2-Prime ideals and their applications

2016 ◽  
Vol 15 (03) ◽  
pp. 1650051 ◽  
Author(s):  
Charef Beddani ◽  
Wahiba Messirdi
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
The Other ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Other Hand ◽  
Valuation Rings ◽  
Integrally Closed

This paper introduces the notion of [Formula: see text]-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain [Formula: see text] is a valuation ring if and only if every ideal of [Formula: see text] is [Formula: see text]-prime. On the other hand, we will prove that the normalization [Formula: see text] of [Formula: see text] is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of [Formula: see text] is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

scholarly journals Integrally closed factor domains

1988 ◽  
Vol 37 (3) ◽  
pp. 353-366 ◽  
Author(s):  
Valentina Barucci ◽  
David E. Dobbs ◽  
S.B. Mulay
Keyword(s):  
Prime Ideal ◽  
The Other ◽  
Integral Domains ◽  
Other Hand ◽  
Dedekind Domains ◽  
Integrally Closed

This paper characterises the integral domains R with the property that R/P is integrally closed for each prime ideal P of R. It is shown that Dedekind domains are the only Noetherian domains with this property. On the other hand, each integrally closed going-down domain has this property. Related properties and examples are also studied.

Overrings of Prüfer v-multiplication domains

2016 ◽  
Vol 16 (08) ◽  
pp. 1750147 ◽  
Author(s):  
Shiqi Xing ◽  
Fanggui Wang
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Integral Ideal ◽  
Ideal Formula ◽  
Similar Work ◽  
Star Operation ◽  
Flat Modules ◽  
Torsion Free

Let [Formula: see text] be an integral domain, [Formula: see text] and [Formula: see text] the set of fractional ideals of [Formula: see text]. Let [Formula: see text] a finitely generated ideal with [Formula: see text]. For a torsion-free [Formula: see text]-module [Formula: see text], define [Formula: see text] for some [Formula: see text]. Call [Formula: see text] a [Formula: see text]-module if [Formula: see text]. On [Formula: see text], the function [Formula: see text] is a star-operation of finite character. An integral ideal [Formula: see text] maximal with respect to being a proper [Formula: see text]-ideal is a prime ideal called a maximal [Formula: see text]-ideal. A torsion-free [Formula: see text]-module [Formula: see text] is called [Formula: see text]-flat, if [Formula: see text] is a flat [Formula: see text]-module for each [Formula: see text], the set of maximal [Formula: see text]-ideals of [Formula: see text]. [Formula: see text] is called a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD), if [Formula: see text] is a valuation ring for each [Formula: see text]. We characterize [Formula: see text]-flat modules in a manner similar to the characterization of flat modules, study them when they are rings [Formula: see text] with [Formula: see text] and characterize P[Formula: see text]MDs using them and compare our work with similar work in the literature.

Primary Ideals in Prüfer Domains

1966 ◽  
Vol 18 ◽  
pp. 1024-1030 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Quotient Ring ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Primary Ideals

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

The Equality (A∩B)n=An∩Bnfor Ideals

1972 ◽  
Vol 24 (5) ◽  
pp. 792-798 ◽  
Author(s):  
Robert Gilmer ◽  
Anne Grams
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Valuation Ring ◽  
Integral Domain ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Integrally Closed

LetDbe an integral domain with identity, and letRbe a commutative ring. Ifnis a positive integer,Rwill be said tohave property (n), (n)′,or(n)″according asproperty(n):For anyx, y∊R,(x,y)n= (xn, yn).property(n)′: For anyx ∊ Rand any idealAofRsuch that xn∊ An, it follows thatx ∊ A.property(n)′: For any idealsA, BofR, (A∩ B)n= An∩ Bn.J. Ohm introduced property (n) in [7] in connection with the question: Ifn≦ 2 and ifDhas property (n), mustDbe a Prüfer domain? (The integral domainDwith identity is a Prüfer domain if each nonzero finitely generated ideal ofDis invertible; equivalently,DPis a valuation ring for each proper prime idealPofD.)Prior to Ohm's paper, it was known that ifDhas property (2) and ifDis integrally closed, thenDis Prüfer.

Some rings of interest in the study of places

1970 ◽  
Vol 68 (2) ◽  
pp. 255-264
Author(s):  
J. T. Knight
Keyword(s):  
Integral Domain ◽  
Automorphism Groups ◽  
The Other ◽  
Elementary Property ◽  
Unique Factorization ◽  
Precise Sense ◽  
Other Hand ◽  
Integrally Closed

The rings studied in this paper were first constructed in connexion with the problem (Proposition 1) of extending a place from an integral domain to its field of fractions. They provide (Proposition 3) a series of examples of places not so extendable, generalizing an example implicit in the work of Samuel(3); and by the use of ultraproducts they show that such extendability is (in certain precise sense) not an elementary property (Proposition 4). On the other hand they have some unexpected properties, especially in connexion with the relation 1/y0 + … + 1/yr = 1, and they have very small automorphism groups (Proposition 2 and Corollaries); and they are not unique factorization rings (Proposition 5), though they are integrally closed (Proposition 6).

A Characterization of the Topological Dimension

1982 ◽  
Vol 25 (4) ◽  
pp. 487-490
Author(s):  
Gerd Rodé
Keyword(s):  
Hausdorff Space ◽  
The Other ◽  
Natural Numbers ◽  
Topological Dimension ◽  
Other Hand ◽  
The One

AbstractThis paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N1,..., NK), where N1,..., NK, K are natural numbers. It is shown that a space X contains a system of faces of type (N1,..., NK) if and only if dim(X) ≥ N1 + … + NK. The two limit cases of the theorem, namely Nk = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

scholarly journals Maqui berry vs Sloe berry – Liquor-based Beverage for New Development

2015 ◽  
Vol 10 (1) ◽  
pp. 1934578X1501000
Author(s):  
Amadeo Gironés-Vilaplana ◽  
Diego A. Moreno ◽  
Cristina García-Viguera
Keyword(s):  
The Other ◽  
Northern Spain ◽  
Organoleptic Properties ◽  
Beneficial Effects ◽  
Other Hand ◽  
New Development ◽  
Aristotelia Chilensis ◽  
Maqui Berry ◽  
Over Time

“Pacharán” is an aniseed liquor-based beverage made with sloe berry ( Prunus spinosa L.) that has been produced in northern Spain. On the other hand, maqui berry ( Aristotelia chilensis) is a common edible berry from Chile, and currently under study because of its multiple beneficial effects on health. The aim of this work was to design a new aniseed liquor-based beverage with maqui berry, as an industrial alternative to a traditional alcoholic product with bioactive berries. The characterization of its composition, compared with the traditional “Pacharán”, and its evolution during maceration (6 and 12 months) showed that the new maqui liquor had significantly-higher anthocyanin retention over time. More studies on the organoleptic properties and bioactivity are underway.

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

scholarly journals Ontology, Ontologies and the “I” of FAIR

2020 ◽  
Vol 2 (1-2) ◽  
pp. 181-191 ◽  
Author(s):  
Giancarlo Guizzardi
Keyword(s):  
Value Of Information ◽  
Direct Consequence ◽  
Semantic Interoperability ◽  
The Other ◽  
Added Value ◽  
Computational Tools ◽  
Other Hand ◽  
Guiding Principles

According to the FAIR guiding principles, one of the central attributes for maximizing the added value of information artifacts is interoperability. In this paper, I discuss the importance, and propose a characterization of the notion of Semantic Interoperability. Moreover, I show that a direct consequence of this view is that Semantic Interoperability cannot be achieved without the support of, on one hand, (i) ontologies, as meaning contracts capturing the conceptualizations represented in information artifacts and, on the other hand, of (ii) Ontology, as a discipline proposing formal meth- ods and theories for clarifying these conceptualizations and articulating their representations. In particular, I discuss the fundamental role of formal ontological theories (in the latter sense) to properly ground the construction of representation languages, as well as methodological and computational tools for supporting the engineering of ontologies (in the former sense) in the context of FAIR.

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