When is (D, K) an S-accr pair?

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Subramanian Visweswaran
Keyword(s):  
Closed Subset ◽  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Ideal Theory ◽  
Prüfer Domain ◽  
Content Type ◽  
Ring Of Fractions ◽  
Intermediate Domain ◽  
Necessary And Sufficient

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLet S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.Originality/valueMotivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

WHEN IS THE INTEGRAL CLOSURE COMPARABLE TO ALL INTERMEDIATE RINGS

2016 ◽  
Vol 95 (1) ◽  
pp. 14-21 ◽  
Author(s):  
MABROUK BEN NASR ◽  
NABIL ZEIDI
Keyword(s):  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Necessary And Sufficient ◽  
Quotient Rings

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

When is a fixed ring comparable to all overrings?

2020 ◽  
pp. 2250050
Author(s):  
Ahmed Ayache
Keyword(s):  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Fixed Ring ◽  
Necessary And Sufficient

An overring [Formula: see text] of an integral domain [Formula: see text] is said to be comparable if [Formula: see text], [Formula: see text], and each overring of [Formula: see text] is comparable to [Formula: see text] under inclusion. We do provide necessary and sufficient conditions for which [Formula: see text] has a comparable overring. Several consequences are derived, specially for minimal overrings, or in the case where the integral closure [Formula: see text] of [Formula: see text] is a comparable overring, or also when each chain of distinct overrings of [Formula: see text] is finite.

FINITELY VALUATIVE DOMAINS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250112 ◽  
Author(s):  
PAUL-JEAN CAHEN ◽  
DAVID E. DOBBS ◽  
THOMAS G. LUCAS
Keyword(s):  
Nonnegative Integer ◽  
Integral Domain ◽  
Nonzero Element ◽  
Integral Closure ◽  
Quotient Field ◽  
Prüfer Domain ◽  
Maximal Ideals ◽  
Saturated Chain

For a pair of rings S ⊆ T and a nonnegative integer n, an element t ∈ T\S is said to be within n steps of S if there is a saturated chain of rings S = S0 ⊊ S1 ⊊ ⋯ ⊊ Sm = S[t] with length m ≤ n. An integral domain R is said to be n-valuative (respectively, finitely valuative) if for each nonzero element u in its quotient field, at least one of u and u-1 is within n (respectively, finitely many) steps of R. The integral closure of a finitely valuative domain is a Prüfer domain. Moreover, an n-valuative domain has at most 2n + 1 maximal ideals; and an n-valuative domain with 2n + 1 maximal ideals must be a Prüfer domain.

An answer to a problem about the number of overrings

2016 ◽  
Vol 15 (06) ◽  
pp. 1650022 ◽  
Author(s):  
M. Ben Nasr
Keyword(s):  
Open Problem ◽  
Integral Domain ◽  
Integral Closure ◽  
Closed Domain ◽  
Prüfer Domain ◽  
Finite Spectrum ◽  
Integrally Closed

Let [Formula: see text] be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure [Formula: see text] is a Prüfer domain with finite spectrum and there are only finitely many rings between [Formula: see text] and [Formula: see text]. Jaballah solved the problem of counting the overrings in the case [Formula: see text] but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math. 23 (2005) 353–360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.

The importance of the political order for peace

2017 ◽  
Vol 9 (1) ◽  
pp. 24-37 ◽  
Author(s):  
Janja Mikulan Kildi ◽  
Victor Cepoi
Keyword(s):  
Rule Of Law ◽  
Sufficient Conditions ◽  
Theoretical Framework ◽  
Political Order ◽  
Political Factors ◽  
Content Type ◽  
The Rule Of Law ◽  
Depth Analysis ◽  
Fuzzy Set Analysis ◽  
Necessary And Sufficient

Purpose The purpose of this paper is to address the issue of security challenges in the selected states in Eurasia and Central and Eastern Europe from the perspective of peace and politics. Since the security situation differs significantly across the sub-regions under scrutiny, the main objective of the current research is to establish and test a theoretical model of relevant combinations of political factors and their relation to peace. The theoretical framework has been designed by following Fukuyama’s idea of political order and upgraded with the newly constructed index of social exclusion. Design/methodology/approach According to the objective and methodological gaps in this field, the paper applies comparative fuzzy set analysis. The method relies on the theoretical framework and empirical data and allows a constant communication between the two. It has allowed the research to focus on what conditions are necessary and/or sufficient for peace in the selected countries. Findings It is evident there are several different paths to achieve peace. However, the rule of law clearly constitutes both the necessary and sufficient conditions for peace in the selected regions. Moreover, the results of the research reveal that the concepts of interest are highly interlinked with each other, especially when it comes to the rule of law, state, and democratic accountability. Thus, further research should focus on in-depth analysis of each particular case to explore which conditions or combinations are decisive in particular setting. Originality/value Such findings can contribute to improving the areas where Eurasian countries are still lagging behind. Further research should focus on small but significant differences within these four groupings of countries, to improve the understanding of prerequisites for peace and contribute to the development of state and human security in the EU neighborhood.

Observer-based control for a singular Markovian jumping system

2017 ◽  
Vol 34 (1) ◽  
pp. 33-52
Author(s):  
Ching-Min Lee
Keyword(s):  
Control Problem ◽  
Control Systems ◽  
Discrete Time ◽  
Networked Control Systems ◽  
Sufficient Conditions ◽  
Networked Control ◽  
Markovian Jumping ◽  
Content Type ◽  
Robot Systems ◽  
Necessary And Sufficient

Purpose For most practical control system problems, the state variables of a system are not often available or measureable due to technical or economical constraints. In these cases, an observer-based controller design problem, which is involved with using the available information on inputs and outputs to reconstruct the unmeasured states, is desirable, and it has been wide investigated in many practical applications. However, the investigation on a discrete-time singular Markovian jumping system is few so far. This paper aims to consider an observer-based control problem for a discrete-time singular Markovian jumping system and provides a set of easy-used conditions to the proposed control law. Design/methodology/approach According to the connotation of the separation principle extended from linear systems, a mode-dependent observer and a state-feedback controller is designed and carried out independently via two sets of derived necessary and sufficient conditions in terms of linear matrix inequalities (LMIs). Findings A set of necessary and sufficient conditions for an admissibility analysis problem related to a discrete-time singular Markovian jumping system is derived to be a doctrinal foundation for the proposed design problems. A mode-dependent observer and a controller for such systems could be designed via two sets of strictly LMI-based synthesis conditions. Research limitations/implications The proposed method can be applied to discrete-time singular Markovian jumping systems with transition probability pij > 0 rather than the ones with pii = 0. Practical implications The formulated problem and proposed methods have extensive applications in various fields such as power systems, electrical circuits, robot systems, chemical systems, networked control systems and interconnected large-scale systems. Take robotic networked control systems for example. It is recognized that the variance phenomena derived from network transmission, such as packets dropout, loss and disorder, are suitable for modeling as a system with Markovian jumping modes, while the dynamics of the robot systems can be described by singular systems. In addition, the packets dropout or loss might result in unreliable transmission signals which motivates an observer-based control problem. Originality/value Both of the resultant conditions of analysis and synthesis problems for a discrete-time singular Markovian jumping system are necessary and sufficient, and are formed in strict LMIs, which can be used and implemented easily via MATLAB toolbox.

Perceived value, satisfaction and future intentions in sport services

2019 ◽  
Vol 32 (4) ◽  
pp. 566-579 ◽  
Author(s):  
Mario Alguacil ◽  
Juan Núñez-Pomar ◽  
Carlos Pérez-Campos ◽  
Vicente Prado-Gascó
Keyword(s):  
Linear Models ◽  
Qualitative Comparative Analysis ◽  
Sufficient Conditions ◽  
Perceived Value ◽  
Hierarchical Regression ◽  
Brand Trust ◽  
Predictive Capability ◽  
Content Type ◽  
Necessary And Sufficient

Purpose The purpose of this paper is to analyze the role of brand-related variables as congruence and brand trust on the traditional model formed by perceived quality, perceived value (PV) and satisfaction, in order to compare predictive models for the variables of PV, satisfaction and future intentions of 683 users of sports services. Design/methodology/approach The analysis has been carried out using two different methodologies. First, three models have been proposed to be analyzed by hierarchical regression models, in order to subsequently propose a fuzzy-set qualitative comparative analysis (fsQCA) to verify the existence or not of necessary and sufficient conditions. Findings The results indicate that both the classic service variables and the elements related to the brand significantly predict PV, satisfaction and future intentions, in some cases with greater predictive weight being given to congruence and trust than the classic service variables. In addition, linear models have been shown to improve their predictive capability by including brand-related variables, especially the future intentions model. After the fsQCA, congruence and trust have proved to be sufficient combinations to achieve high levels of PV and future intentions, while this is not the case for satisfaction. Originality/value The importance of the aspects related to the brand, either on their own or in combination with the classic service variables, is demonstrated, contributing to the literature on brand image in sports services, which is practically non-existent.

scholarly journals Fixed point theorems for non-self maps I

1994 ◽  
Vol 17 (4) ◽  
pp. 713-716 ◽  
Author(s):  
Troy L. Hicks ◽  
Linda Marie Saliga
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Supposef:C→XwhereCis a closed subset ofX. Necessary and sufficient conditions are given forfto have a fixed point. All results hold whenXis complete metric space. Several results hold in a much more general setting.

scholarly journals IRREDUCIBLE ACTIONS AND COMPRESSIBLE MODULES

2011 ◽  
Vol 10 (01) ◽  
pp. 101-117 ◽  
Author(s):  
INÊS BORGES ◽  
CHRISTIAN LOMP
Keyword(s):  
Endomorphism Ring ◽  
Operator Algebra ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Injective Hull ◽  
Module Structure ◽  
Ring Of Fractions ◽  
Finite Set ◽  
Group Gradings ◽  
Necessary And Sufficient

Any finite set of linear operators on an algebra A yields an operator algebra B and a module structure on A, whose endomorphism ring is isomorphic to a subring AB of certain invariant elements of A. We show that if A is a critically compressible left B-module, then the dimension of its self-injective hull  over the ring of fractions of AB is bounded by the uniform dimension of A and the number of linear operators generating B. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions.

scholarly journals Cardinalities of prime spectra of precompletions

2021 ◽  
pp. 133-152
Author(s):  
Erica Barrett ◽  
Emil Graf ◽  
S. Loepp ◽  
Kimball Strong ◽  
Sharon Zhang
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Local Domain ◽  
Content Type ◽  
Necessary And Sufficient

Given a complete local (Noetherian) ring T T , we find necessary and sufficient conditions on T T such that there exists a local domain A A with | A | > | T | |A| > |T| and A ^ = T \widehat {A} = T , where A ^ \widehat {A} denotes the completion of A A with respect to its maximal ideal. We then find necessary and sufficient conditions on T T such that there exists a domain A A with A ^ = T \widehat {A} = T and | S p e c ( A ) | > | S p e c ( T ) | |\mathrm {Spec}(A)| > |\mathrm {Spec}(T)| . Finally, we use “partial completions” to create local rings A A with A ^ = T \widehat {A} = T such that S p e c ( A ) \mathrm {Spec}(A) has varying cardinality in different varieties.

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