# Noncommutative Algebra

In Math Insight

Noncommutative Algebra (Graduate Texts in Mathematics) 1993rd Edition

ByAndrei Prokopiwon March 7, 2004

Format: Hardcover

This is a very nice introduction to the theory of semisimple modules and rings, central simple algebras, and the Brauer group. It starts off with some review of basic concepts such as modules, tensor products, and field extensions – much of which is in the exercises. Indeed, a lot of the material is developed in the exercises so it is very useful to go through them. Most of the exercises aren’t exceedingly difficult either, and hints are provided for the more tricky ones.

Chapter one gets the ball rolling with the theory of simple modules and rings. The Wedderburn structure theorem is proved, as well as the structure theorem for simple artinian rings. These ideas are examined further in chapter two by analysing the Jacobson radical, thereby giving another characterization of semisimple rings. Chapter three goes on to study central simple algebras. Here they prove the famous Wedderburn theorem that every finite division ring is commutative, as well as the famous Frobenius theorem on real division algebras. Chapter four finally gives the definition of the Brauer group with a discussion of group cohomology. It proves that for a Galois extension K/k the second Galois cohomology group H^2(Gal(K/k),K*) is isomorphic to the Brauer group Br(K/k). The naturality of this isomorphism is discussed later on and developed through the exercises, although they never actually define what natuarality really is.

One way I think the book could be improved would be to introduce some more abstract homological algebra. The definition of group cohomology given is in terms of n-cochains. I think it would be useful to include the notion of derived functors here, and the general definition of the cohomolgy homology of a derived functor through the use of injective and projective resolutions. Then the group cohomology could be defined using the Ext functors, and the cocycle complexes obtained by examined the particular projective resolution called the Eilenberg-MacLane bar resolution. This method would help introduce the reader to homological algebra, and at the same time give some concrete use of such abstract constructions. The authors generally shy away from categorical language though, which might appeal to some readers.

The later chapters discuss primitive rings, some representation theory of finite groups, dimension theory of rings, and the Brauer group of a commutative ring. I have not gone through this later material in detail (yet), so I cannot give comment on it.

Overall, if you want to learn some non-commutative algebra, by all means buy this book and work out as many exercises as you can. I found the exposition to be quite illuminating and overall very well written.

Chapter one gets the ball rolling with the theory of simple modules and rings. The Wedderburn structure theorem is proved, as well as the structure theorem for simple artinian rings. These ideas are examined further in chapter two by analysing the Jacobson radical, thereby giving another characterization of semisimple rings. Chapter three goes on to study central simple algebras. Here they prove the famous Wedderburn theorem that every finite division ring is commutative, as well as the famous Frobenius theorem on real division algebras. Chapter four finally gives the definition of the Brauer group with a discussion of group cohomology. It proves that for a Galois extension K/k the second Galois cohomology group H^2(Gal(K/k),K*) is isomorphic to the Brauer group Br(K/k). The naturality of this isomorphism is discussed later on and developed through the exercises, although they never actually define what natuarality really is.

One way I think the book could be improved would be to introduce some more abstract homological algebra. The definition of group cohomology given is in terms of n-cochains. I think it would be useful to include the notion of derived functors here, and the general definition of the cohomolgy homology of a derived functor through the use of injective and projective resolutions. Then the group cohomology could be defined using the Ext functors, and the cocycle complexes obtained by examined the particular projective resolution called the Eilenberg-MacLane bar resolution. This method would help introduce the reader to homological algebra, and at the same time give some concrete use of such abstract constructions. The authors generally shy away from categorical language though, which might appeal to some readers.

The later chapters discuss primitive rings, some representation theory of finite groups, dimension theory of rings, and the Brauer group of a commutative ring. I have not gone through this later material in detail (yet), so I cannot give comment on it.

Overall, if you want to learn some non-commutative algebra, by all means buy this book and work out as many exercises as you can. I found the exposition to be quite illuminating and overall very well written.

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