Divisorial Prime Ideals in Prüfer Domains

1984 ◽  
Vol 27 (3) ◽  
pp. 324-328 ◽  
Author(s):  
Marco Fontana ◽  
James A. Huckaba ◽  
Ira J. Papick
Keyword(s):  
Prime Ideal ◽  
Recent Work ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Divisorial Ideal ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractGiven a Prüfer domain R and a prime ideal P in R, we study some conditions which force P to be a divisorial ideal of R. This paper extends some recent work of Huckaba and Papick.

On 1-absorbing prime ideals of commutative rings

2020 ◽  
pp. 2150175
Author(s):  
A. Yassine ◽  
M. J. Nikmehr ◽  
R. Nikandish
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Valuation Domains ◽  
Prufer Domains ◽  
Prüfer Domains

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

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.

The Complete Quotient Ring of Images of Semilocal Prüfer Domains

1977 ◽  
Vol 29 (5) ◽  
pp. 914-927 ◽  
Author(s):  
John Chuchel ◽  
Norman Eggert
Keyword(s):  
Homomorphic Image ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Local Rings ◽  
Topological Ring ◽  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Quotient Rings ◽  
Prüfer Rings

It is well known that the complete quotient ring of a Noetherian ring coincides with its classical quotient ring, as shown in Akiba [1]. But in general, the structure of the complete quotient ring of a given ring is largely unknown. This paper investigates the structure of the complete quotient ring of certain Prüfer rings. Boisen and Larsen [2] considered conditions under which a Prüfer ring is a homomorphic image of a Prüfer domain and the properties inherited from the domain. We restrict our investigation primarily to homomorphic images of semilocal Prüfer domains. We characterize the complete quotient ring of a semilocal Prüfer domain in terms of complete quotient rings of local rings and a completion of a topological ring.

FACTORIZATION IN PRÜFER DOMAINS

2017 ◽  
Vol 60 (2) ◽  
pp. 401-409 ◽  
Author(s):  
JIM COYKENDALL ◽  
RICHARD ERWIN HASENAUER
Keyword(s):  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe construct a norm on the nonzero elements of a Prüfer domain and extend this concept to the set of ideals of a Prüfer domain. These norms are used to study factorization properties Prüfer of domains.

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 MODULES OVER PRÜFER DOMAINS WHICH SATISFY THE RADICAL FORMULA

2007 ◽  
Vol 49 (1) ◽  
pp. 127-131
Author(s):  
DILEK BUYRUK ◽  
DILEK PUSAT-YILMAZ
Keyword(s):  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains

Abstract.In this paper we prove that if R is a Prüfer domain, then the R-module R⊕ R satisfies the radical formula.

scholarly journals Some properties of divisorial prime ideals in Prüfer domains

1986 ◽  
Vol 39 ◽  
pp. 95-103 ◽  
Author(s):  
Marco Fontana ◽  
James A. Huckaba ◽  
Ira J. Papick
Keyword(s):  
Prime Ideals ◽  
Prufer Domains ◽  
Prüfer Domains

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.

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