Module Theory Reading Group

This reading group focuses on "Module Theory: An approach to linear algebra" by T. S. Blyth. The text explores the theory of modules as a generalization of vector spaces, providing the algebraic foundation for advanced linear algebra and multilinear algebra. The group will cover the fundamental properties of modules, including submodules, morphisms, and quotient modules, before moving into more specialized topics such as chain conditions (Noetherian and Artinian modules), projective and injective modules, and tensor products. A central objective is to reach the structure theorem for finitely generated modules over a principal ideal domain (PID) and apply it to the study of canonical forms for matrices. Specific topics include: * Basic module and vector space constructions. * Chain conditions and Jordan-Hölder towers. * Products, coproducts, and free modules. * Duality, transposition, and multilinear algebra. * Exterior algebras and the algebraic definition of determinants. * Modules over a PID and their application to abelian groups and vector space decomposition. Prerequisites: Participants should have a working knowledge of standard undergraduate-level groups, rings, and fields. Format: The group will proceed through the 20 sections of the book, utilizing the exercises provided at the end of each chapter to consolidate the theory. The first half of the book (Sections 1–11) establishes core module theory, while the second half addresses more advanced ring-theoretic and multilinear results.