On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

2018 ◽  
Vol 85 (3-4) ◽  
pp. 356
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Integral Closure ◽  
Prime Ideals ◽  
Local Rings ◽  
Finite Sets ◽  
Maximal Ideals ◽  
Valuation Rings ◽  
Canonical Bijection ◽  
Minimal Prime

<p>It is shown that, for all local rings (R,M), there is a canonical bijection between the set <em>DO(R)</em> of depth one minimal prime ideals ω in the completion <em><sup>^</sup>R</em> of <em>R</em> and the set <em>HO(R/Z)</em> of height one maximal ideals <em>̅M'</em> in the integral closure <em>(R/Z)'</em> of <em>R/Z</em>, where <em>Z </em>:<em>= Rad(R)</em>. Moreover, for the finite sets <strong>D</strong> := {<em>V*/V* </em>:<em>= (<sup>^</sup>R/ω)'</em>, ω ∈ DO(R)} and H := {<em>V/V := (R/Z)'<sub><em>̅M'</em></sub>, <em>̅M'</em> ∈ HO(R/Z)</em>}:</p><p>(a) The elements in <strong>D</strong> and <strong>H</strong> are discrete Noetherian valuation rings.</p><p>(b) <strong>D</strong> = {<em><sup>^</sup>V</em> ∈ <strong>H</strong>}.</p>

Prime Ideals in GCD-Domains

1974 ◽  
Vol 26 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Philip B. Sheldon
Keyword(s):  
General Setting ◽  
Chain Condition ◽  
Integral Closure ◽  
Prime Ideals ◽  
Unique Factorization Domain ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Complete Integral Closure ◽  
Minimal Prime ◽  
Bezout Domains

A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.

Notes on Local Integral Extension Domains

1978 ◽  
Vol 30 (01) ◽  
pp. 95-101 ◽  
Author(s):  
L. J. Ratliff
Keyword(s):  
Prime Ideal ◽  
Maximal Chain ◽  
Integral Closure ◽  
Prime Ideals ◽  
Local Domain ◽  
Important Paper ◽  
Integral Extension ◽  
Local Integral ◽  
Maximal Ideals ◽  
Extension Domain

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3]. In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P &lt; height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

scholarly journals Noetherian Rings—Dimension and Chain Conditions

2005 ◽  
Vol 4 (3) ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Sufficient Conditions ◽  
Artinian Ring ◽  
Small Dimension ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Local Rings ◽  
Direct Products ◽  
Finite Set ◽  
Minimal Prime ◽  
Necessary And Sufficient

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime ideals of a noetherian ring. In particular, we express noetherian rings with certain properties as finite direct products of noetherian rings with a unique minimal prime ideal, as an analogue to the expression of an artinian ring as a finite direct product of artinian local rings. Besides, we also consider the set of ideals I in R such that M ≠ I M for a given module M and show that a maximal element among these is prime. In Section Four, we deal with dimensions of prime ideals, Krull’s Small Dimension Theorem and generalize it (and its converse) to the case of a finite set of prime ideals. Towards the end of the paper, we also consider the sets of linear dependencies that might hold between the generators of an ideal and consider the ideals generated by the coefficients in such linear relations.

Maximal d-Ideals in a Riesz Space

1983 ◽  
Vol 35 (6) ◽  
pp. 1010-1029 ◽  
Author(s):  
Charles B. Huijsmans ◽  
Ben de Pagter
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Prime Ideals ◽  
Topological Properties ◽  
Order Unit ◽  
Structure Space ◽  
Maximal Ideals ◽  
Minimal Prime ◽  
The Ideal

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e > 0 is a weak order unit).

scholarly journals Essential ideals in subrings of C(X) that contain C*(X)

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1631-1637
Author(s):  
A. Taherifar
Keyword(s):  
Prime Ideals ◽  
Intermediate Ring ◽  
Maximal Ideals ◽  
Minimal Prime

Let A(X) be a subring of C(X) that contains C*(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences ZA and ?A are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C*(X) and C(X), respectively, to any intermediate ring A(X). Moreover, the inverse map Z-1A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, first, we characterize essential ideals in A(X). Afterwards, we show that Z-1A maps essential (resp., free) z-filters on X to essential (resp., free) ideals in A(X) and Z-1A maps essential ?A-filters to essential ideals. Similar to C(X) we observe that the intersection of all essential minimal prime ideals in A(X) is equal to the socle of A(X). Finally, we give a new characterization for the intersection of all essential maximal ideals of A(X).

On the dimension of the non-Cohen–Macaulay locus of local rings admitting dualizing complexes

1991 ◽  
Vol 109 (3) ◽  
pp. 479-488 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Prime Ideals ◽  
Local Rings ◽  
Large Power ◽  
Systems Of Parameters ◽  
The Difference ◽  
Image Position ◽  
Minimal Prime ◽  
Positive Integers ◽  
Dualizing Complexes

In this paper we mainly consider local rings admitting dualizing complexes. It is well-known that if a Noetherian local ring A admits a dualizing complex, then the non-Cohen–Macaulay (abbreviated CM) locus of A is closed in the Zariski topology (cf. [8, 10]). If the dimension of this locus is zero and A is equidimensional, i.e. the punctured spectrum of A is locally CM and dim(A/P) = dim (A) for all minimal prime ideals P ∈ Ass (A), then A is a generalized CM ring and its structure is well-understood (see [2, 12]). For instance, one of the characterizations of generalized CM rings is the conditions that for any parameter ideal q contained in a large power of the maximal ideal m of A, the difference between length and multiplicityis independent of the choice of q. However, if the dimension of the non-CM locus is larger than zero, little is known about how this dimension is related to the structure of the local ring A. The purpose of this paper is to show that if M is a finitely generated A-module, then there exist systems of parameters x = (x1, …, xd) (where d = dim M) such that the differenceis a polynomial in n1, …, nd for all positive integers n1, …, nd and the degree of IM(n1, …, nd;x) is independent of the choice of x. We shall also give various characterizations of this degree by using the notion of reducing systems of parameters of Auslander and Buchsbaum[l]. In particular, if the module M is equidimensional we shall show that the degree of IM(n1, …, nd;x) is equal to the dimension of the non-CM locus of M.

scholarly journals Semi-local formal fibers of minimal prime ideals of excellent reduced local rings

2012 ◽  
Vol 4 (1) ◽  
pp. 29-56 ◽  
Author(s):  
N. Arnosti ◽  
R. Karpman ◽  
C. Leverson ◽  
J. Levinson ◽  
S. Loepp
Keyword(s):  
Prime Ideals ◽  
Local Rings ◽  
Minimal Prime
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