scholarly journals Commutative rings with comparable regular elements

1996 ◽  
Vol 60 (1) ◽  
pp. 90-108 ◽  
Author(s):  
Paolo Zanardo
Keyword(s):  
Valuation Ring ◽  
Commutative Rings ◽  
The Other ◽  
Regular Elements ◽  
Zero Divisors ◽  
Valuation Rings ◽  
A Chain ◽  
Prüfer Rings ◽  
The Ideal

AbstractLet ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

2-Prime ideals and their applications

2016 ◽  
Vol 15 (03) ◽  
pp. 1650051 ◽  
Author(s):  
Charef Beddani ◽  
Wahiba Messirdi
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
The Other ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Other Hand ◽  
Valuation Rings ◽  
Integrally Closed

This paper introduces the notion of [Formula: see text]-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain [Formula: see text] is a valuation ring if and only if every ideal of [Formula: see text] is [Formula: see text]-prime. On the other hand, we will prove that the normalization [Formula: see text] of [Formula: see text] is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of [Formula: see text] is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

scholarly journals Injective Modules Over Prufer Rings

1959 ◽  
Vol 15 ◽  
pp. 57-69 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Integral Domains ◽  
Injective Modules ◽  
Valuation Rings ◽  
Weakened Form ◽  
Prüfer Rings

The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.

NEUROTIC SYMPTOMATICS RELATED TO PERSONAL CHARACTERISTICS IN YOUTH

2019 ◽  
pp. 2014-2019
Author(s):  
Milen Dimov
Keyword(s):  
Personal Characteristics ◽  
The Self ◽  
The Other ◽  
Social Responsiveness ◽  
Training Process ◽  
Self Assessment ◽  
School Year ◽  
Real Behavior ◽  
Low Levels ◽  
The Ideal

The present study traces the dynamics of personal characteristics in youth and the manifested neurotic symptoms in the training process. These facts are the reason for the low levels of school results in the context of the existing theoretical statements of the problem and the empirical research conducted among the trained teenagers. We suggest that the indicators of neurotic symptomatology in youth – aggression, anxiety, and neuroticism, are the most demonstrated, compared to the other studied indicators of neurotic symptomatology. Studies have proved that there is a difference in the act of neurotic symptoms when tested in different situations, both in terms of expression and content. At the beginning of the school year, neurotic symptoms, more demonstrated in some aspects of aggressiveness, while at the end of school year, psychotism is more demonstrated. The presented summarized results indicate that at the beginning of the school year, neurotic symptoms are strongly associated with aggression. There is a tendency towards a lower level of social responsiveness, both in the self-assessment of real behavior and in the ideal “I”-image of students in the last year of their studies. The neurotic symptomatology, more demonstrated due to specific conditions in the life of young people and in relation to the characteristics of age.

KONSTRUKSI ISLAM DALAM KEKUASAAN NEGARA

1970 ◽  
Vol 6 (2) ◽  
Author(s):  
Nurul Aini Musyarofah
Keyword(s):  
Islamic Law ◽  
Social Condition ◽  
The Other ◽  
Good Society ◽  
Ideal Form ◽  
Other Hand ◽  
Islamic Politics ◽  
The Relationship ◽  
The Ideal

The relationship between Islam and state raises a controversy that includes two main groups;formalists and substantialists. Both of them intend to achieve a good social condition which is inaccordance with Islamic politics. The ideal form of good society to be achieved is principallydescribed in the main source of Islamic law, Al Qur’an and As Sunnah, as follows. A form of goodsociety should supprot equality and justice, egalitarianism, and democracy in its social community.The next problem is what the needed methods and instruments to achieve the ideal Islamic politicsare. In this case, the debate on the formalization and substance of Islamic teaching is related to therunning formal political institution.Each group claims itself to be the most representative to the ideal Islam that often leads to anescalating conflict. On the other hand thr arguments of both groups does not reach the wholeMuslims. As a result, the discourse of Islam and state seems to be elitist and political. As a result,Both groups suspect each other each other and try to utilize the controversy on the relationshipbetween Islam and state to get their own benefit which has no relation with the actualization ofIslamic teaching.

scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

Adorno’s Philistine: the Dialectic of Art and its Other

2019 ◽  
pp. 1-31
Author(s):  
Paul Ingram
Keyword(s):  
The Other ◽  
Critical Potential ◽  
The Social ◽  
The Aesthetic ◽  
Critical Turn ◽  
The Ideal

Abstract Theodor Adorno’s philistine functions as the other of art, or as the ideal embodiment of everything that the bourgeois aesthetic subject is not. He insists on the truth-content of the derogation, while recognising its unjust social foundation, and seeking to reflect that tension in a self-critical turn. His model of advanced art is negatively delimited by the philistinism of art with a cause and the philistinism of art for enjoyment, which represent the poles of the aesthetic and the social. The philistine is also the counterpart to the connoisseur, with the interplay between them pointing to his preferred approach to aesthetics, in which an affinity for art and alienness to it are combined without compromise. However, Adorno fails to realise fully the critical potential of the philistine as the immanent negation of art and aesthetics.

scholarly journals XXIII.—On the Disruptive Discharge of Electricity: An Experimental Thesis for the Degree of Doctor of Science, Department A

1878 ◽  
Vol 28 (2) ◽  
pp. 633-671 ◽  
Author(s):  
Alexander Macfarlane
Keyword(s):  
Copper Wire ◽  
Late Summer ◽  
The Other ◽  
Straight Line ◽  
Summer Session ◽  
The Difference ◽  
A Chain ◽  
The One ◽  
Metallic Parts ◽  
The University

The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,—Messrs H. A. Salvesen, G. M. Connor, and D. E. Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburgh in 1876 in the names of Mr J. A. Paton, M. A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems mentioned below.The above sketch which I took of the apparatus in situ may facilitate tha description of the method. The receiver of an air-pump, having a rod capable of being moved air-tight up and down through the neck, was attached to one of the conductors of a Holtz machine in such a manner that the conductor of the machine and the rod formed one conducting system. Projecting from the bottom of the receiver was a short metallic rod, forming one conductor with the metallic parts of the air-pump, and by means of a chain with the uninsulated conductor of the Holtz machine. Brass balls and discs of various sizes were made to order, capable of being screwed on to the ends of the rods. On the table, and at a distance of about six feet from the receiver, was a stand supporting two insulated brass balls, the one fixed, the other having one degree of freedom, viz., of moving in a straight line in the plane of the table. The fixed insulated ball A was made one conductor with the insulated conductor of the Holtz and the rod of the receiver, by means of a copper wire insulated with gutta percha, having one end stuck firmly into a hole in the collar of the receiver, and having the other fitted in between the glass stem and the hollow in the ball, by which it fitted on to the stem tightly. A thin wire similarly fitted in between the ball B and its insulating stem connected the ball with the insulated half ring of a divided ring reflecting electrometer.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

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