A survey of integral representation theory
Irving Reiner · Bulletin of the American Mathematical Society · 1970
1. Introduction.Notation and definitions 2. General remarks.Jordan-Zassenhaus Theorem 3. Extensions 4. Higman ideal 5. Representations over local domains 6. Genus 7. Maximal orders 8. Further results on genera 9. Projective modules and relative projective modules 10.Grothendieck groups and Whitehead groups 11.Commutative orders and related results 12. Divisibility of modules 13.Hereditary orders and related results 14. Finiteness of the number of indecomposable representations 15.Representations of specific groups and orders 16.Representation rings 17.Group rings 18. Algebraic number theory 19.Krull-Schmidt and Cancellation Theorems Introduction. Notation and definitions.First of all I wish to acknowledge with thanks the many helpful conversations I have had with Professors Olga Taussky, Peter Roquette and Hans Zassenhaus, when I first began studying the subject of integral representations.Historically, the subject received its main impetus from two branches of algebra.One branch is algebraic number theory, especially that part concerned with ideal theory; and the other is matrix theory, mainly that portion dealing with matrix representations of associative algebras.Methods of homological algebra have played an increasingly important role in the subject in recent years.An expanded version of an address delivered before the Chicago meeting of the Society by invitation of the Committee to Select Hour Speakers for Western Sectional Meetings, April 20, 1968, under the title Recent progress in the theory of integral representations; received by the editors September 17, 1969.