Quintessential-modulated ideals

doi:10.1016/j.jalgebra.2020.07.024

Journal of Algebra

Volume 564, 15 December 2020, Pages 119-150

https://doi.org/10.1016/j.jalgebra.2020.07.024 Get rights and content

Abstract

Let R denote a commutative Noetherian ring and I an ideal of R. The concept of quintessential sequences over zero ideal was introduced by McAdam and Ratliff (1985) [8]. They showed that these sequences enjoy many of the basic properties of asymptotic sequences over zero ideal. It was shown, that quintessential sequences over ideals $I \neq (0) R$ are not a good analogue of asymptotic sequences over $I \neq (0) R$ . By making use of the new concept of quintessential grade of an ideal over another ideal, we show that there exists a class of ideals I for which quintessential sequences over I are an excellent analogue of asymptotic sequences over I. Also, we give more results on quintessential sequences over an ideal and derive generalizations of some McAdam-Ratliff's results (1985) [8], [13].

Introduction

Let us first set the convention that all rings in this paper are commutative Noetherian with a unit $1 \neq 0$ . The first published result involving asymptotic sequences was by Rees [15]. Let I be an ideal of a ring R. A sequence $x = x_{1}, \dots, x_{n}$ of elements of R is called an asymptotic sequence over I if $(I, (x)) R \neq R$ and for all $1 \leq i \leq n$ , we have $x_{i} \notin ⋃_{P \in \overline{A^{⁎}} ((I, (x_{1}, \dots, x_{i - 1})) R)} P .$ Recall that $\overline{A^{⁎}} (J) = ⋃_{n \in N} {Ass}_{R} (R / \overline{J^{n}})$ is the set of asymptotic prime divisors of the ideal J of R, where $\overline{J^{n}} = {x \in R | there exists a positive integer h and elements j_{k} \in J^{n k}, for k = 1, \dots, h, such that x^{h} + j_{1} x^{h - 1} + \dots + j_{h} = 0}$ . Then in [12] it was shown that asymptotic sequences in R (that is, asymptotic sequences over $I = (0) R$ ) have many of the basic properties of regular sequences.

The interesting concepts of quintessential prime divisors of an ideal and quintessential sequences in R were introduced and studied by McAdam and Ratliff [8]. There, they showed that these concepts are an excellent analogue of, respectively, asymptotic prime divisors of an ideal and asymptotic sequences in R. A sequence $x = x_{1}, \dots, x_{n}$ of elements of R is called a quintessential sequence in R if $(x) R \neq R$ and for all $1 \leq i \leq n$ , we have $x_{i} \notin ⋃_{P \in Q ((x_{1}, \dots, x_{i - 1}) R)} P,$ where $Q (J) = {P \in Spec (R) | J \subseteq P and there exists z \in {Ass}_{\hat{R_{P}}} (\hat{R_{P}}) such that J \hat{R_{P}} + z is P \hat{R_{P}} -primary}$ is the set of quintessential prime divisors of the ideal J of R and $\hat{R_{P}}$ is the completion of $R_{P}$ . It should be noted that the notation and terminology concerning these prime divisors for ideals was changed in [3]. In papers before [3], what we termed quintessential was called essential and E was used in place of Q. Some reasons for these changes are given in the appendix in [3]. The reader should keep these changes in mind when checking references.

Furthermore, Ratliff in [13] introduced the concept of quintessential sequence over I. A sequence $x = x_{1}, \dots, x_{n}$ of elements of R is called a quintessential sequence over I if $(I, (x)) R \neq R$ and for all $1 \leq i \leq n$ , we have $x_{i} \notin ⋃_{P \in Q ((I, (x_{1}, \dots, x_{i - 1})) R)} P .$

It was shown in [13] that quintessential sequences over ideals $I \neq (0) R$ are not a good analogue of asymptotic sequences over $I \neq (0) R$ . Thus the analogy between asymptotic properties and quintessential properties breaks down when it comes to sequences over a nonzero ideal. In this paper, we show that there exists a class of ideals I, called quintessential-modulated ideals for which quintessential sequences over I are an excellent analogue of asymptotic sequences over I; see Section 7. For this purpose we introduce the concept of quintessential grade of an ideal over another ideal. More precisely, let I and J be ideals of a ring R, it is shown all maximal quintessential sequences over I in J have the same length and denoted ${qegd}_{I} (J)$ . Also, using this new grade, we give more results on quintessential sequences over an ideal and derive generalizations of some McAdam-Ratliff's results [8], [13]. Finally, for any unexplained notation or terminology we refer the reader to [5], [16].

Section snippets

On quintessential grade over an ideal

In this section, we show that ${qegd}_{I} (J)$ is unambiguously defined for ideals I and J in a ring R. Then we give several results concerning this grade.

Definition 2.1

Let I and J be ideals of a ring R.

(i) $Q (I) : = {P \in Spec (R) | I \subseteq P and there exists z \in {Ass}_{\hat{R_{P}}} (\hat{R_{P}}) such that I \hat{R_{P}} + z is P \hat{R_{P}} -primary}$ . The members of $Q (I)$ are called the quintessential prime divisors of I.

(ii) A sequence $x = x_{1}, \dots, x_{n}$ of elements of R is called a quintessential sequence over I if $(I, (x)) R \neq R$ and for all $1 \leq i \leq n$ , we have $x_{i} \notin ⋃_{P \in Q ((I, (x_{1}, \dots, x_{i - 1})) R)} P .$

Quintessential component of an ideal

The concept of quintessential component of an ideal introduced by McAdam and Ratliff [8]. In this section we give some new results concerning quintessential component of an ideal.

Definition 3.1

Let I be an ideal of a ring R.

(i) If $P_{1}, . . ., P_{k}$ are maximal in $Q (I)$ and $S = R ∖ ⋃_{i = 1}^{k} P_{i}$ . Then $I_{qe} = I R_{S} \cap R$ is the quintessential component of I.

(ii) $Q_{qe} (I) : = ⋃_{n \in N} {Ass}_{R} (R / {(I^{n})}_{qe})$ .

(iii) If n is a non-negative integer, then $Q_{n} (I) : = {P \in V (I) | {qegd}_{I} (P) = n}$ .

In [8, Proposition 5.10], McAdam and Ratliff showed that if $x = x_{1}, \dots, x_{n}$ is a

The locally unmixed ring over an ideal

A ring R is locally unmixed if for any $P \in Spec (R)$ , $R_{P}$ is an unmixed ring. In this section we introduce a generalization of locally unmixed rings and give several characterizations of these rings.

Definition 4.1

Let I be an ideal of a local ring R. We say R is unmixed over I, if $\dim \hat{R} / (I \hat{R} + z) = \dim R / I$ , for all $z \in {Ass}_{\hat{R}} (\hat{R})$ . More generally, if R is not necessarily local, R is a locally unmixed over I, if for any $P \in V (I)$ , $R_{P}$ is unmixed over $I R_{P}$ .

Theorem 4.2

Let I be an ideal of a ring R. The following are equivalent:

(i) ${qegd}_{(I, (x}$

Quintessential grade over an ideal and certain related rings

In this section it is shown that ${qegd}_{I} (J)$ behaves nicely when passing to certain rings related to R. We begin with faithfully flat extensions.

Theorem 5.1

Let S be a faithfully flat extension of a ring R. Then ${qegd}_{I} (J) = {qegd}_{I S} (J S)$ , for all ideals I and J in R.

Proof

Let I and J be ideals in R, let ${qegd}_{I} (J) = n$ , and let $x = x_{1}, \dots, x_{n}$ be a maximal quintessential sequence over I from J. Then there exists $P \in Q ((I, (x)) R)$ such that $J \subseteq P$ . Let $P$ be a minimal prime divisor of PS, then by [8, Proposition 3.6], $P \in Q ((I, (x)) S)$ , and by

Some bounds for quintessential grade over an ideal

In this section, we give several bounds on quintessential grade over an ideal. We begin with a nice inequality.

Theorem 6.1

Let $I \subseteq J \subseteq K$ be ideals in a ring R and $x = x_{1}, \dots, x_{h}$ be a quintessential sequence over I in J. Then there exists a maximal quintessential sequence over J in K, say $y = y_{1}, \dots, y_{n}$ such that $x, y$ is a quintessential sequence over I in K. In particular ${qegd}_{I} (J) + {qegd}_{J} (K) \leq {qegd}_{I} (K) .$

Proof

Let ${qegd}_{J} (K) = n$ . If $n = 0$ , we are done. If $n > 0$ , then $K ⊈ ⋃_{P \in Q (J)} P,$ and so by [13, Lemma 5.2], $K ⊈ ⋃_{P \in Q ((I, (x)) R)} P .$ Pick $y_{1} \in K$ with $y_{1} \notin ⋃_{P \in}$

Quintessential-modulated ideals

In this section we show that there exists a class of ideals I for which quintessential sequences over I are an excellent analogue of asymptotic sequences over I. A good overview of the concept of asymptotic sequences is [6]. We begin with a definition.

Definition 7.1

(i) An ideal I of a ring R is said to be quintessential-modulated if for every ideal J of R and element b in Jacobson radical of R, we have ${qegd}_{I} ((J, (b)) R) \leq {qegd}_{I} (J) + 1$ .

(ii) An ideal I of a ring R is said to be locally quintessential-modulated if $I R_{P}$

Acknowledgements

We are deeply grateful to L.J. Ratliff for his careful reading and useful comments during the preparation of this work.

References (16)

D. Katz et al.
Essential prime divisors and projectively equivalent ideal
J. Algebra
(1987)
S. McAdam et al.
Essential sequences
J. Algebra
(1985)
L.J. Ratliff
Asymptotic sequences
J. Algebra
(1983)
M. Brodmann
Asymptotic stability of $Ass (M / I^{n} M)$
Proc. Am. Math. Soc.
(1979)
I. Kaplansky
Commutative Rings
(1970)
D. Katz et al.
U-essential prime divisors and sequences over and ideal
Nagoya Math. J.
(1986)
H. Matsumura
Commutative Ring Theory
(1986)
S. McAdam
Asymptotic Prime Divisors
(1983)

There are more references available in the full text version of this article.

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^☆: The first author was supported by a grant from IPM.

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Quintessential-modulated ideals☆

Abstract

Introduction

Section snippets

On quintessential grade over an ideal

Quintessential component of an ideal

The locally unmixed ring over an ideal

Quintessential grade over an ideal and certain related rings

Some bounds for quintessential grade over an ideal

Quintessential-modulated ideals

Acknowledgements

J. Algebra

J. Algebra

J. Algebra

Asymptotic stability of Ass(M/InM)

Proc. Am. Math. Soc.

Commutative Rings

U-essential prime divisors and sequences over and ideal

Nagoya Math. J.

Commutative Ring Theory

Asymptotic Prime Divisors

Asymptotic stability of $Ass (M / I^{n} M)$