Generated submodule, cyclic and finitely generated modules, module basis and free module

Depends on

• Submodule of a module
• The one-step submodule criterion; intersections and sums of submodules are submodules
• The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups

Used by

• Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators Corollary
• Over a Noetherian ring the homomorphism module between two finitely generated modules is finitely generated Corollary
• ℤ/nℤ is generated but not free as a ℤ-module for n>1 Counterexample
• Finitely presented modules and finitely presented algebras Definition
• Invariant factors and elementary divisors of a finitely generated module over a PID Definition
• Minimal Free Resolution Over A Local Ring Definition
• Noetherian commutative rings and modules Definition
• Noetherian modules: every submodule is finitely generated Definition
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras Definition
• The free module on a set and its standard basis Definition
• The free-group functor F:Set toGrp and free-module functor R⁽⁻⁾:Set→ R-Mod Example
• FALSE: every finitely generated module over a domain is a direct sum of cyclic modules False statement
• FALSE: every torsion-free module over a PID is free False statement
• A finite module has the union of its generator-cyclic supports Lemma
• Determinant trick for Nakayama Lemma
• Every generating set of a finitely generated module contains a finite generating subset Lemma
• Finite modules over Noetherian rings are Noetherian Lemma
• Local Koszul Acyclicity Inductive Converse Lemma
• Local Koszul H One Detects First Regularity Failure Lemma
• The localised Hom map is an isomorphism for finite free sources Lemma
• The submodule generated by a subset consists of the finite R-linear combinations of that subset Lemma
• Abelian groups and ℤ-modules have the same objects and morphisms Proposition
• For finite-dimensional V, V_T is finitely generated and torsion, with annihilator generated by the minimal polynomial Proposition
• A finitely generated semisimple module is a finite direct sum of simple modules Theorem
• A module-valued coend is the direct sum of the diagonal values modulo the dinaturality submodule Theorem
• Finite generation, ACC, and maximal-condition characterizations of Noetherian modules Theorem
• Finitely generated modules over a left Noetherian ring are Noetherian Theorem
• For every ring R, the category R-Mod is complete and cocomplete Theorem
• Integrality and finite-module characterizations for one element Theorem
• Localisation of Hom for finite and finitely presented modules Theorem
• Noetherian and Artinian conditions are each exact in short exact sequences Theorem
• Over a Noetherian ring a module is Noetherian exactly when it is finitely generated, exactly when it is finitely presented Theorem
• Support of a tensor product of finite modules is the intersection of the supports Theorem
• The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem
• Under Choice, every finitely generated nonzero module has a maximal proper submodule Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)