"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.