Proposition \[coherent-7\]. However, the difference is that Proposition \[coherent-7\] addresses consequences on the tight closure of ideals on Noetherian rings, while the following does so on perfect rings rather than Noetherian rings.

Let $R$ be a Noetherian domain of characteristic $p>0$. Suppose that $S$ is any perfect domain that is integral over $R$. If there is a finitely generated ideal of $S$ that is not tightly closed, then $S$ is not coherent.

Note first that $R \subseteq S \subseteq R^{+}$. Let $I$ be any finitely generated ideal of $S$. Then extend $R$ to its module-finite extension domain $R'$ such that the generators of $I$ are contained in the Noetherian domain $R'$. Then applying Lemma \[Krull\], we obtain $\bigcap_{n>0} I^{n}S \subseteq \bigcap_{n>0} I^{n}R^{+}=0$. The proposition now follows from Lemma \[perfect\].

We end this section with the following example to use the results in this section.

Recently, an example of a Noetherian domain $R$ of characteristic 2 together with an ideal $I$ such that $I^{+} \ne I^{*}$, was constructed. Here $I^{+}=I R^{+} \cap R$ is the plus closure of $I$. Let $L=\overline{(\mathbb{Z}/2\mathbb{Z})}$, the algebraic closure of $\mathbb{Z}/2\mathbb{Z}$, and $$R:=R_{t}=L[x,y,z,t]/(g_{t}),$$ where $g_{t}=z^{4}+xyz^{2}+x^{3}z+y^{3}z+tx^{2}y^{2}$. Let also $S=L[t]-\{0\}$. The ring $R$ is a three-dimensional normal hypersurface 