Rings and Things and a Fine Array of Twentieth Century Associative Algebra
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The Center of Topologically Primitive Exponentially Galbed Algebras
THE CENTER OF TOPOLOGICALLY PRIMITIVE EXPONENTIALLY GALBED ALGEBRAS MART ABEL AND MATI ABEL Received 29 September 2004; Revised 11 December 2005; Accepted 18 December 2005 Let A be a unital sequentially complete topologically primitive exponentially galbed Haus- dorff algebra over C, in which all elements are bounded. It is shown that the center of A is topologically isomorphic to C. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction (1) Let A be an associative topological algebra over the field of complex numbers C with separately continuous multiplication. Then A is an exponentially galbed algebra (see, e.g., [1–4, 19, 20]) if every neighbourhood O of zero in A defines another neighbourhood U of zero such that n a k a ... a ∈ U ⊂ O k : 0, , n (1.1) k=0 2 for each n ∈ N. Herewith, A is locally pseudoconvex,ifithasabase{Uλ : λ ∈ Λ} of neigh- bourhoods of zero consisting of balanced and pseudoconvex sets (i.e., of sets U for which μU ⊂ U,whenever|μ| 1, and U + U ⊂ ρU for a ρ 2). In particular, when every Uλ in {Uλ : λ ∈ Λ} is idempotent (i.e., UλUλ ⊂ Uλ), then A is called a locally m-pseudoconvex algebra, and when every Uλ in {Uλ : λ ∈ Λ} is A-pseudoconvex (i.e., for any a ∈ A there is a μ>0suchthataUλ,Uλa ⊂ μUλ), then A is called a locally A-pseudoconvex algebra.Itiswell known (see [21,page4]or[6, page 189]) that the locally pseudoconvex topology on A is given by a family {pλ : λ ∈ Λ} of kλ-homogeneous seminorms, where kλ ∈ (0,1] for each λ ∈ Λ.Thetopologyofalocallym-pseudoconvex (A-pseudoconvex) algebra -
Free Ideal Rings and Localization in General Rings
This page intentionally left blank Free Ideal Rings and Localization in General Rings Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization, which is treated for general rings, but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note. paul cohn is Emeritus Professor of Mathematics at the University of London and Honorary Research Fellow at University College London. NEW MATHEMATICAL MONOGRAPHS Editorial Board B´ela Bollob´as William Fulton Frances Kirwan Peter Sarnak Barry Simon For information about Cambridge University Press mathematics publications visit http://publishing.cambridge.org/stm/mathematics Already published in New Mathematical Monographs: Representation Theory of Finite Reductive Groups Marc Cabanes, Michel Enguehard Harmonic Measure John B. Garnett, Donald E. Marshall Heights in Diophantine Geometry Enrico Bombieri, Walter Gubler Free Ideal Rings and Localization in General Rings P. M. COHN Department of Mathematics University College London Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge ,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521853378 © Cambridge University Press 2006 This publication is in copyright. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title One-sided prime ideals in noncommutative algebra Permalink https://escholarship.org/uc/item/7ts6k0b7 Author Reyes, Manuel Lionel Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Koushik Sen Spring 2010 One-sided prime ideals in noncommutative algebra Copyright 2010 by Manuel Lionel Reyes 1 Abstract One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplan- sky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bi- jectively to the classes of cyclic right R-modules that are closed under extensions. -
TWO RESULTS on PROJECTIVE MODULES We Wish to Present Two Results. the First Is from the Paper Pavel Prıhoda, Projective Modules
TWO RESULTS ON PROJECTIVE MODULES ANDREW HUBERY We wish to present two results. The first is from the paper Pavel Pˇr´ıhoda, Projective modules are determined by their radical factors, J. Pure Applied Algebra 210 (2007) 827{835. Theorem A. Let Λ be a ring, with Jacobson radical J, and let P and Q be two projective (right) Λ-modules. Then we can lift any isomorphism f¯: P=P J −!∼ Q=QJ to an isomorphism f : P −!∼ Q. In particular, projective modules are determined by their tops. Special cases have been known for a long time. For example, we say that Λ is right perfect if every right Λ-module admits a projective cover. Then P P=P J is a projective cover, so the result follows (Bass 1960). In general, though, P ! P=P J will not be a projective cover. Beck (1972) showed that if P=P J is a free Λ=J-module, then P is free. Note that if J = 0 then the theorem tells us nothing! The second result is from the appendix to the paper Michael Butler and Alastair King, Minimal resolutions of algebras, J. Algebra 212 (1999) 323{362. Theorem B. Let Λ = TA(M) be an hereditary tensor algebra, where A is semisim- ple and M is an A-bimodule. Then every projective module is isomorphic to an induced module, so one of the form X ⊗A Λ; equivalently, every projective module is gradable. This is trivial if there are no oriented cycles; that is, if the graded radical Λ+ is nilpotent, so Λ is semiprimary and J(Λ) = Λ+. -
Jacobson Radical and on a Condition for Commutativity of Rings
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 4 Ver. III (Jul - Aug. 2015), PP 65-69 www.iosrjournals.org Jacobson Radical and On A Condition for Commutativity of Rings 1 2 Dilruba Akter , Sotrajit kumar Saha 1(Mathematics, International University of Business Agriculture and Technology, Bangladesh) 2(Mathematics, Jahangirnagar University, Bangladesh) Abstract: Some rings have properties that differ radically from usual number theoretic problems. This fact forces to define what is called Radical of a ring. In Radical theory ideas of Homomorphism and the concept of Semi-simple ring is required where Zorn’s Lemma and also ideas of axiom of choice is very important. Jacobson radical of a ring R consists of those elements in R which annihilates all simple right R-module. Radical properties based on the notion of nilpotence do not seem to yield fruitful results for rings without chain condition. It was not until Perlis introduced the notion of quasi-regularity and Jacobson used it in 1945, that significant chainless results were obtained. Keywords: Commutativity, Ideal, Jacobson Radical, Simple ring, Quasi- regular. I. Introduction Firstly, we have described some relevant definitions and Jacobson Radical, Left and Right Jacobson Radical, impact of ideas of Right quasi-regularity from Jacobson Radical etc have been explained with careful attention. Again using the definitions of Right primitive or Left primitive ideals one can find the connection of Jacobson Radical with these concepts. One important property of Jacobson Radical is that any ring 푅 can be embedded in a ring 푆 with unity such that Jacobson Radical of both 푅 and 푆 are same. -
Basically Full Ideals in Local Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Algebra 250, 371–396 (2002) provided by Elsevier - Publisher Connector doi:10.1006/jabr.2001.9099, available online at http://www.idealibrary.com on Basically Full Ideals in Local Rings William J. Heinzer Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 and Louis J. Ratliff Jr. and David E. Rush Department of Mathematics, University of California, Riverside, California 92521 Communicated by Craig Huneke Received September 4, 2001 Let A be a finitely generated module over a (Noetherian) local ring R M.We say that a nonzero submodule B of A is basically full in A if no minimal basis for B can be extended to a minimal basis of any submodule of A properly containing B. We prove that a basically full submodule of A is M-primary, and that the fol- lowing properties of a nonzero M-primary submodule B of A are equivalent: (a) B is basically full in A; (b) B =MBA M; (c) MB is the irredundant intersection of µB irreducible ideals; (d) µC≤µB for eachcover C of B. Moreover, if B ∗ is an M-primary submodule of A, then B = MBA M is the smallest basically full submodule of A containing B and B → B∗ is a semiprime operation on the set of nonzero M-primary submodules B of A. We prove that all nonzero M-primary ideals are closed withrespect to thisoperation if and only if M is principal. In rela- tion to the closure operation B → B∗, we define and study the bf-reductions of an M-primary submodule D of A; that is, the M-primary submodules C of D suchthat C ⊆ D ⊆ C∗.IfGM denotes the form ring of R withrespect to M and G+M its maximal homogeneous ideal, we prove that Mn =Mn∗ for all (resp. -
A Prime Ideal Principle in Commutative Algebra
Journal of Algebra 319 (2008) 3006–3027 www.elsevier.com/locate/jalgebra A Prime Ideal Principle in commutative algebra T.Y. Lam ∗, Manuel L. Reyes Department of Mathematics, University of California, Berkeley, CA 94720, USA Received 11 May 2007 Available online 18 September 2007 Communicated by Luchezar L. Avramov Abstract In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. © 2007 Elsevier Inc. All rights reserved. Keywords: Commutative algebra; Commutative rings; Prime ideals; Ideal families; Prime ideal principles; Module categories 1. Introduction One of the most basic results in commutative algebra, given as the first theorem in Kaplansky’s book [Ka2], is (1.1) below, which guarantees that certain kinds of ideals in a commutative ring are prime. (In the following, all rings are assumed to be commutative with unity, unless otherwise specified.) * Corresponding author. E-mail addresses: [email protected] (T.Y. -
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. -
January 2002 Prizes and Awards
January 2002 Prizes and Awards 4:25 p.m., Monday, January 7, 2002 PROGRAM OPENING REMARKS Ann E. Watkins, President Mathematical Association of America BECKENBACH BOOK PRIZE Mathematical Association of America BÔCHER MEMORIAL PRIZE American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION Association for Women in Mathematics ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN Association for Women in Mathematics CHAUVENET PRIZE Mathematical Association of America FRANK NELSON COLE PRIZE IN NUMBER THEORY American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society CERTIFICATES OF MERITORIOUS SERVICE Mathematical Association of America LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America CLOSING REMARKS Hyman Bass, President American Mathematical Society MATHEMATICAL ASSOCIATION OF AMERICA BECKENBACH BOOK PRIZE The Beckenbach Book Prize, established in 1986, is the successor to the MAA Book Prize. It is named for the late Edwin Beckenbach, a long-time leader in the publica- tions program of the Association and a well-known professor of mathematics at the University of California at Los Angeles. The prize is awarded for distinguished, innov- ative books published by the Association. Citation Joseph Kirtland Identification Numbers and Check Digit Schemes MAA Classroom Resource Materials Series This book exploits a ubiquitous feature of daily life, identification numbers, to develop a variety of mathematical ideas, such as modular arithmetic, functions, permutations, groups, and symmetries. -
Regular Ideals in Commutative Rings, Sublattices of Regular Ideals, and Prtifer Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 111, 404426 (1987) Regular Ideals in Commutative Rings, Sublattices of Regular Ideals, and Prtifer Rings D. D. ANDERSON* AND J. PASCUAL+ Departmenr of Muthematics, The University qf Iowa, Iowa City, Iowa 52242 Communicated by J. DieudonnC Received April 9, 1986 In this paper we investigate regularly generated. regular, semiregular, and faithful ideals in a commutative ring R and the sublattices they determine. Connections with multiplicative lattice theory are given. Given a Priifer ring R we show that there is a Priifer domain D with the sublattice of regular ideals of R isomorphic to the lattice of ideals of D. Numerous examples of rings with zero divisors having cer- tain properties are given. A Priifer ring with an invertible ideal that is not generated by regular elements is constructed. An example is given to show that the intersec- tion of two regular principal ideals need not be generated by regular elements. (I3 1987 Academic Press. lnc 1. INTRODUCTION Throughout this paper, R will be a commutative ring with identity. We are interested in how parts of the multiplicative theory of ideals for integral domains extend to rings with zero divisors; in particular, to what extent the regular ideals of a ring R behave like the ideals of an integral domain. Section 2 contains the necessary definitions and some preliminary remarks. Regularly generated, regular, semiregular, and faithful ideals are defined along with several “regularity” conditions for rings. A brief outline of the method of idealization is given. -
Idempotent Pairs and PRINC Domains
Idempotent pairs and PRINC domains Giulio Peruginelli, Luigi Salce and Paolo Zanardo∗ August 28, 2018 Dedicated to Franz Halter-Koch on the occasion of his 70th birthday Abstract A pair of elements a,b in an integral domain R is an idempotent pair if either a(1 a) bR, − ∈ or b(1 b) aR. R is said to be a PRINC domain if all the ideals generated by an idempotent − ∈ pair are principal. We show that in an order R of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if R is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders Z[√ d], d > 0 − square-free, that are PRINC and not integrally closed, are for d = 3, 7. Keywords: Orders, Conductor, Primary decomposition, Dedekind domains, Principal ideals, Projective- free. 2010 Mathematics Subject Classification: 13G05, 13F05, 13C10, 11R11. Introduction. Let R be an integral domain, Mn(R) the ring of matrices of order n with entries in R, T any singular matrix in Mn(R) (i.e. det T = 0). A natural question is to find conditions on R to ensure that T is always a product of idempotent matrices. This problem was much investigated in past years, see [17] for comprehensive references. The case when R is a B´ezout domain (i.e. the finitely generated ideals of R are all principal) is crucial. In fact, we recall three fundamental results, valid for matrices with entries in a B´ezout domain. -
Stably Free Modules Over Innite Group Algebras
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UCL Discovery Department of Mathematics University College London Stably free modules over infinite group algebras Pouya Kamali A thesis submitted for the degree of Doctor of Philosophy Supervisor: Professor F.E.A. Johnson June 2010 1 I, Pouya Kamali, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. ..................................................... 2 Abstract We study finitely generated stably-free modules over infinite integral group algebras by using the language of cyclic algebras and relating it to well-known results in K-theory. For G a free or free abelian group and Q8n, the quaternionic group of order 8n, we show that there exist infinitely many isomorphically distinct stably-free modules of rank 1 over the integral group algebra of the group Γ = Q8n × G whenever n admits an odd divisor. This result implies that the stable class of the augmentation ideal Ω1(Z) displays infinite splitting at minimal level whenever G is the free abelian group on at least 2 generators. This is of relevance to low dimensional topol- ogy, in particular when computing homotopy modules of a cell complex with fundamental group Γ. 3 Acknowledgements First and foremost, I would like to thank my supervisor, Professor F.E.A. Johnson, who accepted me as his student and introduced me to the fascinat- ing subject of algebraic K-theory. My mathematical personality has greatly been shaped by his generous attitude towards research, but also by the man- ner in which he communicated it.