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An Introduction to Noncommutative Noetherian Rings

K. R. Goodearl and R. B. Warfield, Jr
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Darren Glass
, on

"A ring is said to be noetherian if every family of ideals has a maximal  element".  This definition may seem unnatural when one first runs across  it, but soon becomes quite natural - number theorists use it as a crucial  fact about rings of integers, algebraic geometers use it as a critical  feature of coordinate rings, and it shows up throughout mathematics.  But  most of the examples that come to mind - or at least come to this  reviewer's mind - are of commutative rings which satisfy the noetherian  condition.  An Introduction to Noncommutative Noetherian Rings by K.R.  Goodearl and R.B. Warfield Jr is a rich source for examples of  noncommutative rings which are noetherian - such as rings of differential  algebras and quantum groups - and also develops much of the theory of such  rings.

The first edition of the book was published in 1989, shortly after which  Warfield passed away.  Goodearl has recently revised the book and a new  edition has been released in the London Mathematical Society's Student  Texts series.  Readers of the first edition will notice a number of  changes in this edition, including a restructuring of the topics, many  more explicit examples, and an increased emphasis on quantum groups, a  subject which has flourished in the decade that has passed since the first  edition.  The new edition also discusses the examples of skew polynomial  rings in much more explicit detail, going through the examples of  twistings by automorphisms first and then twistings by derivations before  considering the general case.  This level of explicitness would make large  parts of the book accessible even to an undergraduate who has only  completed a first course in algebra, although the target audience is  primarily at the graduate student level.

One thing that has not changed in the second edition is the excellent  exposition.   This book should be held up as a great example of exposition  of highly technical material.  Each chapter starts by telling you what it  is going to tell you, and in particular how that topic relates to the  analogous ideas in the theories of abelian groups and commutative rings  which the reader may already be familiar with.  There are copious  exercises sprinkled throughout the chapter which give even more explicit  examples of the theory as it is developed, and the end of each chapter has  yet more exercises.  Each chapter ends with a 'notes' section which gives  the historical details of the various topics - this is a nice solution to  the problem of wanting to give those details for interested readers  without slowing down the exposition of the body of the material, and is a  solution that I wish more authors would adopt.

One of the problems that many books have when covering this type of  material is that it ends up reading like a series of disconnected  definitions followed by theorems showing that they are not really  disconnected which one only gets to after slogging through many technical  details.  Goodearl and Warfield avoid this problem and succeed in  motivating definitions - often by giving the "right" way to think about  something alongside the technical definition - and giving colorful  examples throughout which make the theory come alive and quite  comprehensible.  I wouldn't go as far as to say that this book is good  relaxing bedtime reading, but it is comes as close as one could imagine for a book on noncommutative noetherian rings.

Darren Glass teaches at Gettysburg College.


1. A few Noetherian rings; 2. Skew polynomial rings; 3. Prime ideals; 4. Semisimple modules, Artinian modules, and torsionfree modules; 5. Injective hulls; 6. Semisimple rings of fractions; 7. Modules over semiprime Goldie rings; 8. Bimodules and affiliated prime ideals; 9. Fully bounded rings; 10. Rings and modules of fractions; 11. Artinian quotient rings; 12. Links between prime ideals; 13. The Artin-Rees property; 14. Rings satisfying the second layer condition; 15. Krull dimension; 16. Numbers of generators of modules; 17. Transcendental division algebras.