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 Commutative ideal theory without finiteness conditions: Primal ideals

2004 ◽  
Vol 357 (7) ◽  
pp. 2771-2798 ◽  
Author(s):  
Laszlo Fuchs ◽  
William Heinzer ◽  
Bruce Olberding
Keyword(s):  
Ideal Theory ◽  
Finiteness Conditions ◽  
Commutative Ideal

scholarly journals INTEGRAL DOMAINS WHOSE SIMPLE OVERRINGS ARE INTERSECTIONS OF LOCALIZATIONS

2005 ◽  
Vol 04 (02) ◽  
pp. 195-209 ◽  
Author(s):  
MARCO FONTANA ◽  
EVAN HOUSTON ◽  
THOMAS LUCAS
Keyword(s):  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

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.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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