Moduletheoretic

Module theory is a branch of abstract algebra that studies modules over rings. A module M over a ring R with identity is an abelian group equipped with an R-action, satisfying compatibility with ring operations and distributivity. When R is a field, R-modules are precisely vector spaces. The theory generalizes linear algebra to non-field scalars and to noncommutative or more general rings.

Submodules, quotient modules, and module homomorphisms form the basic language. A submodule N ≤ M is a subgroup closed under the R-action. Homomorphisms f: M → N preserve addition and scalar multiplication, and exact sequences encode how modules fit together. Short exact sequences 0 → A → B → C → 0 measure extensions of A by C and play a central role in homological perspectives.

Common classes include free modules (isomorphic to R^n), projective modules (lifting property for surjections), injective modules (extension property), and flat modules (preserving tensor products). Simple modules have no nontrivial submodules; semisimple modules decompose into direct sums of simple modules. Finitely generated modules over rings with appropriate conditions are central to many structural questions.

A fundamental result is the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M is finitely generated, then M ≅ R^n ⊕ ⊕i R/(d_i) with nonzero d_i and d_i dividing d_{i+1}. The d_i are invariants determined by M and provide a classification in this setting.

Homological methods introduce functors such as Hom_R(-, M), tensor products, and derived functors Ext^i_R(-, M) and Tor_i^R(-, M), arising from projective resolutions. These tools reveal extensions, relations, and depth properties. The subject interacts with algebraic geometry, representation theory, and number theory, and continues to develop through localization, categorical approaches, and advanced homological techniques.