Contributions to the Taxonomy of Rings
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Subset Semirings
University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2013 Subset Semirings Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebraic Geometry Commons, Analysis Commons, and the Other Mathematics Commons Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. Subset Semirings. Ohio: Educational Publishing, 2013. This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Subset Semirings W. B. Vasantha Kandasamy Florentin Smarandache Educational Publisher Inc. Ohio 2013 This book can be ordered from: Education Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 Copyright 2013 by Educational Publisher Inc. and the Authors Peer reviewers: Marius Coman, researcher, Bucharest, Romania. Dr. Arsham Borumand Saeid, University of Kerman, Iran. Said Broumi, University of Hassan II Mohammedia, Casablanca, Morocco. Dr. Stefan Vladutescu, University of Craiova, Romania. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm ISBN-13: 978-1-59973-234-3 EAN: 9781599732343 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SUBSET SEMIRINGS OF TYPE I 9 Chapter Three SUBSET SEMIRINGS OF TYPE II 107 Chapter Four NEW SUBSET SPECIAL TYPE OF TOPOLOGICAL SPACES 189 3 FURTHER READING 255 INDEX 258 ABOUT THE AUTHORS 260 4 PREFACE In this book authors study the new notion of the algebraic structure of the subset semirings using the subsets of rings or semirings. -
Artinian Rings Having a Nilpotent Group of Units
JOURNAL OF ALGEBRA 121, 253-262 (1989) Artinian Rings Having a Nilpotent Group of Units GHIOCEL GROZA Department of Mathematics, Civil Engineering Institute, Lacul Tei 124, Sec. 2, R-72307, Bucharest, Romania Communicated by Walter Feit Received July 1, 1983 Throughout this paper the symbol R will be used to denote an associative ring with a multiplicative identity. R,[R”] denotes the subring of R which is generated over R, by R”, where R, is the prime subring of R and R” is the group of units of R. We use J(R) to denote the Jacobson radical of R. In Section 1 we prove that if R is a finite ring and at most one simple component of the semi-simple ring R/J(R) is a field of order 2, then R” is a nilpotent group iff R is a direct sum of two-sided ideals that are homomorphic images of group algebras of type SP, where S is a particular commutative finite ring, P is a finite p-group, and p is a prime number. In Section 2 we study the artinian rings R (i.e., rings with minimum con- dition) with RO a finite ring, R finitely generated over its center and so that every block of R is as in Lemma 1.1. We note that every finite-dimensional algebra over a field F of characteristic p > 0, so that F is not the finite field of order 2 is of this type. We prove that R is of this type and R” is a nilpotent group iff R is a direct sum of two-sided ideals that are particular homomorphic images of crossed products. -
Arxiv:0704.2561V2 [Math.QA] 27 Apr 2007 Nttto O Xeln Okn Odtos H Eoda Second the Conditions
DUAL FEYNMAN TRANSFORM FOR MODULAR OPERADS J. CHUANG AND A. LAZAREV Abstract. We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich’s dual construction producing graph cohomology classes from a contractible differential graded Frobenius alge- bra. The dual Feynman transform of a modular operad is indeed linear dual to the Feynman transform introduced by Getzler and Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman transform, the dual notion admits an extremely simple presentation via generators and relations; this leads to an explicit and easy description of its algebras. We discuss a further generalization of the dual Feynman transform whose algebras are not neces- sarily contractible. This naturally gives rise to a two-colored graph complex analogous to the Boardman-Vogt topological tree complex. Contents Introduction 1 Notation and conventions 2 1. Main construction 4 2. Twisted modular operads and their algebras 8 3. Algebras over the dual Feynman transform 12 4. Stable graph complexes 13 4.1. Commutative case 14 4.2. Associative case 15 5. Examples 17 6. Moduli spaces of metric graphs 19 6.1. Commutative case 19 6.2. Associative case 21 7. BV-resolution of a modular operad 22 7.1. Basic construction and description of algebras 22 7.2. Stable BV-graph complexes 23 References 26 Introduction arXiv:0704.2561v2 [math.QA] 27 Apr 2007 The relationship between operadic algebras and various moduli spaces goes back to Kontse- vich’s seminal papers [19] and [20] where graph homology was also introduced. -
Representations of Finite Rings
Pacific Journal of Mathematics REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON Vol. 53, No. 2 April 1974 PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 2, 1974 REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON In this paper we extend the concept of the Szele repre- sentation of finite rings from the case where the coefficient ring is a cyclic ring* to the case where it is a Galois ring. We then characterize completely primary and nilpotent finite rings as those rings whose Szele representations satisfy certain conditions. 1* Preliminaries* We first note that any finite ring is a direct sum of rings of prime power order. This follows from noticing that when one decomposes the additive group of a finite ring into its pime power components, the component subgroups are, in fact, ideals. So without loss of generality, up to direct sum formation, one needs only to consider rings of prime power order. For the remainder of this paper p will denote an arbitrary, fixed prime and all rings will be of order pn for some positive integer n. Of the two classes of rings that will be studied in this paper, completely primary finite rings are always of prime power order, so for the completely primary case, there is no loss of generality at all. However, nilpotent finite rings do not need to have prime power order, but we need only classify finite nilpotent rings of prime power order, the general case following from direct sum formation. If B is finite ring (of order pn) then the characteristic of B will be pk for some positive integer k. -
A Unified Approach to Various Generalizations of Armendariz Rings
Bull. Aust. Math. Soc. 81 (2010), 361–397 doi:10.1017/S0004972709001178 A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS GREG MARKS ˛, RYSZARD MAZUREK and MICHAŁ ZIEMBOWSKI (Received 5 February 2009) Abstract Let R be a ring, S a strictly ordered monoid, and ! V S ! End.R/ a monoid homomorphism. The skew generalized power series ring RTTS;!UU is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the .S; !/-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of .S; !/- Armendariz rings and obtain various necessary or sufficient conditions for a ring to be .S; !/-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal. 2000 Mathematics subject classification: primary 16S99, 16W60; secondary 06F05, 16P60, 16S36, 16U80, 20M25. Keywords and phrases: skew generalized power series ring, .S; !/-Armendariz, semicommutative, 2-primal, reversible, reduced, uniserial. 1. Introduction In 1974 Armendariz noted in [3] that whenever the product of two polynomials over a reduced ring R (that is, a ring without nonzero nilpotent elements) is zero, then the products of their coefficients are all zero, that is, in the polynomial ring RTxU the following holds: for any f .x/ D P a xi ; g.x/ D P b x j 2 RTxU; i j (∗) if f .x/g.x/ D 0; then ai b j D 0 for all i; j: Nowadays the property (∗) is known as the Armendariz condition, and rings R that satisfy (∗) are called Armendariz rings. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst
Volume-9 • Number-1 Jan -June 2017 pp. 9-13 available online at www.csjournalss.com A UGC Recommended Journal http://ugc.ac.in/journallist vide letter Dated: 28/03/2017 On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst. Prof. D.A.V. College (Lahore), Ambala City Abstract: A group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and satisfy certain axioms while a ring is an algebraic structure with two binary operations namely addition and multiplication. I have tried to discuss prime ideals, prime radical and m- system in this paper. Keywords: Prime ideals, semi-prime ideals, m-system, n-system. 1. INTRODUCTION In this paper, I have tried to explain the concept of prime ideals in an arbitrary ring, Radical of a ring and few properties of m-system. Besides these some theorems and lemma have been raised such as “If A is an ideal in ring R then B(A) coincide with intersection of all prime ideals in R which contain A”. Also some theorems and lemmas based on m-system and n-system have been established. 1.1 Prime Integer: An integer p is said to be prime integer if it has following property that if a and b are integers such that ab is divisible by p then a is divisible by p or b is divisible by p. 1.2 Prime Ideal [1]: An ideal P in ring R is said to be a prime ideal if and only if it has the following property: If A and B are ideals in R such that AB P, then A P or B P 1.3 m-system: A set M of elements of a ring R is said to be an m-system if and only if it has following property: If a, b M, these exists x R such that axb M 1.4 Theorem: If P is an ideal in ring R, then following conditions are equivalent: (i) P is a prime ideal (ii) If a, b R such that aRb P then a P or b P (iii) If (a) and (b) are principal ideals in R such that (a) (b) P then a P or b P. -
Rings Whose Quasi-Injective Modules Are Injecttve K
proceedings of the american mathematical society Volume 33, Number 2, June 1972 RINGS WHOSE QUASI-INJECTIVE MODULES ARE INJECTTVE K. A. BYRD Abstract. A ring R is called a V-ring, respectively SSI-ring, respectively QH-ring if simple, respectively semisimple, respectively quasi-injective, right Ä-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian. The rings R which have the property that quasi-injective right R- modules are injective (herein called QII-rings) have been little mentioned in the literature though related classes of rings have been extensively examined. Consider the classes of R-modules consisting of (1) all R- modules, (2) injective R-modules, (3) projective R-modules, (4) quasi- injective R-modules, and (5) quasi-projective R-modules. Of all the nontrivial inclusions one can assume between two of these classes, all but one has been shown to characterize the ring R as either semisimple artinian, quasi-Frobenius, or uniserial. That one excepted is the class of QII-rings whose internal structure, it seems, has not been completely determined. This paper does not attempt a general structure theorem but investi- gates the connection between QII-rings and rings whose semisimple right modules are injective (SSI-rings). -
The Projective Line Over the Finite Quotient Ring GF(2)[X]/⟨X3 − X⟩ and Quantum Entanglement I
The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background Metod Saniga, Michel Planat To cite this version: Metod Saniga, Michel Planat. The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background. 2006. hal-00020182v1 HAL Id: hal-00020182 https://hal.archives-ouvertes.fr/hal-00020182v1 Preprint submitted on 7 Mar 2006 (v1), last revised 6 Jun 2006 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background Metod Saniga† and Michel Planat‡ †Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´aLomnica, Slovak Republic ([email protected]) and ‡Institut FEMTO-ST, CNRS, D´epartement LPMO, 32 Avenue de l’Observatoire F-25044 Besan¸con, France ([email protected]) Abstract 3 The paper deals with the projective line over the finite factor ring R♣ ≡ GF(2)[x]/hx −xi. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As R♣ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. -
Encyclopaedia of Mathematical Sciences Volume 135 Invariant
Encyclopaedia of Mathematical Sciences Volume 135 Invariant Theory and Algebraic Transformation Groups VI Subseries Editors: R.V. Gamkrelidze V.L. Popov Martin Lorenz Multiplicative Invariant Theory 123 Author Martin Lorenz Department of Mathematics Temple Universit y Philadelphia, PA 19122, USA e-mail: [email protected] Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze Mathematics Subject Classification (2000): Primary: 13A50 Secondary: 13H10, 13D45, 20C10, 12F20 ISSN 0938-0396 ISBN 3-540-24323-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media GmbH springeronline.com ©Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover Design: E. Kirchner, Heidelberg, Germany Printed on acid-free paper 46/3142 YL 5 4 3 2 1 0 To my mother, Martha Lorenz, and to the memory of my father, Adolf Lorenz (1925 – 2001) Preface Multiplicative invariant theory, as a research area in its own right, is of relatively recent vintage: the systematic investigation of multiplicative invariants was initiated by Daniel Farkas in the 1980s. -
Symmetric Ideals and Numerical Primary Decomposition
SYMMETRIC IDEALS AND NUMERICAL PRIMARY DECOMPOSITION A Thesis Presented to The Academic Faculty by Robert C. Krone In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mathematics Georgia Institute of Technology August 2015 Copyright c 2015 by Robert C. Krone SYMMETRIC IDEALS AND NUMERICAL PRIMARY DECOMPOSITION Approved by: Professor Anton Leykin, Advisor Professor Stavros Garoufalidis School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Professor Josephine Yu Professor Santosh Vempala School of Mathematics College of Computing Georgia Institute of Technology Georgia Institute of Technology Professor Greg Blekherman Date Approved: 26 May 2015 School of Mathematics Georgia Institute of Technology ACKNOWLEDGEMENTS There are many people who deserve recognition for their parts in my completion of the Ph.D. thesis. First I thank my advisor Anton Leykin for his guidance and support, for supplying interesting and fruitful research problems, for pushing me to go to conferences, give talks and meet potential collaborators, and for putting up with my procrastination and habitual tardiness. I would also like to acknowledge the other research collaborators who contributed to the work that appears in this thesis. These are Jan Draisma, Rob Eggermont, Jon Hauenstein, Chris Hillar and Thomas Kahle. Thanks to Jan Draisma for giving the me the opportunity to work with him and his group at TU Eindhoven during the spring of 2013. Thanks to my thesis committee members Greg Blekherman, Stavros Garoufalidis, Anton Leykin, Santosh Vempala and Josephine Yu, for taking the time to be a part of this process. Gratitude goes to all of my friends at Georgia Tech who worked very hard to make my graduate school experience interesting, both mathematically and otherwise. -
On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals
Canad. Math. Bull. Vol. 14 (3), 1971 ON PRIME RINGS WITH ASCENDING CHAIN CONDITION ON ANNIHILATOR RIGHT IDEALS AND NONZERO INFECTIVE RIGHT IDEALS BY KWANGIL KOH AND A. C. MEWBORN If / is a right ideal of a ring R91 is said to be an annihilator right ideal provided that there is a subset S in R such that I={reR\sr = 0, VseS}. lis said to be injective if it is injective as a submodule of the right regular i£-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective. LEMMA 1. Let M and T be right R-modules such that M is injective and T has zero singular submodule [4] and no nonzero injective submodule. Then Hom# (M, T)={0}. Proof. Suppose fe Hom^ (M, T) such that /=£ o. Let Kbe the kernel off. Then K is a proper submodule of M and there exists me M such that f(m)^0. Let (K:m)={reR\mreK}. Since the singular submodule of T is zero and f(m)(K:m)={0} the right ideal (K:m) has zero intersection with some nonzero right ideal / in R. Then ra/#{0} and K n mJ={0}. Let mJ be the injective hull of m J. Since M is injective, mJ is a submodule of M. mJC\ K={0} since m J has nonzero intersection with each submodule which has nonzero intersection with mJ (See [4, p.