Submodule of a module

Depends on

• Unital left and right modules over a ring; unqualified module means left module
• Subgroup

Used by

• Modular Specht modules need not be simple, and form heads can vanish Counterexample
• Artinian modules by the descending chain condition Definition
• Composition series and length of a module Definition
• Generated submodule, cyclic and finitely generated modules, module basis and free module Definition
• Noetherian commutative rings and modules Definition
• Noetherian modules: every submodule is finitely generated Definition
• Quantum Verma modules M(λ), highest weight vectors, and the simple highest weight quotient L(λ) Definition
• Quotient module M/N with scalar multiplication on additive cosets Definition
• Simple module: a nonzero module with no proper nonzero submodule Definition
• Span, finite generation and generation degree of an FI-module Definition
• The category FI and FI-modules over a commutative ring Definition
• The coinvariant functors Φₐ and the transition maps T Definition
• The direct sum of an indexed family of modules Definition
• The socle as the sum of all simple submodules Definition
• Weight of an FI-module over a field of characteristic zero Definition
• A nonsplit simple has Hom-pairing diagonal two Example
• Left ideals are exactly the submodules of the regular left module _RR Example
• All initial forms define the tangent cone Lemma
• Coinvariants detect vanishing of subquotients of bounded weight, and padded irreducibles stabilise Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• Finite modules over Noetherian rings are Noetherian Lemma
• Finite-variable polynomial algebras over fields are Noetherian by finite generators Lemma
• Kernels, images, quotients, and matrix coefficients of comodules Lemma
• Submodules of finite modules over a Noetherian ring are finite by induction Lemma
• The one-step submodule criterion; intersections and sums of submodules are submodules Lemma
• The scheme-theoretic linear span of the tangent cone Lemma
• Abelian groups and ℤ-modules have the same objects and morphisms Proposition
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• Integrable bounded-above quantum modules with finite-dimensional weight spaces are direct sums of simple highest weight modules Proposition
• The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel Proposition
• Torsion elements and p-primary elements form submodules over a domain Proposition
• 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
• A module-valued coend is the direct sum of the diagonal values modulo the dinaturality submodule Theorem
• For every ring R, the category R-Mod is complete and cocomplete Theorem
• Hopf–Maschke theorem by integral averaging Theorem
• The quotient action is well defined and makes M/N a module Theorem
• Under Choice, every finitely generated nonzero module has a maximal proper submodule Theorem

Dependency tree · two levels

7 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)