The chief purpose of this book is the development for 
the first time of a general theory of the arithmetics of 
algebras, which furnishes a direct generalization of the 
classic theory of algebraic numbers. The book should 
appeal not merely to those interested in either algebra 
or the theory of numbers, but also to those interested in 
the foundations of mathematics. Just as the final 
stage in the evolution of number was reached with 
the introduction of hypercomplex numbers (which make 
up a linear algebra), so also in arithmetic, which began 
with integers and was greatly enriched by the introduc 
tion of integral algebraic numbers, the final stage of its 
development is reached in the present new theory of 
arithmetics of linear algebras. 
Since the book has interest for wide classes of readers, 
no effort has been spared in making the presentation 
clear and strictly elementary, requiring on the part of 
the reader merely an acquaintance with the simpler 
parts of a first course in the theory of equations. Each 
definition is illustrated by a simple example. Each 
chapter has an appropriate introduction and summary. 
The author’s earlier brief book, Linear Algebras 
(Cambridge University Press, 1914), restricted attention 
to complex algebras. But the new theory of arithmetics 
of algebras is based on the theory of algebras over a 
general field. The latter theory was first presented by 
Wedderburn in his memoir in the Proceedings of the 
London Mathematical Society for 1907. The proofs of