[...]...xii 5 Contents of Basic Algebra 1 REAL P O L Y N O M I A L EQUATIONS A N D INEQUALITIES 5.1 5.2 5.3 5.4 5.5 5.6 6 n 405 Definition and examples of associative algebras 406 Exterior algebras Application to determinants 411 Regular matrix representations of associative algebras Norms and traces 422 Change of base field Transitivity of trace and norm 426 Non-associative algebras Lie and Jordan algebras... sections 8. 1-8 .3, 8.6, 8.12, Chapter 9 to section 9.13 A slight modification of this program would be to trade off sections 4. 6-4 .8 for sections 5. 1-5 .5 and 5.9 F o r students who have had no Galois theory it will be desirable to supplement section 8.3 with some of the material of Chapter 4 of Basic Algebra I If an important objective of a course in algebra is an understanding of the foundations of algebraic... Cheng, Professor Florence Jacobson, and James Reid Florence Jacobson assisted in compiling the index Finally we should mention the fine j o b of typing that was done by Joyce Harry and D o n n a Belli I am greatly indebted to all of these individuals, and I take this opportunity to offer them my sincere thanks January 1980 Nathan Jacobson Introduction In the Introduction to Basic Algebra I (abbreviated... an example.) 11 R-mod, the category of left modules for a fixed ring R (We assume lx = x for x in a left R-module M.) obR-mod is the class of left modules for R and the morphisms are R-module homomorphisms Products are composites of maps If R = A is a division ring (in particular, a field), then R-mod is the category of (left) vector spaces over A In a similar manner one defines mod-R as the category... translators of this book, Professors Cao Xi-hua and W a n g Jian-pan, pointed out some errors in the first edition which are now corrected We are indeed grateful for their interest and their important inputs to the new edition O u r heartfelt thanks are due also to F D Jacobson, for reading the proofs of this text and especially for updating the index January 1989 Nathan Jacobson Preface to the First Edition... one-semester course on representation theory of finite groups, and Chapter 7 and parts of Chapters 8, 9, and 10 could be used for a one-semester course in commutative algebras Chapters 1, 3, and 6 could be used for an introductory course in homological algebra Chapter 11 on real fields is somewhat isolated from the remainder of the book However, it constitutes a direct extension of Chapter 5 of Basic. .. A morphism f in R-mod or in Grp is monic (epic) if and only if the map of the underlying set is injective (surjective) Proof Let / : A - B in R-mod or Grp Iff is injective (surjective) as a m a p of sets, then it is left (right) cancellable in Set It follows that / is monic (epic) in R-mod or Grp N o w suppose the set m a p / is not injective Then C = k e r / ^ 0 in the case of R-mod and # 1 in the... Elimination procedures Resultants 322 Decision method for an algebraic curve 327 Generalized Sturm's theorem Tarski's principle 335 306 455 Partially ordered sets and lattices 456 Distributivity and modularity 461 The theorem of Jordan-Holder-Dedekind 466 The lattice of subspaces of a vector space Fundamental theorem of projective geometry 468 Boolean algebras 474 The Mobius function of a partialy ordered... understanding of the foundations of algebraic structures and the relation be- xvi Preface to the First Edition tween algebra and mathematical logic, then all of Chapter 2 should be included in the course This, of course, will necessitate thinning down other parts, e.g., homological algebra There are many other possibilities for a one-year course based on this text The material in each chapter is treated... sections in Chapter 8: 8.13, Transcendency Bases for Domains; 8.18, Tensor Products of Fields; and 8.19, Free Composites of Fields The latter two are taken from Volume III of our Lectures in Abstract Algebra (D Van N o s t r a n d 1964; Springer-Verlag, 1980) The most notable addition to Chapter 9 is Krasner's lemma, used to give an improved proof of a classical theorem of Kurschak's lemma (1913) We also