Section
Algebra: Module Theory
Foundational definitions and theorems in module theory over rings.
Core idea
This section collects definitions and results on modules over rings: submodules, quotient modules, homomorphisms, exact sequences, free and projective modules, tensor products, and structure theorems over principal ideal domains.
Definitions
Basic Structures
Homomorphisms
Exact Sequences
Direct Constructions
Generation and Bases
- Cyclic module
- Finitely generated module
- Free module
- Basis (of a free module)
- Rank (of a free module)
Torsion
Annihilators
Simple and Semisimple Modules
Composition Series and Length
Chain Conditions
Projective, Injective, and Flat Modules
Tensor Products
Hom and Duality
Algebras
Graded and Filtered Structures
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Axioms
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Theorems
Isomorphism Theorems
- First isomorphism theorem (modules)
- Second isomorphism theorem (modules)
- Third isomorphism theorem (modules)
- Correspondence theorem (modules)
Structure Theorems
- Structure theorem for f.g. modules over PID
- Elementary divisor theorem
- Smith normal form theorem
- Rational canonical form theorem
- Jordan canonical form theorem
- Krull-Schmidt-Azumaya theorem
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Lemmas
- Splitting lemma
- Projective iff every s.e.s. ending in it splits
- Projective is direct summand of free
- Baer's criterion (injectivity)
- Tensor product preserves direct sums
- Tensor-Hom adjunction lemma
- Universal property of quotient modules
- Kernels are submodules
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Propositions
- Submodule criterion
- Kernel and image are submodules
- M/ker(f) ≅ im(f)
- Exactness via kernels and images
- Direct sum universal property
- Free module universal property
- Tensor product universal property
- Tensor commutes with direct limits/sums
- Hom turns sums into products
- Projective implies flat
- f.g. projective are locally free
- Semisimple iff every submodule is direct summand
- Artinian + Noetherian implies finite length
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