Let R {\displaystyle {}R} be a noetherian domain of positive characteristic, let
be the Frobenius homomorphism and
(mit q = p e {\displaystyle {}q=p^{e}} ) its e {\displaystyle {}e} th iteration. Let I {\displaystyle {}I} be an ideal and set
Then define the tight closure of I {\displaystyle {}I} to be the ideal
be a noetherian domain of positive characteristic, let
)
its 
th iteration. Let 
be an ideal and set
![{\displaystyle {}I^{[q]}={\text{ extended ideal of }}I{\text{ under }}F^{e}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16c15b18043ea4cd331ed8f76c002ab05846c9d)
to be the ideal
![{\displaystyle {}I^{*}={\left\{f\in R\mid {\text{ there exists }}z\neq 0{\text{ such that }}zf^{q}\in I^{[q]}{\text{ for all }}q=p^{e}\right\}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73e19d08b5e9712f4297e86be0cf8df5a73c9dd3)