Tight closure
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In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Mel Hochster and Craig Huneke in the 1980s.
Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p is a prime.
Let I be an ideal of R. The tight closure of I, denoted by I * , is another ideal of R containing I. It is defined as follows.
if and only if there exists a 
, where c is not contained in any minimal prime ideal of R, such that ![c z^{p^e} \in I^{[p^e]}](../../../math/5/9/4/5949151a7e10151e91e5d1ae2964bc73.png)
for all 
. If $R$ is reduced, then one can instead consider e > 0.
Here ![I^{[p^e]}](../../../math/8/5/0/8501aadf23edcf9f9718702f140a76d6.png)
is used to denote the ideal of R generated by the pe'th powers of elements of I, called the eth Frobenius power of I.
