## Brian Hartley - *Rings, modules and linear algebra*

Brian Hartley wrote only one book, namely

*Rings, modules and linear algebra. A further course in algebra describing the structure of Abelian groups and canonical forms of matrices through the study of rings and modules*. It was written jointly with Trevor Hawkes and first published in 1970. We give below three extracts of reviews of this book:**Rings, modules and linear algebra (1970), by Brian Hartley and Trevor Hawkes.**

**Review by: Robert Eugene MacRae.**

*Mathematical Reviews*MR0267997**(42 #2897).**

The authors have written a very attractive little book whose purpose is to introduce undergraduates to modern abstract algebra. To this end they introduce the notion of a ring and quickly restrict their attention to principal ideal domains. The usual examples are given and the phenomenon of unique factorization is covered. Next, modules are introduced and a discussion of various types is undertaken - in particular, finitely generated, free, torsion and torsion free modules. They then prove in detail the classical structure theorems for finitely generated modules over a principal ideal domain.-
**Review by: Edward McWilliam Patterson.**

*The Mathematical Gazette***56**(395) (1972), 73.

In these days of over-production of text-books on undergraduate mathematics, it is pleasant to come across a book which really is needed, and which can be heartily recommended. The present work possesses this rare combination of qualities. It should be regarded as an essential possession for any self- respecting undergraduate specialising in mathematics. Not only does it give a detailed account of the basic properties of rings and modules, in particular proving the central theorems about the decomposition of modules over a principal ideal domain, but also it provides an excellent illustration of the way in which mathematicians can unify different theories through generalisation, without losing sight of the applications of their abstract theories. ... The style of writing is clear, instructive, informative and never dull. Full consideration is given to the reader's possible doubts and misunderstandings; yet he is given every encouragement to do what every mathematician must do-think for himself. ... All in all, this is a first-rate book. **Review by: Allan George Heinicke.**

*Amer. Math. Monthly***79**(2) (1972), 192-193.

The main goals of this book are one theorem (the decomposition of a finitely generated module over a principal ideal domain into a direct sum of cyclic sub- modules) and two types of applications of this theorem (the study of finitely generated abelian groups and the discovery of canonical forms for linear transformations on finite dimensional vector spaces). The book gives, to the properly prepared reader, a highly readable account of these results. The authors aim at the reader who has a "well-developed facility with the language of sets, operations, and mappings, as well as a working knowledge of vector spaces, linear transformations, and matrices". In addition, some prior knowledge of elementary ring theory (in particular the unique factorization property in the ring of integers and in the ring of polynomials over a field) would not be amiss, for it seems to this reviewer that the early chapters of the book would be a bit stiff for a beginner.