Module VIII·Article I·~1 min read
Modules over Rings
Tensor Algebra and Sylow Theorems (Advanced)
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Generalization of Vector Spaces
A vector space is a module over a field. If we replace the field with a ring, we obtain a module—a more general structure in which “scalar multiplication” might not possess inverses.
Left R-module M: an abelian group (M, +) with an operation r·m (r ∈ R, m ∈ M), satisfying the axioms of distributivity and associativity.
Examples: ℤ-modules are simply abelian groups; K-modules = K-vector spaces; ideals of a ring R are R-modules.
Free and Projective Modules
A free module: has a basis (like a vector space). $R^n = R \times ... \times R$.
Not every module is free: ℤ/2ℤ is a ℤ-module that does not have a basis in the usual sense.
Projective module: a direct summand of a free module. An analog of a “direct complement”—an algebraic-geometric concept (Stiefel–Whitney bundles are projective modules).
Structure Theorem for Finitely Generated Modules over a PID
If R is a PID (principal ideal domain: ℤ, K[x]), M is a finitely generated R-module, then:
$ M \cong R^r \oplus R/(d_1) \oplus R/(d_2) \oplus ... \oplus R/(d_k), \text{ where } d_1 \mid d_2 \mid ... \mid d_k. $
Applications: classification of finite abelian groups (R=ℤ); Jordan normal form (R=K[x], M is the operator's space). Both classifications are special cases of a single theorem!
Exact Sequences
A sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ is exact if the image of each map equals the kernel of the next.
A short exact sequence: $M'$ is a submodule of $M$, $M'' = M/M'$. It splits if $M \cong M' \oplus M''$.
Long exact sequences appear in homological algebra and topology (Mayer–Vietoris sequence, exact sequence of a pair).
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