Basic Algebra I

Second Edition

Nathan Jacobson

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486135229
• Edition: 2
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.
Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as Lie and Jordan algebras, lattices, and Boolean algebras. Exercises appear throughout the text, along with insightful, carefully explained proofs. Volume II comprises all subjects customary to a first-year graduate course in algebra, and it revisits many topics from Volume I with greater depth and sophistication.

1

Monoids and Groups

The theory of groups is one of the oldest and richest branches of algebra. Groups of transformations play an important role in geometry, and, as we shall see in Chapter 4, finite groups are the basis of Galois’ discoveries in the theory of equations. These two fields provided the original impetus for the development of the theory of groups, whose systematic study dates from the early part of the nineteenth century.

A more general concept than that of a group is that of a monoid. This is simply a set which is endowed with an associative binary composition and a unit—whereas groups are monoids all of whose elements have inverses relative to the unit. Although the theory of monoids is by no means as rich as that of groups, it has recently been found to have important “external” applications (notably to automata theory). We shall begin our discussion with the simpler and more general notion of a monoid, though our main target is the theory of groups. It is hoped that the preliminary study of monoids will clarify, by putting into a better perspective, some of the results on groups. Moreover, the results on monoids will be useful in the study of rings, which can be regarded as pairs of monoids having the same underlying set and satisfying some additional conditions (e.g., the distributive laws).

A substantial part of this chapter is foundational in nature. The reader will be confronted with a great many new concepts, and it may take some time to absorb them all. The point of view may appear rather abstract to the uninitiated. We have tried to overcome this difficulty by providing many examples and exercises whose purpose is to add concreteness to the theory. The axiomatic method, which we shall use throughout this book and, in particular, in this chapter, is very likely familiar to the reader: for example, in the axiomatic developments of Euclidean geometry and of the real number system. However, there is a striking difference between these earlier axiomatic theories and the ones we shall encounter. Whereas in the earlier theories the defining sets of axioms are categorical in the sense that there is essentially only one system satisfying them—this is far from true in the situations we shall consider. Our axiomatizations are intended to apply simultaneously to a large number of models, and, in fact, we almost never know the full range of their applicability. Nevertheless, it will generally be helpful to keep some examples in mind.

The principal systems we shall consider in this chapter are: monoids,...