Free ideal rings can be regarded as a natural and non-commutative generalization of principal ideal domains. Recently there has been a surprising increase in non-commutative theories, and in this book, the author has reorganized his previous book ‘Free rings and their relations’ [2nd ed., 1985; Zbl 0659.16001] with many improvements. The theory of skew field extensions has been omitted, since a fuller account is now available in the same author’s book ‘Skew fields. Theory of general division rings’ [Encycl. Math. Appl. 57. Cambridge: Cambridge Univ. Press (1995; Zbl 0840.16001)].
The central part of the book is Chapter 7, which studies the homomorphisms of rings into skew fields. For a commutative ring such homomorphisms are described completely in terms of prime ideals, and the reader will find a similar but less obvious non-commutative description here.
In the remaining 6 chapters the theory of free ideal rings is developed, and their similarity to principal ideal domains is stressed. Therefore the author devotes Chapter 1 to recalling the properties of principal ideal domains. Chapter 2 introduces the main topic, free ideal rings, which frequently satisfy a weak algorithm relative to a filtration. Chapter 3 discusses a non-commutative analogue of the unique factorization property of principal ideal domains. Chapter 4 argues that the factors of any element form a modular lattice, which is even distributive in the case of free algebras. Chapter 5 deals with a non-commutative generalization of finitely generated modules over principal ideal domains. Chapter 6 examines centers, centralizers and subalgebras.