On ideal theory in high prufer domains

Moshe Jarden

doi:10.1007/bf01169264

Ideal theory in Prüfer domains —An unconventional approach

Journal of Algebra ◽

10.1016/s0021-8693(02)00040-6 ◽

2002 ◽

Vol 252 (2) ◽

pp. 411-430 ◽

Cited By ~ 8

Author(s):

Laszlo Fuchs ◽

Edward Mosteig

Keyword(s):

Ideal Theory ◽

Prufer Domains ◽

Prüfer Domains

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On Prüfer-like properties of Leavitt path algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501224 ◽

2019 ◽

Vol 19 (07) ◽

pp. 2050122 ◽

Cited By ~ 1

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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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.

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Prüfer domains and pure submodules of direct sums of ideals

Mathematika ◽

10.1112/s0025579300007889 ◽

1999 ◽

Vol 46 (2) ◽

pp. 425-432 ◽

Cited By ~ 2

Author(s):

Bruce Olberding

Keyword(s):

Direct Sums ◽

Prufer Domains ◽

Pure Submodules ◽

Prüfer Domains

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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.

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