Arithmetic Rings and their generalizations

R. Strano

doi:10.35352/gioenia.v51i381.1

Arithmetic Rings and their generalizations

Bullettin of the Gioenia Academy of Natural Sciences of Catania ◽

10.35352/gioenia.v51i381.1 ◽

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.

Download Full-text

The Complete Quotient Ring of Images of Semilocal Prüfer Domains

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-092-x ◽

1977 ◽

Vol 29 (5) ◽

pp. 914-927 ◽

Cited By ~ 1

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.

Download Full-text

FACTORIZATION IN PRÜFER DOMAINS

Glasgow Mathematical Journal ◽

10.1017/s0017089517000179 ◽

2017 ◽

Vol 60 (2) ◽

pp. 401-409 ◽

Cited By ~ 1

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.

Download Full-text

A Gorenstein analogue of a result of Bertin

Journal of Algebra and Its Applications ◽

10.1142/s021949881550019x ◽

2014 ◽

Vol 14 (02) ◽

pp. 1550019 ◽

Cited By ~ 3

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.

Download Full-text

MODULES OVER PRÜFER DOMAINS WHICH SATISFY THE RADICAL FORMULA

Glasgow Mathematical Journal ◽

10.1017/s0017089507003485 ◽

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.

Download Full-text

On flat finitely generated ideals

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011357 ◽

1980 ◽

Vol 21 (1) ◽

pp. 131-135 ◽

Cited By ~ 4

Author(s):

David E. Dobbs

Keyword(s):

Commutative Ring ◽

Finitely Generated ◽

Homological Characterization ◽

Prufer Domains ◽

Prüfer Domains

It is shown that if I is a finitely generated ideal of a commutative ring R such that the multiplication map I ⊗RI → I is an injection, then I is locally principal. As a corollary, one obtains a new homological characterization of Prüfer domains.

Download Full-text

Divisorial Prime Ideals in Prüfer Domains

Canadian Mathematical Bulletin ◽

10.4153/cmb-1984-049-9 ◽

1984 ◽

Vol 27 (3) ◽

pp. 324-328 ◽

Cited By ~ 11

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.

Download Full-text

On 1-absorbing prime ideals of commutative rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501759 ◽

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.

Download Full-text

Decomposition and classification of length functions

Forum Mathematicum ◽

10.1515/forum-2018-0168 ◽

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.

Download Full-text