Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets

Depends on

• Submodule of a module
• The quotient group $G/N$ and coset product $(gN)(hN)=ghN$
• Unital left and right modules over a ring; unqualified module means left module

Used by

• Associated primes of a cyclic quotient are colon primes Corollary
• Every module is a quotient of a free module Corollary
• M⊗_RR/I≅ M/IM naturally Corollary
• ℤ/nℤ is generated but not free as a ℤ-module for n>1 Counterexample
• Associated graded algebra of a filtered algebra Definition
• Composition series and length of a module Definition
• Module homomorphism and isomorphism, kernel, image and cokernel Definition
• Primary submodules and primary ideals Definition
• Regular Sequence On A Module Definition
• Subrepresentations, quotient representations, and intertwiners Definition
• Symmetric and exterior powers over an arbitrary field Definition
• The radical, socle, head, and Loewy series of a finite-dimensional module Definition
• A module-valued coend computed as a quotient of a direct sum Example
• The Prüfer p-group is Artinian but not Noetherian Example
• An irreducible submodule of a Noetherian module is primary Lemma
• Coprime cyclic quotients over a PID split by the Chinese remainder map Lemma
• Finite modules over Noetherian rings are Noetherian Lemma
• Primary submodules are exactly quotients with nilpotent zero divisors Lemma
• The quotient Lie-algebra bracket is well-defined Lemma
• Abelian groups and ℤ-modules have the same objects and morphisms Proposition
• A module-valued coend is the direct sum of the diagonal values modulo the dinaturality submodule Theorem
• Correspondence theorem for submodules of a quotient module Theorem
• Equivalent characterizations of injective modules Theorem
• Finite modules over Noetherian rings admit prime filtrations Theorem
• For every ring R, the category R-Mod is complete and cocomplete Theorem
• Invariant-factor decomposition of a finitely generated module over a PID Theorem
• Localisation commutes with quotient modules and arbitrary direct sums Theorem
• Second isomorphism theorem for modules Theorem
• Tensoring is right exact Theorem
• The quotient action is well defined and makes M/N a module Theorem
• Third isomorphism theorem for modules 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)