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.

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 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.

A Gorenstein analogue of a result of Bertin

2014 ◽  
Vol 14 (02) ◽  
pp. 1550019 ◽  
Author(s):  
Lei Qiao ◽  
Fanggui Wang
Keyword(s):  
Integral Domain ◽  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Gorenstein Flat

An integral domain R is called a Gorenstein Prüfer (G-Prüfer) domain if it is coherent and every submodule of a flat R-module is Gorenstein flat. In this paper, we show that every n-FC domain is an intersection of local G-Prüfer domains. We also give several characterizations of G-Prüfer domains.

scholarly journals π-domains, overrings, and divisorial ideals

1978 ◽  
Vol 19 (2) ◽  
pp. 199-203 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Prufer Domains ◽  
Interesting Class ◽  
Prüfer Domains ◽  
Krull Domains

In this paper we study several generalizations of the concept of unique factorization domain. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. Theorem 1 shows that the class of π-domains forms a rather natural subclass of the class of Krull domains. In Section 3 we consider overrings of π-domains. In Section 4 generalized GCD-domains are introduced: these form an interesting class of domains containing all Prüfer domains and all π-domains.

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.

Intersections of quotient rings and Prüfer v-multiplication domains

2016 ◽  
Vol 15 (08) ◽  
pp. 1650149 ◽  
Author(s):  
Said El Baghdadi ◽  
Marco Fontana ◽  
Muhammad Zafrullah
Keyword(s):  
Integral Domain ◽  
Algebraic Approach ◽  
Quotient Field ◽  
Topological Characterization ◽  
Locally Finite ◽  
Star Operation ◽  
Star Operations ◽  
Quotient Rings ◽  
Special Case

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call an overring [Formula: see text] of [Formula: see text] a subring of [Formula: see text] containing [Formula: see text] as a subring. A family [Formula: see text] of overrings of [Formula: see text] is called a defining family of [Formula: see text], if [Formula: see text]. Call an overring [Formula: see text] a sublocalization of [Formula: see text], if [Formula: see text] has a defining family consisting of rings of fractions of [Formula: see text]. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535–2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer [Formula: see text]-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer [Formula: see text]-multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d’anneaux intègers, J. Algebra 43 (1976) 55–60] and Proposition 3.2 of [N. Dessagnes, Intersections d’anneaux de Mori — exemples, Port. Math. 44 (1987) 379–392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P vMDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer [Formula: see text]-multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]).

On ideal theory in high prufer domains

1975 ◽  
Vol 14 (4) ◽  
pp. 303-336 ◽  
Author(s):  
Moshe Jarden
Keyword(s):  
Ideal Theory ◽  
Prufer Domains ◽  
Prüfer Domains

scholarly journals Arithmetic Rings and their generalizations

2018 ◽  
Vol 51 (381) ◽  
pp. FP1-FP6
Author(s):  
R. Strano
Keyword(s):  
Commutative Ring ◽  
Natural Generalization ◽  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains

Prüfer domains are characterized by various properties regarding ideals and operations between them. In this note we consider six of these properties. The natural generalization of the notion of Prüfer domain to the case of a commutative ring with unit, not necessarily a domain, is the notion of arithmetic ring. We ask if the previous properties characterize arithmetic ring in the case of a general commutative ring with unit. We prove that four of such properties characterize arithmetic rings while the remaining two are weaker and give rise to two different generalizations.

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