Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

ON LCM-STABLE MODULES

2014 ◽  
Vol 13 (04) ◽  
pp. 1350133 ◽  
Author(s):  
HWANKOO KIM ◽  
FANGGUI WANG
Keyword(s):  
Finite Type ◽  
Integral Domain ◽  
Free Module ◽  
Torsion Free Module ◽  
Polynomial Module ◽  
Torsion Free ◽  
Krull Domains

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.

scholarly journals Perinormalitas di Daerah Krull (Perinormality on Krull Domains)

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Qonita Qurrota A'yun ◽  
Sri Wahyuni
Keyword(s):  
Prime Ideal ◽  
Integral Domain ◽  
Integral Closure ◽  
Normal Domain ◽  
Closedness Property ◽  
Krull Domains

Daerah integral R dikatakan perinormal jika untuk setiap overring (lokal) T dari R yang memenuhi kondisi going-down, maka T merupakan lokalisasi dari R pada ideal prima. Perinormalitas merupakan salah satu sifat ketertutupan integral. Dengan memperhatikan bahwa klosur integral dari daerah normal Noether merupakan daerah Krull, akan ditunjukkan bagaimana sifat perinormalitas di daerah Krull.An integral domain R is said to be perinormal if whenever T is a (local) overring of R such that the inclusion R in T satisfies going-down, it follows that T is a localization of R necessarily at a prime ideal. Perinormality is one of integral closedness property. As the integral closure of any Noetherian normal domain is Krull, it will be shown how perinormality behaves on Krull domains.

Integral Domains Whose w-Integral Closures Are Krull Domains

2020 ◽  
Vol 27 (02) ◽  
pp. 287-298
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Krull Domain ◽  
Integral Closures ◽  
Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

Krull Modules

2013 ◽  
Vol 20 (03) ◽  
pp. 463-474 ◽  
Author(s):  
Hwankoo Kim ◽  
Myeong Og Kim
Keyword(s):  
Integral Domain ◽  
Multiplication Module ◽  
Krull Domain ◽  
Free Modules ◽  
Torsion Free ◽  
Krull Domains

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.

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.

Divisibility Properties of the Semiring of Ideals of an Integral Domain

2020 ◽  
Vol 27 (03) ◽  
pp. 369-380
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
General Setting ◽  
Dedekind Domain ◽  
Unique Factorization ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Commutative Semiring ◽  
Divisibility Properties ◽  
Star Operations

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

Minimizing Carry-Over Effects After Treatment Failure and Maximizing Therapeutic Outcome

2014 ◽  
Vol 222 (3) ◽  
pp. 171-178 ◽  
Author(s):  
Mareile Hofmann ◽  
Nathalie Wrobel ◽  
Simon Kessner ◽  
Ulrike Bingel
Keyword(s):  
Treatment Failure ◽  
Human Subjects ◽  
Subsequent Treatment ◽  
Clinical Samples ◽  
Administration Route ◽  
Healthy Human ◽  
Negative Experiences ◽  
Analgesic Treatment ◽  
Balanced Design ◽  
Carry Over

According to experimental and clinical evidence, the experiences of previous treatments are carried over to different therapeutic approaches and impair the outcome of subsequent treatments. In this behavioral pilot study we used a change in administration route to investigate whether the effect of prior treatment experience on a subsequent treatment depends on the similarity of both treatments. We experimentally induced positive or negative experiences with a topical analgesic treatment in two groups of healthy human subjects. Subsequently, we compared responses to a second, unrelated and systemic analgesic treatment between both the positive and negative group. We found that there was no difference in the analgesic response to the second treatment between the two groups. Our data indicate that a change in administration route might reduce the influence of treatment history and therefore be a way to reduce negative carry-over effects after treatment failure. Future studies will have to validate these findings in a fully balanced design including larger, clinical samples.

Does CBT for Psychosis Have an Impact on Delusions by Improving Reasoning Biases and Negative Self-Schemas?

2018 ◽  
Vol 226 (3) ◽  
pp. 152-163 ◽  
Author(s):  
Stephanie Mehl ◽  
Björn Schlier ◽  
Tania M. Lincoln
Keyword(s):  
Behavioral Therapy ◽  
Theoretical Models ◽  
Self Esteem ◽  
Secondary Outcome ◽  
Intervention Effect ◽  
Produce Change ◽  
Attribution Biases ◽  
Carry Over ◽  
Implicit And Explicit ◽  
Reasoning Biases

Abstract. Cognitive-behavioral therapy for psychosis (CBTp) builds on theoretical models that postulate reasoning biases and negative self-schemas to be involved in the formation and maintenance of delusions. However, it is unclear whether CBTp induces change in delusions by improving these proposed causal mechanisms. This study reports on a mediation analysis of a CBTp effectiveness trial in which delusions were a secondary outcome. Patients with psychosis were randomized to individualized CBTp (n = 36) or a waiting list condition (WL; n = 34). Reasoning biases (jumping to conclusions, theory of mind, attribution biases) and self-schemas (implicit and explicit self-esteem; self-schemas related to different domains) were assessed pre- and post-therapy/WL. The results reveal an intervention effect on two of four measures of delusions and on implicit self-esteem. Nevertheless, the intervention effect on delusions was not mediated by implicit self-esteem. Changes in explicit self-schemas and reasoning biases did also not mediate the intervention effects on delusions. More focused interventions may be required to produce change in reasoning and self-schemas that have the potential to carry over to delusions.

Sign in / Sign up

Export Citation Format

Share Document