Multiplicative Ideal Theory over Integral Domains

2016 ◽  
pp. 403-468
Author(s):  
Fanggui Wang ◽  
Hwankoo Kim
Keyword(s):  
Ideal Theory ◽  
Integral Domains

scholarly journals The monotone catenary degree of monoids of ideals

2019 ◽  
Vol 29 (03) ◽  
pp. 419-457 ◽  
Author(s):  
Alfred Geroldinger ◽  
Andreas Reinhart
Keyword(s):  
Ideal Theory ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Central Topic ◽  
Sets Of Lengths ◽  
Catenary Degree

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper, we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of [Formula: see text]-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions, we establish arithmetical finiteness results, in particular, for the monotone catenary degree and for the structure of sets of lengths and of their unions.

scholarly journals On integrally dependent integral domains

1947 ◽  
Vol 240 (819) ◽  
pp. 295-326 ◽  
Keyword(s):  
Regular Representation ◽  
Ideal Theory ◽  
Separable Extension ◽  
Quotient Field ◽  
Integral Domains ◽  
Closed Set ◽  
Ramification Theory ◽  
Integrally Closed ◽  
The Subject ◽  
Quotient Rings

The subject of this paper is the simultaneous ideal theory of a pair of integral domains R and G ≥ R, of which R is integrally closed, and G integrally dependent on R. It is assumed that the quotient field L of G is a finite separable extension of the quotient field K of R. The device of quotient rings effects a preliminary simplification in many of the proofs; the quotient rings R S and G S , with respect to any existent multiplicatively closed set S of non-zero elements of R, also satisfy the above basic postulates for R and G. Another method of preliminary simplification, valuable in the discussion of ramification theory, is the adjunction of Kronecker indeterminates. Such indeterminates (algebraically independent over K ) are denoted by y or z ; in connexion with the regular representation of L , they are regarded as adjoined to K .

scholarly journals On Prüfer-like properties of Leavitt path algebras

2019 ◽  
Vol 19 (07) ◽  
pp. 2050122 ◽  
Author(s):  
Songül Esin ◽  
Müge Kanuni ◽  
Ayten Koç ◽  
Katherine Radler ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Sufficient Conditions ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Integral Domains ◽  
Ideal Lattices ◽  
Path Algebras ◽  
Dedekind Domains ◽  
Prufer Domains ◽  
Leavitt Path Algebras ◽  
Prüfer Domains

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra [Formula: see text], in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of [Formula: see text] satisfy the distributive law, a property of Prüfer domains and that [Formula: see text] is a multiplication ring, a property of Dedekind domains. In this paper, we first show that [Formula: see text] satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers [Formula: see text], [Formula: see text] and [Formula: see text]. We also show that [Formula: see text] satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which [Formula: see text] satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

Utopophobia

2019 ◽  
Author(s):  
David Estlund
Keyword(s):  
Political Philosophy ◽  
Social Justice ◽  
Political Thought ◽  
Political Action ◽  
Ideal Theory ◽  
Theory Of Justice ◽  
Sound Theory ◽  
Theory Versus ◽  
History Of ◽  
Practical Guidance

Throughout the history of political philosophy and politics, there has been continual debate about the roles of idealism versus realism. For contemporary political philosophy, this debate manifests in notions of ideal theory versus nonideal theory. Nonideal thinkers shift their focus from theorizing about full social justice, asking instead which feasible institutional and political changes would make a society more just. Ideal thinkers, on the other hand, question whether full justice is a standard that any society is likely ever to satisfy. And, if social justice is unrealistic, are attempts to understand it without value or importance, and merely utopian? This book argues against thinking that justice must be realistic, or that understanding justice is only valuable if it can be realized. The book does not offer a particular theory of justice, nor does it assert that justice is indeed unrealizable—only that it could be, and this possibility upsets common ways of proceeding in political thought. The book's author engages critically with important strands in traditional and contemporary political philosophy that assume a sound theory of justice has the overriding, defining task of contributing practical guidance toward greater social justice. Along the way, it counters several tempting perspectives, including the view that inquiry in political philosophy could have significant value only as a guide to practical political action, and that understanding true justice would necessarily have practical value, at least as an ideal arrangement to be approximated. Demonstrating that unrealistic standards of justice can be both sound and valuable to understand, the book stands as a trenchant defense of ideal theory in political philosophy.

The Role of Ideal Theory

2018 ◽  
Author(s):  
Christie Hartley
Keyword(s):  
Ideal Theory ◽  
Racial Minorities ◽  
Gender Inequalities ◽  
Liberal Society ◽  
Social Categories ◽  
Ideal Society ◽  
Gender And Race ◽  
Theories Of Justice ◽  
Comprehensive Doctrines

This chapter discusses whether political liberalism’s commitment to ideal theory makes it ill-suited for theorizing about justice for socially subordinated groups such as women and racial minorities. It is shown that political liberalism’s commitment to ideal theory does not entail assuming away race or gender as social categories that give rise to concerns about justice. Even within a politically liberal well-ordered (ideal) society racial or gender inequalities may arise due to the role that beliefs about race or gender play in some persons’ comprehensive doctrines. Furthermore, it is argued that theories of justice developed for a well-ordered politically liberal society provide important guidance for correcting injustices on the basis of gender and race in nonideal societies.

Introduction

2017 ◽  
Author(s):  
Clare Chambers
Keyword(s):  
Ideal Theory ◽  
Free State ◽  
Regulatory Structure ◽  
Intended Audience ◽  
Political Commitments

The introduction sets out the arguments of Against Marriage: An Egalitarian Defence of the Marriage-Free State. It starts with the question of the intended audience for the book, discussing the implications for those with both personal and political commitments. It outlines the negative thesis against marriage offered in Part One, and the positive thesis in favour of an alternative regulatory structure in Part Two. It summarizes the contents of each chapter. It also includes a discussion of the book’s location within ideal and non-ideal theory.

Humanitarian Intervention and Non-Ideal Theory

2018 ◽  
Author(s):  
Ramon Das
Keyword(s):  
Human Rights ◽  
Civil War ◽  
Humanitarian Intervention ◽  
Ideal Theory ◽  
Military Power ◽  
Moral Integrity ◽  
Philosophical Debate ◽  
Human Rights Abuses ◽  
Target State ◽  
Target States

This chapter argues that the philosophical debate around humanitarian intervention would be improved if it were less ‘ideal-theoretic’. It identifies two ideal-theoretic assumptions. One, in target states where humanitarian intervention is being considered, there are two distinct and easily identified groups: ‘bad guys’ committing serious human rights abuses, and innocent civilians against whom the abuses are being committed. Two, external to the target state in question, there are suitably qualified ‘good guys’—prospective interveners who possess both the requisite military power and moral integrity. If the assumptions hold, the prospects for successful humanitarian intervention are much greater. As a contrast, some possible non-ideal assumptions are that (i) there are many bad guys in a civil war, and (ii) the good guy intervener is itself supporting some of the bad guys. If these non-ideal assumptions hold, prospects for successful humanitarian intervention are small.

scholarly journals A soft set theoretic approach to an AG-groupoid via ideal theory with applications

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Faisal Yousafzai ◽  
Mohammed M. Khalaf
Keyword(s):  
Ideal Theory ◽  
Soft Set ◽  
Theoretic Approach

scholarly journals Conscription and the Color Line: Rawls, Race and Vietnam

2021 ◽  
pp. 1-24
Author(s):  
Brandon M. Terry
Keyword(s):  
Political Philosophy ◽  
Economic Inequality ◽  
Black Americans ◽  
Black Families ◽  
Ideal Theory ◽  
Racial Group ◽  
Color Line ◽  
Racial Injustice ◽  
Black Poverty

This article revisits one of John Rawls's rare forays into activist politics, his proposal presented to the Harvard faculty, calling for a denunciation of the “2-S” system of student deferments from conscription. In little-studied archival papers, Rawls argued that the draft both exposed “background” structural racial injustice and constituted a burdening of black Americans that violated the norms of fair cooperation. Rather than obscuring racial injustice and focusing exclusively on economic inequality, as Charles Mills has claimed, Rawls rejected the ascendant conservative views that naturalized black poverty or else attributed it to cultural pathologies in black families. Thus Rawls found nothing illicit in taking the position of a disadvantaged racial group as a relevant comparison when applying his ideal theory to nonideal circumstances. However, I contend in this article that Rawls's account of political philosophy as an attempt to find a consensus may be similarly ideological, leading him to displace the reality of conflict through begging descriptions, expressivist formulations, and historical romanticism.

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