Noetherian modules: every submodule is finitely generated

Depends on

• Submodule of a module
• Generated submodule, cyclic and finitely generated modules, module basis and free module

Used by

• A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
• Over a Noetherian ring the homomorphism module between two finitely generated modules is finitely generated Corollary
• Finite graded Aₘ-modules, internal shifts and the vertex projectives Definition
• Left and right Noetherian rings Definition
• Regular points of locally Noetherian schemes Definition
• The Prüfer p-group is Artinian but not Noetherian Example
• ℤ is Noetherian but not Artinian as a module over itself Example
• A field has only the zero ideal and itself, hence is Noetherian Lemma
• All initial forms define the tangent cone Lemma
• auslander buchsbaum syzygy projective dimension Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Finite-variable polynomial algebras over fields are Noetherian by finite generators Lemma
• Local Koszul Acyclicity Inductive Converse Lemma
• Local Koszul H One Detects First Regularity Failure Lemma
• Polynomial rings over normal domains are normal Lemma
• Submodules of finite modules over a Noetherian ring are finite by induction Lemma
• The graded horseshoe lemma for finite graded projective resolutions Lemma
• The scheme-theoretic linear span of the tangent cone Lemma
• Conventions for this development and where dependent choice and Zorn's lemma are used Remark
• A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member Theorem
• A module-finite algebra over a Noetherian ring is a Noetherian ring, and so is every ring between the two Theorem
• Coherent higher direct images under proper morphisms Theorem
• Coherent sheaves on a locally Noetherian scheme Theorem
• Finite generation, ACC, and maximal-condition characterizations of Noetherian 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
• Regular equals smooth over a perfect field Theorem

Dependency tree · two levels

6 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

• Keith Conrad, Noetherian Modules, Sections 1-2 (standard reference, not scraped)