> 0$. We will use only the definition to derive the property of coherence on certain perfect rings.

$F$-coherent rings
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A ring $R$ is $\textit{coherent}$ if its every finitely generated ideal is finitely presented. An equivalent definition for $R$ to be coherent is that both $(0:a)$ and $I \cap J$ are finitely generated ideals for any $a \in R$ and finitely generated ideals $I, J \subseteq R$. Another equivalent definition is that every finitely generated submodule of every finitely presented module over $R$ is finitely presented over $R$ ([@Glaz], Theorem 2.3.2 for these definitions). We will use them interchangeably. Let us begin with the definition of $F$-coherent rings and establish some basic properties.

Let $R$ be a Noetherian ring of characteristic $p>0$. Then we say that $R$ is $F$-$\textit{coherent}$ if the perfect closure $R^{\infty}$ is coherent.

A simple observation shows that $R$ is $F$-coherent if and only if $R_{\mathrm{red}}$ is $F$-coherent. In particular, we may assume that $R$ is always reduced. In the following, when we say a $\textit{flat colimit}$ of commutative rings, it means a direct system $\{R_{\alpha}\}_{\alpha \in \Lambda}$ such that, whenever $\alpha < \beta$, the transition map $R_{\alpha} \to R_{\beta}$ is flat in the usual sense.

