($\textit{Peskine-Szpiro}$) $\textit{functor}$ $\mathbf{F}_{R}^{e}(-)$ is defined by $\mathbf{F}_{R}^{e}(M)=R^{(e)} \otimes_{R} M$ for any $R$-module $M$.

Let us denote by $R^{0}$ the complement of the set of all minimal primes of $R$ and let $I$ be an ideal. Then we define $I^{[q]}$ as an ideal generated by $q=p^{e}$-powers of all elements of $I$. Then the $\textit{tight closure}$ $I^{*}$ of $I$ is the set of $x \in R$ such that there exists $c \in R^{0}$ for which $cx^{q} \in I^{[q]}$ for $q=p^{e} \gg 0$. It is easy to see that $I^{*}$ is an ideal containing $I$. An ideal $I$ is $\textit{tightly closed}$ if $I=I^{*}$. A Noetherian ring $R$ is called $\textit{weakly F-regular}$ if every ideal of $R$ is tightly closed. $R$ is called $\textit{F-regular}$ if every localization of $R$ is weakly $F$-regular.

The $\textit{perfect closure}$ of a ring $R$ is defined by adjoining to $R_{\mathrm{red}}$ all higher $p$-power roots of all elements of $R_{\mathrm{red}}$ and denote it by $R^{\infty}$. We also use the following notation. For a ring $R$, we let $R^{1/q}$ denote $(R_{\mathrm{red}})^{1/q}$. The $\textit{Frobenius closure}$ of $I$ is defined as $I^{F}=I R^{\infty} \cap R$. An alternative definition is that $x \in I^{F}$ if and only if $x^{q} \in I^{[q]}$ for all $q=p^{e} 