Prereductions of ideals in local rings

Abstract

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and (Formula presented.) I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set (Formula presented.) of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then: There is a natural one-to-one correspondence between the set (Formula presented.) of the equivalence classes of the type-prereductions of I and the set (Formula presented.) of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring (Formula presented.) (t an indeterminate) such that (Formula presented.) and the Krull dimension of (Formula presented.) is equal to one). (Formula presented.) is the set of maximal elements in (Formula presented.) for all positive integers n, and, for each (Formula presented.) (Formula presented.) for all large integers m. A complete description is given of the prereductions and the elements in (Formula presented.) for each ideal I that is generated by analytically independent elements. If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets (Formula presented.) and (Formula presented.) and (Formula presented.) for all positive integer n.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1080/00927872.2020.1722828

Keywords

13C13, 13C99, 13H99, Algebraically closed residue field, analytically independent elements, ideal cover, integral closure, local ring, maximal relevant ideal, prereduction of an ideal, prereduction of type, Primary: 13B21, Ratliff-Rush closure, reduction of an ideal, reduction of type, Rees ring, Secondary: 12B99

Publication Date

7-2-2020

Journal Title

Communications in Algebra

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