The Algebraic Structure of Crossed Products

Abstract

Preface. Chapters: 1. Preliminaries. Notation and Terminology. Projective, Injective and Flat Modules. Artinian and Noetherian Modules. Group Actions. Cohomology Groups and Group Extensions. Some Properties of Cohomology Groups. Matrix Rings and Related Results. 2. Group-Graded Algebras and Crossed Products: General Theory. Definitions and Elementary Properties. Equivalent Crossed Products. Some Ring-Theoretic Results. The Centre of Crossed Products over Simple Rings. Projective Crossed Representations. Graded and G-Invariant Ideals. Induced Modules. Montgomery's Theorem. 3. The Classical Theory of Crossed Products. Central Simple Algebras. The Brauer Group. Classical Crossed Products and the Brauer Group. 4. Clifford Theory for Graded Algebras. Graded Modules. Restriction to A 1 . Graded Homomorphism Modules. Extension from A 1 . Induction from A 1 . 5. Primitive and Prime Ideals of Crossed Products. Primitive, Prime and Semiprime Ideals. Primitive Ideals in Crossed Products. Prime Coefficient Rings. Incomparability and Going Down. A Going Up Theorem. Chains of Prime and Primitive Ideals. 6. Semiprime and Prime Crossed Products. Coset Calculus. Delta-Methods. The Main Theorem and Its Applications. Sufficient Conditions for Semiprimeness. Twisted Group Algebras. Bibliography. Notation. Index.