# Algebra

## A Teaching and Source Book

• Ernest Shult
• David Surowski
Textbook

1. Front Matter
Pages i-xxii
2. Ernest Shult, David Surowski
Pages 1-19
3. Ernest Shult, David Surowski
Pages 21-71
4. Ernest Shult, David Surowski
Pages 73-103
5. Ernest Shult, David Surowski
Pages 105-136
6. Ernest Shult, David Surowski
Pages 137-161
7. Ernest Shult, David Surowski
Pages 163-184
8. Ernest Shult, David Surowski
Pages 185-230
9. Ernest Shult, David Surowski
Pages 231-277
10. Ernest Shult, David Surowski
Pages 279-332
11. Ernest Shult, David Surowski
Pages 333-354
12. Ernest Shult, David Surowski
Pages 355-441
13. Ernest Shult, David Surowski
Pages 443-469
14. Ernest Shult, David Surowski
Pages 471-527
15. Back Matter
Pages 529-539

### Introduction

This book presents a graduate-level course on modern algebra. It can be used as a teaching book – owing to the copious exercises – and as a source book for those who wish to use the major theorems of algebra.

The course begins with the basic combinatorial principles of algebra: posets, chain conditions, Galois connections, and dependence theories. Here, the general Jordan–Holder Theorem becomes a theorem on interval measures of certain lower semilattices. This is followed by basic courses on groups, rings and modules; the arithmetic of integral domains; fields; the categorical point of view; and tensor products.

Beginning with introductory concepts and examples, each chapter proceeds gradually towards its more complex theorems. Proofs progress step-by-step from first principles. Many interesting results reside in the exercises, for example, the proof that ideals in a Dedekind domain are generated by at most two elements. The emphasis throughout is on real understanding as opposed to memorizing a catechism and so some chapters offer curiosity-driven appendices for the self-motivated student.

### Keywords

Fields Groups Rings Jordan-Hölder Theorem Modern Algebra Modular Lattices Posets Wedderburn's Theorem

#### Authors and affiliations

• Ernest Shult
• 1
• David Surowski
• 2
1. 1.Department of MathematicsKansas State UniversityManhattanUSA
2. 2.(deceased)ManhattanUSA