Noetherian Semigroup Algebras
Within the last decade, semigroup theoretical methods have occurred naturally in many aspects of ring theory, algebraic combinatorics, representation theory and their applications. In particular, motivated by noncommutative geometry and the theory of quan
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Algebra and Applications Volume 7 Managing Editor: Alain Verschoren RUCA, Belgium Series Editors: Christoph Schweigert Hamburg University, Germany Ieke Moerdijk Utrecht University, The Netherlands John Greenlees Sheffield University, UK Mina Teicher Bar-llan University, Israel Eric Friedlander Northwestern University, USA Idun Reiten Norwegian University of Science and Technology, Norway Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C*algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science.
Noetherian Semigroup Algebras by
Eric Jespers Vrije Universiteit Brussel, Brussels, Belgium and
Jan Okniński Warsaw University, Warsaw, Poland
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 ISBN-13 ISBN-10 ISBN-13
1-4020-5809-8 (HB) 978-1-4020-5809-7 (HB) 1-4020-5810-1 (e-book) 978-1-4020-5810-3 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
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To Griet
Eric Jespers
To Iwona
Jan Okni´ nski
Contents 1 Introduction
1
2 Prerequisites on semigroup theory 2.1 Semigroups . . . . . . . . . . . . . 2.2 Uniform semigroups . . . . . . . . 2.3 Full linear semigroup . . . . . . . . 2.4 Structure of linear semigroups . . . 2.5 Closure . . . . . . . . . . . . . . . 2.6 Semigroups over a field . . . . . . .
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7 7 13 17 19 27 32
3 Prerequisites on ring theory 3.1 Noetherian rings and rings satisfying a polynomial identity 3.2 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Group algebras of polycyclic-by-finite groups . . . . . . . . 3.4 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Gelfand-Kirillov dimension . . . . . . . . . . . . . . . . . . 3.6 Maximal orders . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Principal ideal rings
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