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.

scholarly journals A Secular Critique of Religious Ethics and Politics

2017 ◽  
Author(s):  
Paul Cliteur
Keyword(s):  
Extreme Case ◽  
Moral Autonomy ◽  
Religious Ethics ◽  
Religious Leader ◽  
Political Autonomy ◽  
Central Notion ◽  
The Difference ◽  
Do So ◽  
The Ideal ◽  
Ethics And Politics

This chapter discusses the difference between a nonsecular or religious critique of religious ethics and politics and a specifically secular critique. It introduces the central notion of a secular critique, autonomy, and its two types, moral and political. Moral autonomy entails the separation of religion from ethics. The ideal of making that separation is called “moral secularism.” The opposite of moral autonomy is “moral heteronomy.” An extreme case of moral heteronomy is discussed: Abraham’s willingness to sacrifice his own son when God commanded him to do so. Next, the importance of political autonomy and political secularism is illustrated with reference to the conflict between the king Ahab (the model of a secular ruler) and the prophet Elijah (the model of a religious leader). Some stories in the holy scriptures of the monotheist religions held in common by Judaism, Christianity, and Islam are unfavorable toward secularism (both moral and political).

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] $.

Hilbert’s programme and formalism

2018 ◽  
Author(s):  
Michael Detlefsen
Keyword(s):  
Subject Matter ◽  
Reverse Mathematics ◽  
Incompleteness Theorem ◽  
Axiomatic System ◽  
Logical Inference ◽  
Natural Numbers ◽  
Kurt Gödel ◽  
Do So ◽  
The Ideal

In the first, geometric stage of Hilbert’s formalism, his view was that a system of axioms does not express truths particular to a given subject matter but rather expresses a network of logical relations that can (and, ideally, will) be common to other subject matters. The formalism of Hilbert’s arithmetical period extended this view by emptying even the logical terms of contentual meaning. They were treated purely as ideal elements whose purpose was to secure a simple and perspicuous logic for arithmetical reasoning – specifically, a logic preserving the classical patterns of logical inference. Hilbert believed, however, that the use of ideal elements should not lead to inconsistencies. He thus undertook to prove the consistency of ideal arithmetic with its contentual or finitary counterpart and to do so by purely finitary means. In this, ‘Hilbert’s programme’, Hilbert and his followers were unsuccessful. Work published by Kurt Gödel in 1931 suggested that such failure was perhaps inevitable. In his second incompleteness theorem, Gödel showed that for any consistent formal axiomatic system T strong enough to formalize what was traditionally regarded as finitary reasoning, it is possible to define a sentence that expresses the consistency of T, and is not provable in T. From this it has generally been concluded that the consistency of even the ideal arithmetic of the natural numbers is not finitarily provable and that Hilbert’s programme must therefore fail. Despite problematic elements in this reasoning, post-Gödelian work on Hilbert’s programme has generally accepted it and attempted to minimize its effects by proposing various modifications of Hilbert’s programme. These have generally taken one of three forms: attempts to extend Hilbert’s finitism to stronger constructivist bases capable of proving more than is provable by strictly finitary means; attempts to show that for a significant family of ideal systems there are ways of ‘reducing’ their consistency problems to those of theories possessing more elementary (if not altogether finitary) justifications; and attempts by the so-called ‘reverse mathematics’ school to show that the traditionally identified ideal theories do not need to be as strong as they are in order to serve their mathematical purposes. They can therefore be reduced to weaker theories whose consistency problems are more amenable to constructivist (indeed, finitist) treatment.

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.

Units in Group Rings of the Infinite Dihedral Group

1991 ◽  
Vol 34 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Maciej Mirowicz
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Dihedral Group ◽  
Group Rings ◽  
Finitely Generated ◽  
Group Of Units ◽  
Group D

AbstractThis paper studies the group of units U(RD∞) of the group ring of the infinite dihedral group D∞ over a commutative integral domain R. The structures of U(Z2D∞) and U(Z3D∞) are determined, and it is shown that U(ZD∞) is not finitely generated.

Groupings of Metabelian Groups and Extension Categories

1980 ◽  
Vol 32 (2) ◽  
pp. 449-459 ◽  
Author(s):  
K. W. Roggenkamp
Keyword(s):  
Group Ring ◽  
General Result ◽  
Prime Divisor ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Exact Sequences

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall proveTHEOREM. If RG ≌ RH as R-algebras, then G ≌ HThe question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

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.

Carotid Angioplasty

1999 ◽  
Vol 5 (1_suppl) ◽  
pp. 61-65
Author(s):  
G. Duckwiler ◽  
Y.P. Gobin ◽  
F. Viñuela
Keyword(s):  
Insurance Coverage ◽  
Patient Characteristics ◽  
The United States ◽  
Current Status ◽  
Carotid Angioplasty ◽  
Operator Experience ◽  
Angioplasty And Stenting ◽  
Carotid Angioplasty And Stenting ◽  
Do So ◽  
The Ideal

Although no consensus yet exists on the ideal patient characteristics, materials, and indications for carotid angioplasty, it is clear that this procedure which is increasing in popularity will continue to do so. Until such time as the procedure is routinely approved (there are still barriers to insurance coverage for these procedures in the United States), we are highly selective in our application of carotid angioplasty. So far our experience is limited to approximately 40 patients with no major complications and no strokes. However, patient characteristics, operator experience, and patient selection play large roles in the outcomes of these procedures. The current status of carotid angioplasty and stenting wilt be discussed as well as the potential complications and their treatment.

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

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