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
• Euler characteristic in a proper flat family is locally constant Corollary
• Exact finite tensor functors have projective right-module kernels Corollary
• Finite-dimensional coherent cohomology over a field Corollary
• Over a Noetherian ring the homomorphism module between two finitely generated modules is finitely generated Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A Deligne kernel need not be one external tensor factor Counterexample
• A projective generator need not be small Counterexample
• The left-to-right exact equivalence need not preserve the identity Counterexample
• ℤ/nℤ is generated but not free as a ℤ-module for n>1 Counterexample
• Finite graded projective modules Definition
• Finite morphisms of classical varieties Definition
• 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
• Perfect complexes over a ring and its graded version Definition
• Small projective generators and progenerators Definition
• Span, finite generation and generation degree of an FI-module 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
• Closed immersion from a quotient ring Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Matrix-unit rotation for a k–Mₙ(k) Morita pair Example
• The free-group functor F:Set toGrp and free-module functor R⁽⁻⁾:Set→ R-Mod Example
• The kernel end and coend distinguish the regular and co-regular bimodules 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
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Closed immersions are proper Lemma
• Determinant trick for Nakayama Lemma
• Double bar comparison for cyclic bimodule tensor products Lemma
• Elementary ideals are independent of the presentation Lemma
• Every generating set of a finitely generated module contains a finite generating subset Lemma
• Finite is affine and local on its target Lemma
• Finite modules over Noetherian rings are Noetherian Lemma
• Finite normalization commutes with principal localization Lemma
• Finite projective complex for proper flat coherent cohomology Lemma

…and 36 more results.

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)