Document Type
Lecture
Publication Date
2-23-2009
Abstract
A closure operation is an idempotent endomorphism on the set of ideals of a given [commutative Noetherian] ring, which produces an ideal that contains the given ideal, and preserves containment of ideals. If defined element-wise, a closure operation can be seen as a necessary condition for membership in an ideal. I will discuss various closure operations (e.g. integral closure, tight closure, and continuous closure), as well as some of their properties, how they arise, and criteria for containment. I will discuss the related notions of (1) (minimal) reduction with respect to a closure and (2) special parts of closures, including a tie-in with matroid theory. Much of this work is joint with various collaborators (Yao, Hochster, Brennan, and Vraciu).
Relational Format
presentation
Recommended Citation
Epstein, Neil, "Closure Operations on Ideals in Commutative Rings" (2009). Algebra/Number Theory Seminar. 51.
https://egrove.olemiss.edu/math_algebra_number_theory/51
