of $R$ is tightly closed. In order to show that every ideal in every localization of $R$ is tightly closed, it suffices to recall that $F$-coherence is stable under localization, as shown previously. This completes the proof of the proposition.

As a consequence from the above proposition, tight closure commutes with localization in any $F$-coherent ring. Namely, we have the following.

Let $R$ be any reduced $F$-coherent ring of characteristic $p>0$ and let $I \subseteq R$ be any ideal. Then $S^{-1}(I^{*})=(S^{-1}I)^{*}$ for any multiplicative subset $S \subseteq R$.

By the main results of [@BLR], a positive affine semigroup ring $k[C]$ is $F$-pure if and only if it is normal (see also [@HR], Theorem 5.33). In view of Proposition \[coherent-7\] and the example $k[x^{4},x^{2}y,xy^{2},y^{4}]$ treated above, it is interesting to know whether the normality is equivalent to $F$-coherence for affine semigroup rings. If this is true, it would provide many examples of $F$-coherent rings which are not associated with regular rings, $i.e.,$ they are not purely inseparably related.

We state some consequences on the Cohen-Macaulay property for $F$-coherent rings under various conditions.

Suppose that $R$ is a reduced $F$-coherent ring of characteristic $p>0$ and satisfies one of the following conditions:

1.  