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
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
Axioms
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
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
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