Divisorial behavior of prime ideals in Prüfer domains

2015 ◽  
Vol 64 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Thomas G. Lucas
Keyword(s):  
Prime Ideals ◽  
Prufer Domains ◽  
Prüfer Domains

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

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 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.

scholarly journals Toward a classification of prime ideals in Prüfer domains

2010 ◽  
Vol 22 (4) ◽  
Author(s):  
Marco Fontana ◽  
Evan Houston ◽  
Thomas G. Lucas
Keyword(s):  
Prime Ideals ◽  
Prufer Domains ◽  
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.

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.

Prüfer domains and pure submodules of direct sums of ideals

Mathematika ◽  
1999 ◽  
Vol 46 (2) ◽  
pp. 425-432 ◽  
Author(s):  
Bruce Olberding
Keyword(s):  
Direct Sums ◽  
Prufer Domains ◽  
Pure Submodules ◽  
Prüfer Domains

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
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