Closure operations on ideals.

Abstract

We explore new closure operations on sets of ideals in a commutative Noetherian ring of characteristic <italic>p</italic>, related to tight and integral closure. We develop the theories of multiple closure and blowup closure of a set of ideals. We prove that given a set of ideals in a Noetherian ring of characteristic <italic>p</italic>, under mild conditions, the multiple closure of this set agrees with its tight integral closure, a notion introduced by Hochster. We therefore use multiple closure to settle open questions on tight integral closure posed by Hochster. In particular, we show that under mild conditions on the ring, tight integral closure persists under ring maps, and that it commutes with localization if and only if tight closure does. In another part of the thesis we give an effective method for finding normal ideals for graded rings. We give several examples concerning the effectiveness of this bound, and compare it with other possible bounds.