Some Remarks on Noetherian Rings

1957 ◽  
Vol 9 ◽  
pp. 35-37
Author(s):  
Michio Yoshida

In his lecture at the University of Kyoto on September 23, 1955, Professor Artin gave an important theorem on Noetherian rings, which seems to have not a few interesting consequences. It is the purpose of our present note to point out one of them. We begin by quoting a special case of the theorem.

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao ◽  
Y. S. Hamed

<p style='text-indent:20px;'>We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their <inline-formula><tex-math id="M1">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its <inline-formula><tex-math id="M2">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their <inline-formula><tex-math id="M3">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths.</p>


Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1532
Author(s):  
Dong Kyu Kim ◽  
Jung Wook Lim

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.


1949 ◽  
Vol 1 (4) ◽  
pp. 337-343 ◽  
Author(s):  
R. Rado

1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].


1978 ◽  
Vol 1 (4) ◽  
pp. 525-528 ◽  
Author(s):  
A. McD. Mercer

A well-known result due toS. Beatty is that ifαandβare positive irrational numbers satisfyingα−1+β−1=1then each positive integer is to be found in precisely one of the sequences{[kα]},{[kβ]}(k=1,2,3,…)where[x]denotes the integral part ofx. The present note generalizes this result to the case of the pair of sequences{[f(k)]},{[g(k)]}with suitable hypotheses on the functionsfandg. The special casef(x)=αx,g(x)=βxis the result due to Beatty.


2020 ◽  
Vol 42 (1) ◽  
pp. 51-75
Author(s):  
Mustafa Yavuz ◽  
Pilar Herraíz Oliva

The reception of the translations of Aristotelian and pseudo-Aristotelian works at the University of Paris in the thirteenth century promoted a new understanding of the sciences as specialized fields of knowledge. The huge amount of translations required a new organization of knowledge, which included novel subjects and categories. Among these there is a very special case, namely the pseudo-Aristotelian De plantis, translated from Arabic into Latin and then back into Greek to be re-translated into Latin again. De plantis was included in the new curriculum in Ripoll 109 (1230–1240 BCE), and constituted the main source for botanical studies until the sixteenth century. Throughout this paper we will explore the reception and impact of De plantis in both the Arabic and the Latin traditions. We aim to show its foundational role in the development of botany as a theoretical discipline within the natural sciences.


2008 ◽  
Vol 07 (01) ◽  
pp. 21-45 ◽  
Author(s):  
STELIOS CHARALAMBIDES ◽  
JOHN CLARK

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.


1966 ◽  
Vol 9 (2) ◽  
pp. 187-190 ◽  
Author(s):  
A.K. Tiwary

It is well known that Z(p∞) is isomorphic to each of its non-zero homomorphic images [3]. The aim of the present note is to generalize this fact about Z(p∞) to indecomposable infective modules over rings more general than the ring of integers which will include Dedekind domains as a special case.


Author(s):  
Bruno Beltrão Léo

This paper presents the main results of my doctoral thesis written in 2016 as part of the requirements to obtain the Doctor degree in the Doctoral Program in Economics at the University of Brasilia (UnB). It regards the axiomatization of a model of complete preferences under uncertainty that allows for a multiplicity of tastes and beliefs. As in Gilboa and Schmeidler (1989), the analysis is made in an Anscombe-Aumann setup and each act is evaluated by the worst-case scenario. The difference is that while in Gilboa and Schmeidler (1989) the multiple scenarios are composed of a set of priors, here they are composed of a set of probability-utility pairs. Thus, after imposing some postulates on preferences, we come to our main result, described in the form of a representation theorem (Theorem 1). A special case in which the multiple scenarios are a consequence of the multiplicity of tastes only is also axiomatized (Theorem 2).


Author(s):  
Jose´ L. Escalona ◽  
Manuel Gonza´lez ◽  
Khaled E. Zaazaa ◽  
Ahmed A. Shabana

In this paper, a specialized steady state formulation is developed in order to validate the results obtained using multibody wheel/rail two-point contact algorithms developed at the University of Illinois at Chicago. In the specialized formulation, the wheel set is assumed to have three points of contact with the rail. The steady state curving behavior of the wheel set is examined using this specialized formulation. The balanced steady state curving behavior in which the centrifugal force is equal to the lateral component of the gravity force can be obtained as a special case of the specialized steady state formulation developed in this investigation. No derailment analysis or scenarios are considered in this paper in order to validate the results of both the constraint and elastic contact formulations. The results of the simple specialized formulation are compared with the results of the multibody inverse and forward dynamics approaches. In the inverse dynamics, approach it is assumed that the wheel set will maintain three points of contact with the rail. The inverse dynamics formulation model is conceptually different from the specialized steady state formulation since in the inverse dynamics all the degrees of freedom are specified. In the multibody forward dynamics method, no assumptions are made with regard to the existence of the flange contact.


2007 ◽  
Vol 56 (3) ◽  
Author(s):  
Warren T. Reich

A quarant’anni dall’inizio del dialogo fecondo tra scienze della vita e studi umani, che la nascita della bioetica ha reso possibile, si avverte la necessità di valutarne taluni aspetti – tra cui le possibilità future della bioetica, ed anche la sua origine – da un punto di vista particolare: quello dell’outsider. Si tratta di accettare la sfida, per la nostra identità morale, di spostare l’attenzione dallo studio di casi particolari, dal punto di vista dell’insider, allo studio di problemi ai margini della disciplina. L’outsider intellettuale sfida la bioetica convenzionale, prendendo in esame gli avvenimenti deprecabili del nostro tempo, permettendo in questo modo di valutare, a volte ripensandole e riformulandole, le responsabilità delle nostre discipline normative. Un esempio in questa direzione è dato da un Professore di Anestesia all’università di Harvard: Henry Beecher che uscendo dai ranghi della professione medica s’interessò della condizione di vulnerabilità dei soggetti sperimentali denunciando le numerose sperimentazione nelle quali i pazienti non avevano prestato un consenso o che non erano stati avvisati dei rischi. Altri filosofi e scienziati hanno superato, come Beecher, i confini della bioetica convenzionale, approfondendo i c.d. “problemi di frontiera”: la responsabilità di offrire o meno le cure necessarie, mediche e sociali, ai rifugiati disperatamente in cerca di asilo; la cura delle vittime della tortura. Questo modo di identificare le questioni è stato sovente affrontato da filosofi e teologi concordi sulla necessità di ampliare la portata delle domande che la bioetica si pone, per arricchirne la discussione e l’analisi. Questa esperienza di “outsider” permette agli studiosi di offrire risposte razionali alla società, sviluppando al tempo stesso un’etica pubblica credibile arricchendo il contenuto e la riflessione di quella che negli anni ’70 è stata definita come “lo studio delle dimensioni morali di ogni condotta umana, nell’area delle scienze della vita e della salute”. ---------- Forty years from the beginning of the rich dialogue between life sciences and human studies, introduced by bioethics, we feel the necessity of a deeper evaluation of some aspects related to bioethics, among others the future possibilities, the origin, in a particular perspective: that of the outsider. It is to move the attention from the special case study, in the perspective of the insider, to the study of issue related to the “border bioethics”. The intellectual outsider challenges the conventional bioethics. He takes the deplorable events of our time and allows a further and deeper evaluation, recalling or sometimes reformulating, the responsibilities of normative disciplines. In this direction, an example has been given by Henry Beecher Anaesthesia Professor at the University of Harvard, in 1966. Going out of the ranks of the medical profession, he was interested in the vulnerability condition of experimental subjects; he denounced the numerous experimentations where the patients did not give their consent or had not been informed about the risks. Other philosophers and scientists went beyond, as Beecher, the borders of the conventional bioethics, examining in detail the so called “border issues”: for example, the responsibility to offer the necessary cares, medical and social, to the refugees desperately searching for kindergarten; the care of the torture victims. This experience of “outsider” gives the possibility to offer rational answers to the society, developing, at the same time, a believable public ethics, also enriching bioethical contents and reflections, defined, in the 70s, as “the study of the moral dimension of each human conduct, in the field of the science of life and health”.


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