\gg 0$. It is obviously true that $I^{F} \subseteq I^{*}$. A Noetherian ring $R$ is called $\textit{F-pure}$ if $I=I^{F}$ for every ideal $I$ of $R$. If $(R,{{\mathfrak m}})$ is a local Noetherian ring of characteristic $p > 0$, then the Frobenius map on $R$ naturally induces a map on the Čech complex of $R$, hence it gives the Frobenius action on the local cohomology modules $H^{k}_{{{\mathfrak m}}}(R) \to H^{k}_{{{\mathfrak m}}}(R^{(1)})$. $(R,{{\mathfrak m}})$ is $\textit{F-injective}$ if the map $H^{k}_{{{\mathfrak m}}}(R) \to H^{k}_{{{\mathfrak m}}}(R^{(1)})$ is injective for all $0 \le k \le \dim R$. If $(R,{{\mathfrak m}})$ is Cohen-Macaulay, then $R$ is $F$-injective if and only if $I=I^{F}$ for any parameter ideal $I$ of $R$. Here $I$ is a $\textit{parameter ideal}$ if $I$ is generated by a system of parameters. Then it holds that $F$-purity implies $F$-injectivity.

The $\textit{absolute integral closure}$ of an integral domain $R$ is the integral closure of $R$ in an algebraic closure of the field of fractions of $R$ and denote it by $R^{+}$. Then clearly $R^{\infty} \subseteq R^{+}$.

The most part of tight closure theory is concerned with Noetherian rings, while the definition of tight closure itself, at least, makes sense for any ring of characteristic $p 