Chapter 11
Finiteness Conditions for Rings and Modules
A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
—Henri Poincaré
The field 
of complex numbers is a two-dimensional vector space over 
A ring that is also a vector space over a field F is called an algebra over F. In the nineteenth century many attempts were made to describe the division algebras that are finite dimensional over and in particular those (like ) that are fields. Certainly the three-dimensional examples would have had applications to physics, but after looking in vain for such an algebra, the first success came in 1843 when W.R. Hamilton discovered the ring 
of quaternions, a four-dimensional algebra that, surprisingly, was not commutative. It was not until 1878 that G. Frobenius showed that there is no three-dimensional example, and that the only possible associative examples are , and Meanwhile, G. ...
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