Modules

Builds on

Required prerequisites, all the way down: 28

You will learn to

• Work with module homomorphisms, kernels, cokernels and exact sequences.
• Prove the structure theorem for finitely generated modules over a PID, and deduce the classification of finitely generated abelian groups and the Jordan normal form.
• Compute tensor products of modules and describe their effect on exact sequences.

Definition

• A module over a ring R is an abelian group with an action of R satisfying the distributive and associative laws.
• A sequence A → B → C is exact at B if the image of the first map equals the kernel of the second.
• A module P is projective if Hom(P, −) carries surjections to surjections.

Theorem

• A finitely generated module over a PID R is isomorphic to a direct sum R/(d₁) ⊕ ⋯ ⊕ R/(dₖ) with d₁ | d₂ | ⋯ | dₖ.
• Free modules are projective, and over a PID every submodule of a free module is free.
• M ⊗_R − is right exact but in general not exact.

Common misconceptions

• Expecting modules to behave like vector spaces: ℤ/2ℤ is a ℤ-module with no basis.
• Assuming tensoring preserves injectivity: multiplication by 2 on ℤ is injective, but becomes zero after tensoring with ℤ/2ℤ.

Leads to

Optional discoveries

• Adjunctions The tensor–hom adjunction Hom(M ⊗ N, P) ≅ Hom(M, Hom(N, P)), natural in all three variables, is a basic example of an adjunction.

Learning pathways

• Graduate preparation
• Algebra and number theory

Further reading

• David S. Dummit and Richard M. Foote, Abstract Algebra
• Serge Lang, Algebra