# Algebra

## An Approach via Module Theory

Part of the Graduate Texts in Mathematics book series (GTM, volume 136)

Part of the Graduate Texts in Mathematics book series (GTM, volume 136)

This book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields.

Permutation algebra automorphism field homomorphism linear algebra matrices matrix quadratic form transformation

- DOI https://doi.org/10.1007/978-1-4612-0923-2
- Copyright Information Springer-Verlag New York, Inc. 1992
- Publisher Name Springer, New York, NY
- eBook Packages Springer Book Archive
- Print ISBN 978-1-4612-6948-9
- Online ISBN 978-1-4612-0923-2
- Series Print ISSN 0072-5285
- About this book