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.

Distinguished Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 181-193 ◽  
Author(s):  
Raymond C. Heitmann ◽  
Stephen McAdam
Keyword(s):  
Quotient Field ◽  
Zero Divisors ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Krull Domains

This paper introduces a class of domains which we hope to show merits some attention.Definition. The domain R is said to be a distinguished domain if for any 0 ≠ z ∈ K, the quotient field of R, (1 : z) does not consist entirely of zero divisors modulo (1 : z–l). (Note: Here we use the fact that a zero module has no zero divisors. Thus if z–l ∈ R, so that (1 : z–l) = R, then the condition holds trivially.)Section 1 of this paper gives numerous examples of distinguished domains, foremost among them being Krull domains and Prufer domains. In fact Prüfer domains are shown to be exactly those distinguished domains whose prime lattice forms a tree. Other distinguished domains can be constructed by the D + M construction. It is shown that distinguished domains are integrally closed but the converse fails.

scholarly journals Coherent Overrings

1979 ◽  
Vol 22 (3) ◽  
pp. 331-337 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Quotient Field ◽  
Integral Domains ◽  
One Dimensional ◽  
Prufer Domains ◽  
Prüfer Domains

In the study of particular categories of integral domains, it has proved useful to know which overrings of the domains of interest lie within the category, and indeed whether all such overrings do. (Recall: an overring of R is a ring T with R ⊆ T ⊆ quotient field of R.) Two classes of domains classically studied in this setting are Prüfer domains and one-dimensional Noetherian domains. Since both of these classes are contained in the category of coherent domains, it is natural to investigate this category in this setting.

scholarly journals Decomposition and classification of length functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1109-1129
Author(s):  
Dario Spirito
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Bijective Correspondence ◽  
Length Functions ◽  
Standard Representation ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

scholarly journals PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

2003 ◽  
Vol 02 (01) ◽  
pp. 21-50 ◽  
Author(s):  
M. FONTANA ◽  
P. JARA ◽  
E. SANTOS
Keyword(s):  
Characterization Theorem ◽  
Field Extensions ◽  
Star Operation ◽  
Classical Characterization ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Semistar Operation ◽  
Prüfer Domains ◽  
The Given

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

The number of intermediate rings in FIP extension of integral domains

2019 ◽  
Vol 19 (09) ◽  
pp. 2050171 ◽  
Author(s):  
Mabrouk Ben Nasr ◽  
Ali Jaballah
Keyword(s):  
Quotient Field ◽  
Integral Domains ◽  
Integrally Closed

Let [Formula: see text] be an extension of integral domains with only finitely many intermediate rings, where [Formula: see text] is not a field and [Formula: see text] is not necessarily the quotient field of [Formula: see text] or [Formula: see text] is not necessarily integrally closed in [Formula: see text]. In this paper, we exactly determine the number of intermediate rings between [Formula: see text] and [Formula: see text] and give a way to compute it.

scholarly journals Group-theoretic and topological invariants of completely integrally closed Prüfer domains

2016 ◽  
Vol 220 (12) ◽  
pp. 3927-3947 ◽  
Author(s):  
Olivier A. Heubo-Kwegna ◽  
Bruce Olberding ◽  
Andreas Reinhart
Keyword(s):  
Topological Invariants ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

scholarly journals Quasi-projectivity over domains

1999 ◽  
Vol 60 (1) ◽  
pp. 129-135
Author(s):  
Dmitri Alexeev
Keyword(s):  
Tensor Product ◽  
Integral Domain ◽  
Quotient Field ◽  
Projective Ideal ◽  
Prufer Domains ◽  
Prüfer Domains

Let R be an integral domain with quotient field Q. We investigate quasi- and Q-projective ideals, and properties of domains all ideals of which are quasi-projective. It is shown that the so-called l½-generated ideals are quasi-projective, moreover, projective. A module M is quasi-projective if and only if, for a projective ideal P of R, the tensor product M ⊗RP is quasi-projective. Domains whose all ideals are quasi-projective are characterised as almost maximal Prüfer domains. Q is quasi-projective if and only if every proper submodule of Q is complete in its R-topology.

Bezout Domains and Rings with a Distributive Lattice of Right Ideals

1986 ◽  
Vol 38 (2) ◽  
pp. 286-303 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Integral Domains ◽  
Bezout Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided.Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

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 .

Integral Domains Whose w-Integral Closures Are Krull Domains

2020 ◽  
Vol 27 (02) ◽  
pp. 287-298
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Krull Domain ◽  
Integral Closures ◽  
Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

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