many outstanding questions regarding rings of positive characteristic. In fact, a modification of the above problem has a connection with the problem whether tight closure commutes with localization, or not. A typical attempt to establish the localization problem for tight closure was made by its comparison with plus closure. More precisely, if it were true that $R^{+}$ (this notation will be explained later) is coherent, then it would follow that tight closure commutes with localization. Unfortunately, it turns out that this is false, as was shown by the recent work of Brenner and Monsky [@BM]. Except the case when $R$ is a field, the perfect ring $R^{\infty}$ is almost never Noetherian, but it can be better-behaved than $R$ itself. As we shall see below, perfect coherent rings of certain type force their finitely generated ideals to be tightly closed. This result relies on the flatness of the Frobenius map on perfect rings. Also, there are non-trivial cases when the perfect closures of rings, much smaller than $R^{+}$, are not coherent. In this respect, it would be adequate to call a Noetherian ring of positive characteristic $\textit{F-coherent}$ if its perfect closure is coherent. Especially, a subclass of affine semigroup rings forms 