Selected Works of A.I. Shirshov
This book presents translations of selected works of the famous Russian mathematician A.I. Shirshov (1921–1981). He was a pioneer in several directions of associative, Lie, Jordan, and alternative algebras, as well as groups and projective planes. His nam
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Contemporary Mathematicians Gian-Carlo Rota† Joseph P.S. Kung Editors
Selected Works of A.I. Shirshov
Leonid A. Bokut Victor Latyshev Ivan Shestakov Efim Zelmanov Editors
Translated by Murray Bremner and Mikhail V. Kotchetov
Birkhäuser Basel · Boston · Berlin
Editors: Leonid Bokut Sobolev Institute of Mathematics Koptyuga, 4 630090, Novosibirsk Russia Ivan Shestakov IME-USP Ruodo Matao, 1010 Cidade Universitaria CEP 05508-090 Sao Paulo SP, Brasil
Victor Latyshev Mehmat MGU im. Lomonosova Vorob‘evy Gory 119899 Moscow Russia Efim Zelmanov Dep. of Mathematics UCSD 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112 USA
Translators: Murray Bremner University of Saskatchewan Dept. Mathematics & Statistics 106 Wiggins Road Saskatoon SK S7N 5E6 Canada
Mikhail V. Kotchetov Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7 Canada
Library of Congress Control Number: 2009925469 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbiographie; detailed bibliographic data is available in the internet at http://dnb.ddb.de
ISBN 978-3-7643-8857-7 Birkhäuser Verlag, Basel – Boston – Berlin 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Photo on page iii: From the family archive. With the friendly permission of Shirshov’s daughter Trofimova Nina Anatolevna. Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8857-7 987654321
e-ISBN 978-3-7643-8858-4 www.birkhauser.ch
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Publications of A.I. Shirshov [1] Subalgebras of Free Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
[2] On the Representation of Lie Rings in Associative Rings . . . . . . . . . . . . .
15
[3] Subalgebras of Free Commutative and Free Anticommutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
[4] On Special J-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
[5] Some Theorems on Embedding of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
[6] On Some Nonassociative Nil-rings and Algebraic Algebras . . . . . . . . . . .
55
[7] On Rings with Identical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
[8] On Free Lie Rings . . . . . . . . . . . . . . . . . . . . .
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