Engineering↗
Direct Sums, Free Modules and Matrix Representations
Handbook guide to direct sums, free modules and matrix representations with core definitions, structural results, reasoning methods and verification checks.
Tag
modules
Every page in the KEVOS library tagged modules. 20 pages.
Engineering↗
Exact Functors and Projective Modules
Handbook guide to exact functors and projective modules with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Flat Modules, Direct Limits and Inverse Limits
Handbook guide to flat modules, direct limits and inverse limits with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Free, Projective and Injective Modules
Handbook guide to free, projective and injective modules, covering definitions, key results, methods, verification checks and worked examples.
Engineering↗
Injective Modules, Injective Embeddings and Divisible Groups
Injective Modules, Injective Embeddings and Divisible Groups: core definitions, structural results and verification methods in abstract algebra.
Engineering↗
Modules over Noncommutative Rings and Algebra Representations
A representation of an algebra is a module over that algebra. Simple modules, invariant subspaces, composition series and endomorphism rings provide the language for decomposing…
Engineering↗
Modules over Principal Ideal Domains
Handbook guide to modules over principal ideal domains, covering definitions, key results, methods, verification checks and worked examples.
Engineering↗
Modules, Algebras and Module Homomorphisms
Handbook guide to modules, algebras and module homomorphisms with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Modules, Free Modules and Direct Sums
Modules generalise vector spaces by allowing scalars from a ring rather than a field. This introduces free modules, generators, submodules, quotient modules and module homomorph…
Engineering↗
Modules, Submodules and Exact Sequences
Handbook guide to modules, submodules and exact sequences, covering definitions, key results, methods, verification checks and worked examples.
Engineering↗
Semisimple Modules and Key Structure Theorems
Handbook guide to semisimple modules and key structure theorems with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Tensor Products of Modules
Handbook guide to tensor products of modules with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Cofree Modules and Essential Extensions
Cofree modules via Hom from the ring, the adjunction producing enough injectives, essential extensions, and the existence and uniqueness of injective hulls.
Engineering↗
Direct Sums, Products and Split Sequences
Direct sums and direct products of modules, their universal properties, the splitting lemma and equivalent characterisations of split short exact sequences.
Engineering↗
Free and Projective Modules
Free modules and bases, projective modules and the lifting property, the equivalence with direct summands of free modules, and projective resolutions.
Engineering↗
Injective Modules and Dualization
Injective modules, the extension property, Baer's criterion, divisible groups, and the existence of enough injectives.
Engineering↗
Injective Modules over a Principal Ideal Domain
Injective modules over a PID characterised by divisibility, the classification of injective abelian groups, and the resulting short injective resolutions.
Engineering↗
Modules and Module Homomorphisms
Modules over a ring, submodules and quotients, module homomorphisms, exact sequences, the short five lemma and the snake lemma.
Engineering↗
Projective Modules over a Principal Ideal Domain
Structure of projective and finitely generated modules over a PID, the structure theorem, torsion and free parts, and the resulting short projective resolutions.
Engineering↗
The Hom Functor and Left Exactness
The Hom functor in both variables, its covariant and contravariant forms, left exactness, and the precise sense in which it fails to be exact.
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