A Localization of R[x]

1981 ◽  
Vol 33 (1) ◽  
pp. 103-115 ◽  
Author(s):  
James A. Huckaba ◽  
Ira J. Papick
Keyword(s):  
Integral Domain ◽  
Closed Sets ◽  
Interesting Paper ◽  
Object Of Study ◽  
The Ideal

Throughout this paper, R will be a commutative integral domain with identity and x an indeterminate. If ƒ ∈ R[x], let CR(ƒ) denote the ideal of R generated by the coefficients of ƒ. Define SR = {ƒ ∈ R[x]: cR(ƒ) = R} and UR = {ƒ ∈ R(x): cR(ƒ)– 1 = R}. For a,b ∈ R, write . When no confusion may result, we will write c(ƒ), S, U, and (a:b). It follows that both S and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s ⊆ R[x]U.The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work.

scholarly journals Anti-integral extensions $ {R[{\alpha}]/R$

2004 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Kiyoshi Baba ◽  
Ken-Ichi Yoshida
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Arbitrary Positive Integer ◽  
Integral Element ◽  
The Ideal

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

On Representations of Orders Over Dedekind Domains

1960 ◽  
Vol 12 ◽  
pp. 107-125 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Integral Domain ◽  
Homological Algebra ◽  
Dedekind Domain ◽  
Main Concern ◽  
Direct Sums ◽  
Dedekind Domains ◽  
The Ideal

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

The ideal transform and overrings of an integral domain

1968 ◽  
Vol 107 (4) ◽  
pp. 301-306 ◽  
Author(s):  
Jim Brewer
Keyword(s):  
Integral Domain ◽  
The Ideal

THE VIRTUOSIC EXEGESIS OF THE BRAHMAVADIN AND THE RABBI

Numen ◽  
2002 ◽  
Vol 49 (4) ◽  
pp. 427-459 ◽  
Author(s):  
Timothy Lubin
Keyword(s):  
Biblical Text ◽  
Right Action ◽  
The World ◽  
Interpretive Strategies ◽  
True Knowledge ◽  
Object Of Study ◽  
The Temple ◽  
The Sacrifice ◽  
The Ideal

AbstractDespite the vast spatial and theological gulfs separating the Rabbinic and Brahmanic communities, their respective intellectual projects have a number of analogous features. My discussion will (1) outline for each tradition a set of interpretive strategies, showing how these two sets are strikingly similar in approach and logic. Then I will (2) propose that these resemblances are not entirely coincidental. They largely stem from a similar view of the object of study—Torah and the biblical text for the Rabbis, the sacrifice and its verbal articulation for the Brahmins—as eternal, not of human authorship, perfect in form, rich in hidden meanings, the criterion of right action and true knowledge. The exegete aims to fully internalize the sacred word, to perceive the world through it, and to uncover what is hidden in it. This much of my analysis might also be applicable to other traditions that regard themselves as possessing revelation, but (3) I argue that there are further parallels here in the direction these traditions carried their interpretive enterprise. In each tradition, the interpreters continued to build an edifice of ritual knowledge and interpretation even as the central rites were eclipsed by other forms of piety: whether because the cult became inaccessible (in the Diaspora) or unperformable (when the Temple was destroyed), or because it lost patronage (as appears to have happened in India). In tandem with the shift away from priestly sacrifice, each tradition promotes the ideal of study for its own sake, and the transfer of priestly functions to the learned householder.

scholarly journals FILO OF ALEXANDRIA: DISCUSSION TOWARD THE LOGOS IN THE PHILOSOPHY OF EDUCATION

2021 ◽  
Vol 10 (34) ◽  
Author(s):  
A.K EROKHIN ◽  
Keyword(s):  
Educational Activity ◽  
Greek Philosophy ◽  
Philosophical Work ◽  
Philo Of Alexandria ◽  
Object Of Study ◽  
Infinite Power ◽  
The Mind ◽  
Creative Teacher ◽  
The Relationship ◽  
The Ideal

The article considers the influence of Greek philosophy on the ideas of the formation of the Hellenistic philosopher Philo of Alexandria. The object of study was the philosophical work of Philo. This study aimed to discover the ambiguity of the term logos as a central concept that defines in Philo's philosophy the relationship between God (the ideal creative teacher) and the world. In the works of Philo, the Logos appears as the highest, sub-divine, infinite power of the mind, which has no signs, but at the same time is identified with God. The transcendental nature of the Logos, embodied in the image of God's mind, in its paradoxical nature closely corresponds to holiness and higher wisdom. The research methodology is based on an interpretation that allows us to define the allegory and, therefore, the real meaning of Philo's philosophy, the central part of which is the philosophical reflection of the Holy Scriptures as the main source of education and the concept of Scripture, undergoing specific and simultaneously incomparable modifications. To identify these meanings, methods of systematization and hermeneutics are used. The result of the study is expressed in identifying various forms of embodiment and educational activity and the Logos.

scholarly journals IDEAL CONVERGENCE OF DOUBLE SEQUENCES OF CLOSED SETS

2021 ◽  
pp. 605
Author(s):  
Ozer Talo ◽  
Yurdal Sever
Keyword(s):  
Metric Space ◽  
Double Sequence ◽  
Double Sequences ◽  
Ideal Convergence ◽  
Closed Sets ◽  
The Ideal

In the present paper, we introduce the concepts of ideal inner and ideal outer limits which always exist even if empty sets for double sequences of closed sets in Pringsheim's sense. Next, we give some formulas for finding ideal inner and outer limits in a metric space. After then, we define Kuratowski ideal convergence of double sequences of closed sets by means of the ideal inner and ideal outer limits of a double sequence of closed sets. Additionally, we give some examples that our result is more general than the results obtained before.

scholarly journals Proposta de definição interdisciplinar de nome próprio

2021 ◽  
Vol 2 (4) ◽  
pp. 70-94
Author(s):  
Márcia Sipavicius Seide
Keyword(s):  
Final Section ◽  
Proper Names ◽  
Relevance Theory ◽  
The Third ◽  
Proper Name ◽  
Definition Of ◽  
Object Of Study ◽  
The Ideal

Este artigo apresenta uma proposta de definição interdisciplinar do conceito de nome próprio elaborada com base na Onomástica Cognitiva (SJÖBLOM, 2010), na Teoria da Relevância (SPERBER  WILSON ,2001 [1995],  SEIDE SCHULTZ, 2014), na Neurolinguística (VAN LANGENDONCK, 2007), e no conhecimento onomástico do falante ideal (SEIDE, 2021). Na primeira seção deste artigo, descrevem-se o objeto de estudo da Onomástica e as características da subárea da Onomástica em que se insere a pesquisa. Na segunda, são retomadas considerações a respeito dos nomes próprios feitas para sua definição como endereço conceitual. Na terceira seção, apresentam-se as descobertas neurolinguísticas e a descrição do conhecimento onomástico do falante ideal, as quais são integradas resultando na redefinição de nome próprio descrita ao final da terceira seção. Na quarta e última seção do artigo, são descritas algumas implicações dessa redefinição para os estudos onomásticos.  Proposal of interdisciplinary definition of proper nameAbstract: This article makes a proposal of interdisciplinary definition of the concept of proper name based on Cognitive Onomastics (SJÖBLOM, 2010), Theory of Relevance (SPERBER WILSON, 2001 [1995], SEIDE SCHULTZ, 2014, Neurolinguistics (VAN LANGENDONCK, 2007) and the onomastic knowledge of the ideal speaker (SEIDE, 2021). In the first section of this article, the object of study of Onomastics and the characteristics of the onomastic subarea in which the research is included are described. In the second, considerations about proper made by Sperber and Wilson (2001 [1995]) are integrated to the definition of proper names as a conceptual address. In the third section, the neurolinguistic discoveries and the description of the onomastic knowledge of the ideal speaker are presented and integrated and the proper name redefinition is described. In the fourth and final section of the article, some implications of this redefinition for onomastics studies are described.Keywords: Proper name, Cognitive Onomastics, Relevance Theory, Neurolinguistics. Onomastic Knowledge of   Ideal Speaker. 

On the Weak Global Dimension of Pseudovaluation Domains

1978 ◽  
Vol 21 (2) ◽  
pp. 159-164 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Global Dimension ◽  
Valuation Domain ◽  
Valuation Domains ◽  
Dimension 2 ◽  
The Ideal

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

scholarly journals On the automorphisms of the group ring of a unique product group

1987 ◽  
Vol 30 (2) ◽  
pp. 201-205 ◽  
Author(s):  
A. A. Mehrvarz ◽  
D. A. R. Wallace
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Nilpotent Ideal ◽  
Product Group ◽  
Unique Product ◽  
Group 6 ◽  
Do So ◽  
The Ideal

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

Sign in / Sign up

Export Citation Format

Share Document