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 $A \subset$ 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 $\mathbf{P}(I)$ of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then:

1. There is a natural one-to-one correspondence between the set $\mathbf{E}(I)$ of the equivalence classes of the type-prereductions of I and the set $\mathbf{Q}(I)$ of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring $R[t^{-1},It]$ (t an indeterminate) such that $It ⊈ Q$ and the Krull dimension of $R[t^{-1},It]/Q$ is equal to one).

2. $\mathbf{P}(I^n)$ is the set of maximal elements in $\{Q_{[n]}\hspace{0.17em}|\hspace{0.17em}Q\in \mathbf{Q}(I)\}$ for all positive integers n, and, for each $Q \in \mathbf{Q}(I),$ $Q_{[m]}\in \mathbf{P}(I^m)$ for all large integers m.

3. A complete description is given of the prereductions and the elements in $\mathbf{E}(I)$ for each ideal I that is generated by analytically independent elements.

4. 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 $\mathbf{P}(I)$ and $\mathbf{E}(I),$ and $\mathbf{P}(I^n)=\{Q_{[n]}\hspace{0.17em}|\hspace{0.17em}Q\in \mathbf{Q}(I)\}$ for all positive integer n.

Keywords:

• Algebraically closed residue field
• analytically independent elements
• ideal cover
• integral closure
• local ring
• maximal relevant ideal
• prereduction of an ideal
• prereduction of type
• Ratliff-Rush closure
• reduction of an ideal
• reduction of type
• Rees ring

AMS 2010 SUBJECT CLASSIFICATION:

• Primary: 13B21
• 13C13
• Secondary: 12B99
• 13C99
• 13H99

Notes

1 It is shown in Corollary 7.2 (in Section 7) that bM is the unique prereduction of a non-nilpotent proper principal ideal bR in a local ring (R, M).

2 This is a special case of Example 3.2.4.

3 When it is unnecessary to refer to the specific integers k and $n_1,\dots ,n_k,$ then we will generically use type-reduction in place of reduction-of-type $[n_1,\dots ,n_k]$ (and, similarly, type-prereduction in place of prereduction-of-type $[n_1,\dots {,}_k]$).

4 Therefore, since $\{{\beta }_1,\dots ,{\beta }_k\}$ is a prereduction-of-type $[n_1,\dots ,n_k]$ of I, it follows from Remark 5.5.2.d that $\mathit{degree}(\overline{\{{\beta }_1,\dots ,{\beta }_k\}})\le \mathit{max}(\{n_1,\dots ,n_k\}),$ and the strict inequality may hold; this is related to Remark 5.10.3.