-zero divisor $r \in R$ such that $r \cdot \overline{R} \subseteq R$. By iterating the Frobenius map, we get the commutative diagram $$\begin{CD}
R @>>> R^{1/p} 