On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity
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Quantum Groupoids Acting on Semiprime Algebras
Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 546058, 9 pages doi:10.1155/2011/546058 Research Article Quantum Groupoids Acting on Semiprime Algebras Inesˆ Borges1 and Christian Lomp2 1 Instituto Superior De Contabilidade e Administrac¸ao˜ de Coimbra, Quinta Agr´ıcola-Bencanta, 3040-316 Coimbra, Portugal 2 Departamento de Matematica,´ Faculdade de Ciencias,ˆ Universidade do Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal Correspondence should be addressed to Christian Lomp, [email protected] Received 30 March 2011; Accepted 5 June 2011 Academic Editor: Olaf Lechtenfeld Copyright q 2011 I. Borges and C. Lomp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Following Linchenko and Montgomery’s arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime. 1. Introduction Group actions, Lie algebras acting as derivations and finite group gradings are typical examples of Hopf algebra actions which have been studied for many years. Several generali- zations of Hopf algebras have emerged in recent years, like weak Hopf algebras or quantum groupoids introduced by Bohm¨ et al. 1. The action of such objects on algebras, as given by ∗ quantum groupoids acting on C -algebras, 2 or weak Hopf algebras arising from Jones towers 3 are particularly interesting. New examples of weak Hopf algebras arose from double groupoids 4, which were also used to find new weak Hopf actions see 2. -
ON FULLY IDEMPOTENT RINGS 1. Introduction Throughout This Note
Bull. Korean Math. Soc. 47 (2010), No. 4, pp. 715–726 DOI 10.4134/BKMS.2010.47.4.715 ON FULLY IDEMPOTENT RINGS Young Cheol Jeon, Nam Kyun Kim, and Yang Lee Abstract. We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining hs(Matn(R)) = Matn(hs(R)) for any ring R where hs(−) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunow- icz’s simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process. 1. Introduction Throughout this note each ring is associative with identity unless stated otherwise. Given a ring R, denote the n by n full (resp. upper triangular) matrix ring over R by Matn(R) (resp. Un(R)). Use Eij for the matrix with (i, j)-entry 1 and elsewhere 0. Z denotes the ring of integers. A ring (possibly without identity) is called reduced if it has no nonzero nilpotent elements. A ring (possibly without identity) is called semiprime if the prime radical is zero. Reduced rings are clearly semiprime and note that a commutative ring is semiprime if and only if it is reduced. The study of fully idempotent rings was initiated by Courter [2]. -
On Ring Extensions for Completely Primary Noncommutativerings
ON RING EXTENSIONS FOR COMPLETELY PRIMARY NONCOMMUTATIVERINGS BY E. H. FELLER AND E. W. SWOKOWSKI 0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x~\, x an indeterminate where ax = xa for all aeR,isN primary and then constructing the right quotient ring QCR[x]). The details of these results appear in §§1,7 and 8. The correspond- ing results for the commutative case are given by E. Snapper in [7] and [8]. If Re: A, where A is completely JVprimary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(o) when oeA and ao = era for all aER in the cases where a is algebraic or transcendental over R. These structures are determined in §§ 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, JV or N(R) denotes the union of nilpotent ideals^) of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R. -
1. Introduction. the Radical of a Ring Has Been Hitherto Defined by Using Either the Notion of a Nilpotent Or a Nil-Ideal
ON THE RADICAL OF A GENERAL RING JAKOB LEVITZKI 1. Introduction. The radical of a ring has been hitherto defined by using either the notion of a nilpotent or a nil-ideal. In the following we shall ascribe the term specialized radical to the sum Na of all two- sided nilpotent ideals of a ring S, and the term generalized radical to the sum Ny of all two-sided nil-ideals of the ring. In §2 of the present note the notions of a semi-nilpotent ideal and its counterpart the semi- regular ideal are introduced, and the term radical is suggested for the sum N of all two-sided semi-nilpotent ideals of the ring. These nota tions may be justified by the following considerations: (a) Each nilpotent ideal is semi-nilpotent, and each semi-nilpotent ideal is a nil-ideal. (b) The radical N is a two-sided semi-nilpotent ideal which con tains also all one-sided semi-nilpotent ideals of the ring. (c) The radical of S/N is zero. (d) The radical N contains the specialized radical Na and is a sub set of the generalized radical Ny. (e) In the case of an algebra the notions: nilpotent, semi-nilpotent r r and nil-ideal are identical,andiV (r = iV = iV7;butif one turns to general rings, and replaces radical N and semi-nilpotent ideals either by specialized radical Na and nilpotent ideals or generalized radical Ny and nil-ideals, then some restriction has to be imposed on the ring 5 in order to assure the validity of (b) and (c).1 These results are applied in §3 to semi-primary rings (which will be called in short: A -rings). -
Nil Elements and Noncommutative Rings
Acta Ciencia Indica, Vol. XLIII M, No. 1 (2017) Acta Ciencia Indica, Vol. XLIII M, No. 1, 01 (2017)1 NIL ELEMENTS AND NONCOMMUTATIVE RINGS S. K. PANDEY Deptt. of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Daijar-342304, Jodhpur RECEIVED : 31 January, 2017 Nil elements (which are special type of nilpotent elements) have been introduced recently. Here we study the role of nil elements in noncommutative rings and provide some results. Among other things we prove that a noncommutative ring of order n can contain maximum n 1 nil elements and it is noted that if R be a noncommutative ring, N and N are the sets of all nil and nilpotent elements of R respectively then N is a nil ideal of R iff N is a subring of R and x ax N,x xa N, xax 0,x N,a R provided R has non-nil nilpotent elements. KEYWORDS : Even square rings, nil elements, nilpotent elements, noncommutative rings, nil ideals. AMSC2010 : 16D25, 16N40. INTRODUCTION Nil elements do not appear explicitly in the mathematical literatures [1-3] and the notion of nil elements has been introduced in [4]. It has been seen that the set of all nil elements in a commutative ring forms a nil ideal of the ring however the same is not necessarily true in the case of noncommutative rings. Recall that an element a of a ring R is called a nil element if a2 2a 0 and a ring R is called an even square ring if a2 2R,a R. -
Lectures on Non-Commutative Rings
Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address. -
MATH 145 NOTES 1. Meta Stuff and an Overview of Algebraic Geometry
MATH 145 NOTES ARUN DEBRAY AUGUST 21, 2015 These notes were taken in Stanford’s Math 145 class in Winter 2015, taught by Ravi Vakil. I TEXed these notes up using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. CONTENTS 1. Meta Stuff and an Overview of Algebraic Geometry: 1/5/151 2. Projective Space and Fractional Linear Transformations: 1/7/153 3. Diophantine Equations and Classifying Conics: 1/9/155 4. Quadratics and Cubics: 1/12/15 6 5. Cubics: 1/14/15 8 6. The Elliptic Curve Group and the Zariski Topology: 1/16/159 7. Affine Varieties and Noetherian Rings: 1/21/15 11 8. Localization and Varieties: 1/23/15 13 9. Hilbert’s Nullstellensatz: 1/26/15 14 10. Affine Varieties: 1/28/15 15 11. Affine Schemes: 1/30/15 17 12. Regular Functions and Regular Maps: 2/2/15 18 13. Rational Functions: 2/4/15 20 14. Rational Maps: 2/6/15 21 15. Varieties and Sheaves: 2/9/15 23 16. From Rational Functions to Rational Maps: 2/11/15 24 17. Varieties and Pre-Varieties: 2/13/15 25 18. Presheaves, Sheaves, and Affine Schemes: 2/18/15 26 19. Morphisms of Varieties and Projective Varieties: 2/20/15 28 20. Products and Projective Varieties: 2/23/15 29 21. Varieties in Action: 2/25/15 31 22. Dimension: 2/27/15 32 23. Smoothness and Dimension: 3/2/15 33 24. The Tangent and Cotangent Spaces: 3/6/15 35 25. -
The Harvard College Mathematics Review
The Harvard College Mathematics Review Volume 3 Spring 2011 In this issue: CHRISTOPHER POLICASTRO Artin's Conjecture ZHAO CHEN, KEVIN DONAGHUE & ALEXANDER ISAKOV A Novel Dual-Layered Approach to Geographic Profiling in Serial Crimes MICHAEL J. HOPKINS The Three-Legged Theorem MLR A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un found online at the journal's website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors will be listed in the print edition of The HCMR and on a spe http://www.thehcmr.org/ fl) cial page on the The HCMR's website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor Fellow $100-$249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editor-in-Chief Rediet Abebe at hcmr@ hcs . Contributor $500-$1,999 harvard.edu. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad uate students. Articles may highlight any topic in undergrad Contributors ■ Jane Street Capital • The Harvard Uni uate mathematics or in related fields, including computer sci versity Mathematics Department ence, physics, applied mathematics, statistics, and mathemat Cover Image. The image on the cover depicts several ical economics. functions, whose common zero locus describes an algebraic Authors may submit articles electronically, in .pdf, .ps, or .dvi format, to [email protected], or in hard variety (which is in this case an elliptic curve). -
Nil and Jacobson Radicals in Semigroup Graded Rings
Faculty of Science Departement of Mathematics Nil and Jacobson radicals in semigroup graded rings Master thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Carmen Mazijn Promotor: Prof. Dr. E. Jespers AUGUST 2015 Acknowledgements When we started our last year of the Master in Mathematics at VUB, none of us knew how many hours we would spend on the reading, understanding and writing of our thesis. This final product as conclusion of the master was at that point only an idea. The subject was chosen, the first papers were read and the first words were written down. And more words were written, more books were consulted, more questions were asked to our promoters. Writing a Master thesis is a journey. Even though next week everyone will have handed in there thesis, we don’t yet understand clearly where this journey took us, for the future is unknown. First of all I would like to thank professor Eric Jespers, for giving me the chance to grow as mathematician in the past years. With every semester the interest in Algebra and accuracy as mathematician grew. Thank you for the guidance through all the books and papers to make this a consistent dissertation. Secondly I would like to thank all my classmates and compa˜nerosde clase. For frowned faces when we didn’t get something in class, the laughter when we realized it was a ctually quite trivial or sometimes not even at all. For the late night calls and the interesting discussions. It was a pleasure. -
Arxiv:1507.04134V1 [Math.RA]
NILPOTENT, ALGEBRAIC AND QUASI-REGULAR ELEMENTS IN RINGS AND ALGEBRAS NIK STOPAR Abstract. We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial px with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the K¨othe conjecture, namely the integral rings. Key Words: π-algebraic element, nil ring, integral ring, quasi-regular element, Jacobson radical, upper nilradical 2010 Mathematics Subject Classification: 16N40, 16N20, 16U99 1. Introduction Let R be an associative ring or algebra. Every nilpotent element of R is quasi-regular and algebraic. In addition the quasi-inverse of a nilpotent element is a polynomial in this element. In the first part of this paper we will be interested in the connections between these three notions; nilpo- tency, algebraicity, and quasi-regularity. In particular we will investigate how close are algebraic elements to being nilpotent and how close are quasi- regular elements to being nilpotent. We are motivated by the following two questions: Q1. Algebraic rings and algebras are usually thought of as nice and well arXiv:1507.04134v1 [math.RA] 15 Jul 2015 behaved. For example an algebraic algebra over a field, which has no zero divisors, is a division algebra. On the other hand nil rings and algebras, which are of course algebraic, are bad and hard to deal with. -
Solutions to Assignment 5
Solutions to Assignment 5 Throughout, A is a commutative ring with 0 6= 1. 1. We say that a ∈ A is a zerodivisor if there exists b 6= 0 in A such that ab = 0. (This differs from Lang’s definition only to the extent that 0 will be called a zerodivisor.) Let F be the set of all ideals of A in which every element is a zerodivisor. (a) Prove that F has maximal elements. (b) Prove that every maximal element of F is a prime ideal. (c) Conclude that the set of zerodivisors is a union of prime ideals. Solution: (a) Since (0) ∈ F, the set F is nonempty. Given a chain {aα} of elements of F, it S is easily seen that α aα is an ideal as well. Since each element of this ideal is a zerodivisor, it S follows that α aα ∈ F. By Zorn’s Lemma, the set F has maximal elements. (b) Let p ∈ F be a maximal element. If x∈ / p and y∈ / p, then the ideals p + (x) and p + (y) are strictly bigger than p, and hence there exist nonzerodivisors a ∈ p + (x) and b ∈ p + (y). But then ab ∈ p + (xy) is a nonzerodivisor, and so xy∈ / p. Consequently p is a prime ideal. (c) If x ∈ A is a zerodivisor, then every element of the ideal (x) is a zerodivisor. Hence (x) ∈ F, and so (x) ⊆ p for some prime ideal p ∈ F. It follows that [ {x | x ∈ A is a zerodivisor} = p. p∈F 2. Let a be a nilpotent element of A. -
[Nilradical of a Ring] Let N ⊂ a Consist of All Nilpotent Elements. Prove That N
ALGEBRA 1: PROBLEM SET 10 A = a commutative ring in all the problems below. Problem 1. [Nilradical of a ring] Let n ⊂ A consist of all nilpotent elements. Prove that \ n = p p⊂A prime ideal Problem 2. Let a ⊂ A be the set of all zero{divisors of A. Is a an ideal of A? Problem 3. Let n 2 A be a nilpotent element and u 2 A be a unit (that is, u has a multiplicative inverse). Prove that u + n is again a unit. Problem 4. Let S ⊂ A be a set such that 0 62 S and r; s 2 S implies rs 2 S (multiplicatively closed set). Let p be an ideal which is maximal among the ideals not intersecting S. That is, maximal with respect to inclusion, from the following set: IS = fa ⊂ A an ideal such that a \ S = ;g Prove that p is prime. Problem 5. If for every x 2 A there exists an n ≥ 2 such that xn = x, then prove that every prime ideal in A is maximal. Problem 6. Let B be another commutative ring and let f : A ! B be a ring homomorphism. Let p ⊂ A be a prime ideal and define b ⊂ B to be the ideal generated by f(p). Prove or disprove: b is a prime ideal. Problem 7. Let K be a field and R = K[[x]] be the ring of formal power series in a variable x with coefficients from K. Prove that R is a local ring, with unique maximal ideal m = (x).