Noetherian Semigroup Algebras

1. Front Matter

   Pages I-IX

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2. Introduction

   • Eric Jespers, Jan Okniński

   Pages 1-5

3. Prerequisites on semigroup theory

   • Eric Jespers, Jan Okniński

   Pages 7-35

4. Prerequisites on ring theory

   • Eric Jespers, Jan Okniński

   Pages 37-61

5. Algebras of submonoids of polycyclic-by-finite groups

   • Eric Jespers, Jan Okniński

   Pages 63-115

6. General Noetherian semigroup algebras

   • Eric Jespers, Jan Okniński

   Pages 117-147

7. Principal ideal rings

   • Eric Jespers, Jan Okniński

   Pages 149-175

8. Maximal orders and Noetherian semigroup algebras

   • Eric Jespers, Jan Okniński

   Pages 177-195

9. Monoids of I-type

   • Eric Jespers, Jan Okniński

   Pages 197-255

10. Monoids of skew type

    • Eric Jespers, Jan Okniński

    Pages 257-289

11. Examples

    • Eric Jespers, Jan Okniński

    Pages 291-341

12. Back Matter

    Pages 343-362

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The ?rst aim of this work is to present the main results and methods of the theory of Noetherian semigroup algebras. These general results are then applied and illustrated in the context of certain interesting and important concrete classes of algebras that arise in a variety of areas and have been recently intensively studied. One of the main motivations for this project has been the growing int- est in the class of semigroup algebras (and their deformations) and in the application of semigroup theoretical methods. Two factors seem to be the cause for this. First, this ?eld covers several important classes of algebras that recently arise in a variety of areas. Furthermore, it provides methods to construct a variety of examples and tools to control their structure and properties, that should be of interest to a broad audience in algebra and its applications. Namely, this is a rich resource of constructions not only for the noncommutative ring theorists (and not only restricted to Noetherian rings) but also to researchers in semigroup theory and certain aspects of group theory. Moreover, because of the role of new classes of Noetherian algebras in the algebraic approach in noncommutative geometry, algebras of low dimension (in terms of the homological or the Gelfand-Kirillov - mension) recently gained a lot of attention. Via the applications to the Yang-Baxter equation, the interest also widens into other ?elds, most - tably into mathematical physics.

• Algebraic structure
• Gelfand-Kirillov dimension
• Group theory
• Noetherian and PI algebra
• Representation theory
• maximal order
• ring theory
• semigroup algebra
• semigroup, group

From the reviews:

"This book presents the main results and methods of the theory of Noetherian semigroup algebras over fields. … This is a highly technical and specialized monograph that will primarily be of use to researchers in the theory of semigroup algebras … . The many examples given in detail, and the general theory developed, may prove useful to those working in ring or semigroup theory." (Henry E. Heatherly, Mathematical Reviews, Issue 2007 k)

"This work presents a comprehensive treatment of the main results and methods of the theory of Noetherian semigroup algebras. … The main subject in this book is to present when some semigroup algebras are Noetherian and how they are under the Noetherian condition. … This book is a good reference for researchers who are interested in non-commutative algebra and non-commutative geometry through the method of semigroups." (Li Fang, Zentralblatt MATH, Vol. 1135 (13), 2008)

"The book under review is testament to the huge amount and the depth of research on Noetherian semigroup algebras. … The authors have brought together work from a significant number of research papers and made a coherent whole. … it is definitely the place to learn about Noetherian semigroup algebras. Essential reading for people interested in semigroup algebras, it will also be of interest to semigroup theorists, particularly because of the wealth of new examples of semigroups it provides." (John Fountain, Semigroup Forum, Vol. 79, 2009)

• Vrije Universiteit Brussel, Brussels, Belgium

  Eric Jespers

• Warsaw University, Warsaw, Poland

  Jan Okniński

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