'Et moi ... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Inhalt

1. Preliminaries.- 1. Notation and terminology.- 2. Artinian, noetherian and semisimple modules.- 3. Semisimple modules.- 4. The radical and socle of modules and rings.- 5. The Krull-Schmidt theorem.- 6. Matrix rings.- 7. The Wedderburn-Artin theorem.- 8. Tensor products.- 9. Croup algebras.- 2. Frobenius and symmetric algebras.- 1. Definitions and elementary properties.- 2. Frobenius crossed products.- 3. Symmetric crossed products.- 4. Symmetric endomorphism algebras.- 5. Projective covers and injective hulls.- 6. Classical results.- 7. Frobenius uniserial algebras.- 8. Characterizations of Frobenius algebras.- 9. Characters of symmetric algebras.- 10. Applications to projective modular representations.- 11. Külshammer's theorems.- 12. Applications.- 3. Symmetric local algebras.- 1. Symmetric local algebras A with dimFZ(A) ? 4.- 2. Some technical lemmas.- 3. Symmetric local algebras A with dimFZ(A) = 5.- 4. Applications to modular representations.- 4. G-algebras and their applications.- 1. The trace map.- 2. Permutation G-algebras.- 3. Algebras over complete noetherian local rings.- 4. Defect groups in G-algebras.- 5. Relative projective and injective modules.- 6. Vertices as defect groups.- 7. The G-algebra EndR((1H)G).- 8. An application: The Robinson's theorem.- 9. The Brauer morphism.- 10. Points and pointed groups.- 11. Interior G-algebras.- 12. Bilinear forms on G-algebras.