Algebra and Geometry PDF
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PROGRESS IN MATHEMATICS Volume 12 Algebra and Geometry PROGRESS IN MATHEMATICS Translations of /togi Nauki - Seriya Matematika 1968: Volume 1 - Mathematical Analysis Volume 2 - Mathematical Analysis 1969: Volume 3 - Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 4 - Mathematical Analysis Volume 5 - Algebra 1970: Volume 6 - Topology and Geometry Volume 7 - Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 8 - Mathematical Analysis 1971: Volume 9 - Algebra and Geometry Volume 10 - Mathematical Analysis Volume 11 - Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 12 - Algebra and Geometry In preparation: Volume 13 - Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 14 - Algebra, Topology, and Geometry Volume 15 - Mathematical Analysis PROGRESS IN MATHEMATICS Volume 12 Algebra and Geometry Edited by R. V. Gamkrelidze v. A. Steklov Mathematics Institute Academy of Sciences of the USSR, Moscow Translated from Russian by Nasli H. Choksy g:>PLENUM PRESS • NEW YORK-LONDON • 1972 The original Russian text was published for the All-Union Institute of Scientific and Technical Information in Moscow in 1970 as a volume of /togi Nauki - Seriya Maternatika EDITORIAL BOARD R. V. Gamkrelidze, Editor-in-Chief N. M. Ostianu, Secretary P. S. Aleksandrov Yu. V. Linnik N. G. Chudakov E. F. Mishchenko M. K. Kerimov M. A. Naimark A. N. Kolmogorov S. M. Nikol'skii L. D. Kudryavtsev N. Kh. Rozov G. F. Laptev K. A. Rybnikov Library of Congress Catalog Card Number 67-27902 ISBN 978-1-4757-0509-6 ISBN 978-1-4757-0507-2 (eBook) DOI 10.1007/978-1-4757-0507-2 The present translation is published under an agreement with Mezhdunarodnaya Kniga, the Soviet book export agency © 1972 Plenum Press, New York Softcover reprint of the hardcover I st edition 1972 A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011 United Kingdom edition published by Plenum Press, London A Division of Plenum Publishing Company, Ltd. Davis House (4th Floor), 8 Scrubs Lane, Harlesden, NWIO 6SE, England All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher PREFACE This volume contains five review articles, three in the Al- gebra part and two in the Geometry part, surveying the fields of ring theory, modules, and lattice theory in the former, and those of integral geometry and differential-geometric methods in the calculus of variations in the latter. The literature covered is primarily that published in 1965-1968. v CONTENTS ALGEBRA RING THEORY L. A. Bokut', K. A. Zhevlakov, and E. N. Kuz'min § 1. Associative Rings. . . . . . . . . . . . . . . . . . . . 3 § 2. Lie Algebras and Their Generalizations. . . . . . . 13 ~ 3. Alternative and Jordan Rings. . . . . . . . . . . . . . .. 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25 MODULES A. V. Mikhalev and L. A. Skornyakov § 1. Radicals. . . . . . . . . . . . . . . . . . . 59 § 2. Projection, Injection, etc. . . . . . . . . . . . . . . . . .. 62 § 3. Homological Classification of Rings. . . . . . . . . . .. 66 § 4. Quasi-Frobenius Rings and Their Generalizations.. 71 § 5. Some Aspects of Homological Algebra . . . . . . . . .. 75 § 6. Endomorphism Rings . . . . . . . . . . . . . . . . . . . .. 83 § 7. Other Aspects. . . . . . . . . . . . . . . . . .. 87 Bibliography ............................... , 91 LATTICE THEORY M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova § 1. Boolean Algebras ..................... " 111 § 2. Identity and Defining Relations in Lattices . . . . .. 120 § 3. Distributive Lattices. . . . . . . . . . . . . . . . . . . .. 122 vii viii CONTENTS § 4. Geometrical Aspects and the Related Investigations. . . . . . . . . . . . • . . • . . . . . . . . .• 125 § 5. Homological Aspects. . . . . . . . . . . . . . . . . . . . .. 129 § 6. Lattices of Congruences and of Ideals of a Lattice .. 133 § 7. Lattices of Subsets, of Subalgebras, etc. ........ 134 § 8. Closure Operators . . . . . . . . . . . . . . . . . . . . . .. 136 § 9. Topological Aspects. . . . . . . . . . . . . . . . . . . . .. 137 § 10. Partially-Ordered Sets. . . . . . . . . . . . . . . . . . .. 141 § 11. Other Questions. . . . . . . . . . . . . . . . . . . . . . . .. 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 148 GEOMETRY INTEGRAL GEOMETRY G. 1. Drinfel'd Preface ........ . 173 Chapter 1. Measures in Uniform Spaces . . . . . . . . . . . .. 174 § 1. Chern's Program. . . . . . . . . . . . . . . . . . . . . . .. 174 § 2. Invariant Measure of a Point Set and Integral Invariants .......................... " 178 § 3. Measure of a Set of Geometric Elements and Integral Invariants . . . . . . . . . . . . . . . . . .. 183 § 4. Kinematic Measure ... . . . . . . . . . . . . . . . . .. 189 § 5. Abstract Foundations. Some New Directions. . . 191 Chapter II. Actual Problems and Applications . . . . . . . .. 195 § 1. Generalizations of Known Formulas ....... 195 § 2. The Rashevskii Problem . . . . . . . . . . . . . . . . . .. 198 § 3. Applications of Kinematic Measure (Lattices and Coverings). . . . . . . . . . . . . . . . . . . . . . . . .. 198 § 4. Applications of Kinematic Measure (Minkowski Integrals), Moments. . . . . . . . . . . . . . . . . . . . .. 200 § 5. Certain Affine Invariants. . . . . . . . . . . . . . . . . .. 205 § 6. Distribution Functions ofIntersections. . . . . . . . .. 206 § 7. Integral Geometry and Pattern Recognition. . . . . .. 209 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 210 CONTENTS ix DIFFERENTIAL-GEOMETRIC METHODS IN THE CALCULUS OF VARIATIONS N. I. Kabanov introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " 215 § 1. Geometrization of the Simplest n-Dimensional Variational Problem. Finsler Space . . . . . . . . . .. 216 § 2. Variational Problem for Functionals Containing Higher Derivatives. Kawaguchi Spaces ....... " 224 § 3. Variational Problem for Multiple Integrals Spaces with Areal Metric. . . • . . . . . . . . . . . . . .. 229 § 4. Intrinsic Lagrange Problem for Ordinary Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 239 § 5. Intrinsic Lagrange Problem for Multiple Integrals.. 242 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 246 ALGEBRA Ring Theory L. A. Bokut', K. A. Zhevlakov, and E. N. Kuz'min In this survey we cover the papers on ring theory reviewed in the Mathematics section of Referativnyi Zhurnal during 1966- 1968, with the exception of Nos. 11 and 12 of 1968, with which the authors had no opportunity to acquaint themselves. In individual cases earlier results are mentioned, as well as articles which had not been reviewed upto the moment of writing. § 1. Associative Rings The rings being considered in this section are associative. Unless we say otherwise the terminology is referred to the left (modules are left modules, an Artinian ring is a left-Artinian ring, etc.). The following branches of the theory of associative rings do not appear in this survey: commutative rings, extensions of rings, ordered rings, and generalizations of rings. Unless we specit:y otherwise, primality and radicality are to be understood in the sense of Jacobson. 1. Rings with Chain Conditions and Quotient Ri ng s. We call a ring R the (two-sided, left, right) order in a ring Q if Q is the classical full (two-sided, left, right) quotient ring for R. A ring R is called a left (right) Goldie ring if R satis- fies the maximality condition for the left (right) annihilator ideals and if every direct sum of nonzero left (right) ideals contains only a finite number of terms. A left and right Goldie ring is called simply a Goldie ring. The following two theorems of Goldie [305, 306] have evoked large response: A ring R is a two-sided order in a prime ring with minimality condition if and only if R is a prime Goldie ring. A ring R is a right order in a semiprime ring with minimal- ity condition if and only if R is a semiprime right Goldie ring. 3
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