MODULE THEORY An approach to linear algebra Electronic Edition T. S. Blyth Professor Emeritus, University of St Andrews 0 0 0 ? ? ? ? ? ? ? ? ? y y y 0 A B B B=(A B) 0 −−−−−−−! \? −−−−−−−−!?−−−−−−−! ?\ −−−−−! ? ? ? ? ? ? y y y 0 A M M=A 0 −−−−−−−−!?−−−−−−−−−! ? −−−−−−−−! ? −−−−−−−! ? ? ? ? ? ? y y y 0 A=(A B) M=B M=(A + B) 0 ? ? ? −−−−−! ?\ −−−−−! ? −−−−−! ? −−−−−! ? ? ? y y y 0 0 0 PREFACE to the Second O.U.P. Edition 1990 Many branches of algebra are linked by the theory of modules. Since the notion of a module is obtained essentially by a modest generalisation of that of a vector space, it is not surprising that it plays an important role in the theory of linear algebra. Modules are also of great importance in the higher reaches of group theory and ring theory, and are fundamental to the study of advanced topics such as homological algebra, category theory, and algebraic topology. The aim of this text is to develop the basic properties of modules and to show their importance, mainly in the theory of linear algebra. The first eleven sections can easily be used as a self-contained course for first year honours students. Here we cover all the basic material on modules and vector spaces required for embarkation on advanced courses. Concerning the prerequisite algebraic background for this, we mention that any standard course on groups, rings, and fields will suffice. Although we have kept the discussion as self-contained as pos- sible, there are places where references to standard results are unavoidable; readers who are unfamiliar with such results should consult a standard text on abstract alge- bra. The remainder of the text can be used, with a few omissions to suit any particular instructor’s objectives, as an advanced course. In this, we develop the foundations of multilinear and exterior algebra. In particular, we show how exterior powers lead to determinants. In this edition we include also some results of a ring-theoretic nature that are directly related to modules and linear algebra. In particular, we establish the celebrated Wedderburn–Artin Theorem that every simple ring is isomorphic to the ring of endomorphisms of a finite-dimensional module over a division ring. Finally, we discuss in detail the structure of finitely generated modules over a principal ideal domain, and apply the fundamental structure theorems to obtain, on the one hand, the structure of all finitely generated abelian groups and, on the other, important decomposition theorems for vector spaces which lead naturally to various canonical forms for matrices. At the end of each section we have supplied a number of exercises. These provide ample opportunity to consolidate the results in the body of the text, and we include lots of hints to help the reader gain the satisfaction of solving problems. Although this second edition is algebraically larger than the first edition, it is geometrically smaller. The reason is simple: the first edition was produced at a time when rampant inflation had caused typesetting to become very expensive and, re- grettably, publishers were choosing to produce texts from camera-ready material (the synonym of the day for typescript). Nowadays, texts are still produced from camera-ready material but there is an enormous difference in the quality. The inter- vening years have seen the march of technology: typesetting by computer has arrived and, more importantly, can be done by the authors themselves. This is the case with ii PREFACE iii the present edition. It was set entirely by the author, without scissors, paste, or any cartographic assistance, using the mathematical typesetting system TEX developed by Professor Donald Knuth, and the document preparation system LATEX developed by Dr Leslie Lamport. To be more precise, it was set on a Macintosh II computer us- ing the package MacTEX developed by FTL systems Inc. of Toronto. We record here our gratitude to Lian Zerafa, President of FTL, for making this wonderful system available. St Andrews August 1989 T.S.B. Added January 2018 The advance of technology has brought us into the era of electronic books, thus making it possible to resurrect many fine texts that have long been out of print and therefore difficult and expensive to obtain. What is reproduced here is basically the same as the 1990 printed second edition. However, set on my iMac using TeXShop with the package [charter]mathdesign, it takes up fewer pages. In preparing this digital edition I have taken care of typographical errors that were present in the printed second edition. I record here my grateful thanks to those who have been kind enough to communicate them to me. The main difference between this edition and the 1990 printed second edition is of course that this one is free to download! iv PREFACE c 2018 the Author. This book is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as appropriate credit is given to the author. The original source and a link to the Creative Commons license should also be given. Users must also indicate if changes were made. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. CONTENTS 1 Modules; vector spaces; algebras 1 2 Submodules; intersections and sums 7 3 Morphisms; exact sequences 13 4 Quotient modules; isomorphism theorems 28 5 Chain conditions; Jordan-Hölder towers 39 6 Products and coproducts 48 7 Free modules; bases 66 8 Groups of morphisms; projective modules 82 9 Duality ; transposition 98 10 Matrices; linear equations 107 11 Inner product spaces 125 12 Injective modules 138 13 Simple and semisimple modules 146 14 The Jacobson radical 157 15 Tensor products; flat modules; regular rings 164 16 Tensor algebras 183 17 Exterior algebras; determinants 198 18 Modules over a principal ideal domain; finitely generated abelian groups 222 19 Vector space decomposition theorems; canonical forms under similarity 243 20 Diagonalisation; normal transformations 267 Index 301 v 1 MODULES; VECTOR SPACES; ALGEBRAS In this text our objective will be to develop the foundations of that branch of math- ematics called linear algebra. From the various elementary courses that he has fol- lowed, the reader will recognise this as essentially the study of vector spaces and linear transformations, notions that have applications in several different areas of mathematics. In most elementary introductions to linear algebra the notion of a determinant is defined for square matrices, and it is assumed that the elements of the matrices in question lie in some field (usually the field R of real numbers). But, come the consideration of eigenvalues (or latent roots), the matrix whose determinant has to be found is of the form 2 3 x11 λ x12 ... x1n x − x x 6 21 22 λ ... 2n 7 6 . .− . 7 4 . .. 5 xn1 xn2 ... xnn λ − and therefore has its entries in a polynomial ring. This prompts the question of whether the various properties of determinants should not really be developed in a more general setting, and leads to the wider question of whether the scalars in the definition of a vector space should not be restricted to lie in a field but should more generally belong to a ring (which, as in the case of a polynomial ring, may be required at some stage to be commutative). It turns out that the modest generalisation so suggested is of enormous impor- tance and leads to what is arguably the most important structure in the whole of algebra, namely that of a module. The importance of this notion lies in a greatly ex- tended domain of application, including the higher reaches of group theory and ring theory, and such areas as homological algebra, category theory, algebraic topology, etc.. Before giving a formal definition of a module, we ask the reader to recall the following elementary notions. If E is a non-empty set then an internal law of compo- sition on E is a mapping f : E E E. Given (x, y) E E it is common practice to write f (x, y) as x + y, or x× y, except! when it might2 cause× confusion to use such additive or multiplicative notations, in which case notations such as x ? y, x y, x y, etc., are useful. A set on which there is defined an internal law of composi-◦ tion that is associative is called a semigroup. By a group we mean a semigroup with an identity element in which every element has an inverse. By an abelian group we mean a group in which the law of composition is commutative. By a ring we mean 2 Module Theory a set E endowed with two internal laws of composition, these being traditionally denoted by (x, y) x + y and (x, y) x y, such that 7! 7! (1) E is an abelian group under addition; (2) E is a semigroup under multiplication; (3) ( x, y, z E) x(y + z) = x y + xz, (y + z)x = y x + zx. 8 2 A ring R is said to be unitary if it has a multiplicative identity element, such an element being written 1R. By an integral domain we mean a unitary ring in which the non-zero elements form a (cancellative) semigroup under multiplication. By a division ring we mean a unitary ring in which the non-zero elements form a group under multiplication. A ring is commutative if the multiplication is commutative. By a field we mean a commutative division ring. In what follows we shall have occasion to consider mappings of the form f : F E E where F and E are non-empty sets.