scholarly journals A characterisation of semisimple classes

1981 ◽  
Vol 24 (1) ◽  
pp. 5-7 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Homomorphic Image ◽  
Original Work ◽  
Semisimple Class ◽  
Associative Rings ◽  
Independence Relations

Throughout this paper we shall work in the class of associative rings. In (4) it was shown that a class of rings is a semisimple class if and only if it is closed under extensions and ideals and is coinductive. This establishes a duality between radical classes and semisimple classes. This result has been proved also for classes of alternative rings in (2). In the original work by Kuros (1) on this subject two conditions were used for semisimple classes, one of which was weaker than the assumption that the class is closed under ideals. This condition is that every non-zero ideal of a ring in the class should have a non-zero homomorphic image in the class. It is natural to ask whether in the above set of conditions the condition of being closed under ideals can be replaced by this weaker condition. This question is raised in (3) and in (5) but it is suggested there that, in order to compensate, the coinductive condition be replaced by the stronger condition that the class is closed under subdirect sums. In fact we shall show that the weaker condition may be used without needing to replace the coinductive condition. We also give examples to show independence relations among these conditions.

scholarly journals Hereditary semisimple classes

1970 ◽  
Vol 11 (1) ◽  
pp. 7-8 ◽  
Author(s):  
W. G. Leavitt
Keyword(s):  
Homomorphic Image ◽  
The Other ◽  
Radical Class ◽  
Universal Class ◽  
Semisimple Class ◽  
Other Hand ◽  
Associative Rings

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

ON BASE RADICAL OPERATORS FOR CLASSES OF FINITE ASSOCIATIVE RINGS

2018 ◽  
Vol 98 (2) ◽  
pp. 239-250 ◽  
Author(s):  
R. G. MCDOUGALL ◽  
L. K. THORNTON
Keyword(s):  
Maximum Order ◽  
Semisimple Class ◽  
Hereditary Radical ◽  
Complete Listing ◽  
Universal Classes ◽  
General Relations ◽  
Associative Rings

In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.

scholarly journals Kurosh's chains of associative rings

1990 ◽  
Vol 32 (1) ◽  
pp. 67-69 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
E. R. Puczylowski
Keyword(s):  
Natural Number ◽  
Left Ideal ◽  
Homomorphic Image ◽  
Radical Class ◽  
Lower Radical ◽  
Closed Class ◽  
A Chain ◽  
Associative Rings

Let N be a homomorphically closed class of associative rings. Put N1 = Nl = N and, for ordinals a ≥ 2, define Nα (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class IN (lower left strong radical class IsN) determined by N. The chain {Nα} is called Kurosh's chain of N. Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ Nk for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = Nn+l ≠ Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classes N with Nα = Nα for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that Nα = Nα for all α and Nn ≠ Nn+i = Nn+2. This in particular answers Question 6 of [4].

Hereditary Radicals in Associative and Alternative Rings

1965 ◽  
Vol 17 ◽  
pp. 594-603 ◽  
Author(s):  
T. Anderson ◽  
N. Divinsky ◽  
A. Suliński
Keyword(s):  
Homomorphic Image ◽  
Radical Property ◽  
Associative Rings

In the first part of this paper we shall consider associative rings, pointing out where associativity is required. In the second part we shall consider not necessarily associative rings and in particular alternative rings.A property S of rings is said to be a radical property, in the sense of Kurosh (4), if it satisfies the following three conditions:(a) Every homomorphic image of an S-ring (i.e. a ring with property S) is again an S-ring.

scholarly journals Characterizations of semisimple classes

1977 ◽  
Vol 23 (2) ◽  
pp. 172-182 ◽  
Author(s):  
L. C. A. van Leeuwen ◽  
C. Roos ◽  
R. Wiegandt
Keyword(s):  
Semisimple Class ◽  
Associative Rings ◽  
Inductive Property

AbstractTwo characterizations of semisimple classes of associative and alternative rings (and semigroups with 0) are given:(i) A class is a semisimple class if and only if it is hereditary, closed under extensions and subdirect sums;(ii) A class is a semisimple class if and only if it is hereditary, closed under extensions, and has the co-inductive property.The first characterization sharpens Armendariz's (1968) result proved for associative rings, the second one is categorically dual to a characterization of radical classes due to Amitsur (1954).

A Note on Hypernilpotent Radical Properties for Associative Rings

1967 ◽  
Vol 19 ◽  
pp. 447-448 ◽  
Author(s):  
S. E. Dickson
Keyword(s):  
Homomorphic Image ◽  
Lower Radical ◽  
Closed Class ◽  
Radical Construction ◽  
Associative Rings

We work entirely in the category of associative rings. We show that if P1 is a homomorphically closed class which contains the zero rings, then the lower Kurosh radical P of P1 is the class P2 of all rings R such that every non-zero homomorphic image of R has non-zero ideals in P1, provided that P1 is closed under extensions by zero rings (i.e., if I is a P1-ideal of R and (R/I)2 = 0, then R ∈ P1). The latter assumption replaces the hypothesis that P1 be hereditary for ideals in a similar result of Anderson-Divinsky-Sulinsky in (2). This leads to a brief proof that the lower radical construction of Kurosh terminates at Pω0 (where ω0 is the first infinite ordinal) when P1 is a homomorphically closed class of associative rings containing the zero rings. This was proved for arbitrary homomorphically closed classes P1 of associative rings in (2).

Instruction for authors

2018 ◽  
Vol 1 (2) ◽  
pp. 75-79
Author(s):  
Abdulghani Alsamarai
Keyword(s):  
Original Work ◽  
Wide Spectrum ◽  
Case Reports ◽  
Editorial Policy ◽  
High Standard ◽  
Publication Ethics ◽  
Current Interest ◽  
Medical Sciences ◽  
Medical News ◽  
Review Articles

Introduction   The International Journal of Medical Sciences [IJMS], ISSN 2522-7386, is a peer-reviewed, 3 issues published annually. Authors are invited to submit for publication articles with a wide spectrum of coverage reporting original work, in the fields of medicine, nursery, dentistry, and pharmacy sciences. Review articles are usually by invitation only. However, Review articles of current interest and high standard will be considered. Prospective work should not be back dated. There are also sections for Case Reports, Brief Communication, correspondence and medical news items. Authors should read the editorial policy and publication ethics before submitting their manuscripts. Authors should also use the appropriate reporting guidelines in preparing their manuscripts

The Spanish Reception of Le Sacre du Printemps (1913–1936)

2018 ◽  
Vol 36 (1) ◽  
pp. 48-66
Author(s):  
Idoia Murga Castro
Keyword(s):  
Nineteenth Century ◽  
Civil War ◽  
Spanish Civil War ◽  
Original Work ◽  
Silver Age ◽  
Avant Garde ◽  
Ballets Russes ◽  
Rite Of Spring ◽  
Le Sacre Du Printemps ◽  
Cultural Expansion

Centenary celebrations are being held between 2016 and 2018 to mark the first consecutive tours of Diaghilev's Ballets Russes in Spain. This study analyses the Spanish reception of Le Sacre du Printemps (The Rite of Spring) (1913), one of its most avant-garde pieces. Although the original work was never performed in Spain as a complete ballet, its influence was felt deeply in the work of certain Spanish choreographers, composers, painters and intellectuals during the so-called Silver Age, the period of modernisation and cultural expansion which extended from the end of the nineteenth century to the beginning of the Spanish Civil War.

Retroaktive Neuverhandlung

2018 ◽  
Vol 63 (1) ◽  
Author(s):  
Daniel Martin Feige
Keyword(s):  
Original Work ◽  
The Other ◽  
Final Part ◽  
Critical Reading ◽  
The Third ◽  
Works Of Art ◽  
The Relationship

Der Beitrag widmet sich der Frage historischer Folgeverhältnisse in der Kunst. Gegenüber dem Gedanken, dass es ein ursprüngliches Werk in der Reihe von Werken gibt, das späteren Werken seinen Sinn gibt, schlägt der Text vor, das Verhältnis umgekehrt zu denken: Im Lichte späterer Werke wird der Sinn früherer Werke neu ausgehandelt. Dazu geht der Text in drei Schritten vor. Im ersten Teil formuliert er unter der Überschrift ›Form‹ in kritischer Abgrenzung zu Danto und Eco mit Adorno den Gedanken, dass Kunstwerke eigensinnig konstituierte Gegenstände sind. Die im Gedanken der Neuverhandlung früherer Werke im Lichte späterer Werke vorausgesetzte Unbestimmtheit des Sinns von Kunstwerken wird im zweiten Teil unter dem Schlagwort ›Zeitlichkeit‹ anhand des Paradigmas der Improvisation erörtert. Der dritte und letzte Teil wendet diese improvisatorische Logik unter dem Label ›Neuaushandlung‹ dann dezidiert auf das Verhältnis von Vorbild und Nachbild an. The article proposes a new understanding of historical succession in the realm of art. In contrast to the idea that there is an original work in the series of works that gives meaning to the works that come later, the text proposes to think it exactly the other way round: in the light of later works, the meanings of earlier works are renegotiated. The text proceeds in three steps to develop this idea. Under the heading ›Form‹ it develops in the first part a critical reading of Danto’s and Eco’s notion of the constitution of the artworks and argues with Adorno that each powerful work develops its own language. In the second part, the vagueness of the meaning of works of art presupposed in the idea of renegotiating earlier works in the light of later works is discussed under the term ›Temporality‹ in terms of the logic of improvisation. The third and final part uses this improvisational logic under the label ›Renegotiation‹ to understand the relationship between model and afterimage in the realm of art.

Value Meanings of the Modern Textbook of the Russian Language

2018 ◽  
Vol 79 (8) ◽  
pp. 7-10 ◽  
Author(s):  
A. D. Deikina
Keyword(s):  
Original Work ◽  
School Education ◽  
Russian Language ◽  
Educational Strategy ◽  
Personal Meanings ◽  
Current Trends ◽  
Modern Textbook ◽  
The Russian Language

Analysis of the current trends in the teaching of the Russian language allows to assert the value of the category of values in the educational strategy. In the context of orientation of the textbook to modern requirements the role of the text in the characteristic of language as an expression of value and personal meanings is emphasized. Providing personal and humanistic thinking and the formation of value view of students in the Russian language is more successful on a wide background of text material by stimulating a variety of ways of original work of students. Its predicted results are closely related to the awareness of the value of the Russian language. Attention is paid to the resources associated with the organization of open educational space on the basis of axiological ideas as the leading in the theory of school education and textbook.

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