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.

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.

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.

scholarly journals On the Structure of Complete Local Rings

1950 ◽  
Vol 1 ◽  
pp. 63-70 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Structure Theorem ◽  
Finite Basis ◽  
Local Rings ◽  
Topological Ring

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

Classical Quotient Rings and Ordinary Extensions of 2-Primal Rings

2006 ◽  
Vol 13 (03) ◽  
pp. 513-523 ◽  
Author(s):  
Yong Uk Cho ◽  
Nam Kyun Kim ◽  
Mi Hyang Kwon ◽  
Yang Lee
Keyword(s):  
Local Ring ◽  
Division Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Quotient Rings

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

On Prufer Rings as Images of Prufer Domains

1973 ◽  
Vol 40 (1) ◽  
pp. 87
Author(s):  
Monte B. Boisen ◽  
Max D. Larsen
Keyword(s):  
Prufer Domains ◽  
Prüfer Domains ◽  
Prüfer Rings

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 On two-sided artinian quotient rings

1972 ◽  
Vol 13 (2) ◽  
pp. 159-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Regular Element ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Zero Divisors ◽  
Quotient Rings

Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving theTheorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal.

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.

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