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

Depends on

• Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
• Group and abelian group

Used by

• (S,R)-bimodules and commuting left and right scalar actions Definition
• A semisimple ring as a ring whose left regular module is semisimple Definition
• An R-linear action of G on a left R-module, and a G-module over R Definition
• Annihilators, torsion elements and the torsion subset of a module Definition
• Balanced maps from a right module and a left module, and bilinear maps over a commutative ring Definition
• Divisible modules over an integral domain Definition
• Left and right Artinian rings Definition
• Left and right Noetherian rings Definition
• Localisation of a module at a multiplicative subset Definition
• Module homomorphism and isomorphism, kernel, image and cokernel Definition
• Noetherian commutative rings and modules Definition
• Quotient module M/N with scalar multiplication on additive cosets Definition
• Simple module: a nonzero module with no proper nonzero submodule Definition
• Submodule of a module Definition
• Support of a module Definition
• The direct sum of an indexed family of modules Definition
• The F[x]-module V_T of an endomorphism Definition
• The submodule IM generated by products of elements of an ideal I with elements of a module M Definition
• Left ideals are exactly the submodules of the regular left module _RR Example
• The free-abelian-group monad sends a set to its finite formal integer combinations Example
• The residue classes ℤ/n as a ℤ-module Example
• Coextension of scalars Hom_R(S,M) carries its canonical left S-module structure Lemma
• In a module, 0_Rm=0_M, r0_M=0_M, (-r)m=-(rm) and r(-m)=-(rm) Lemma
• Module finiteness is transitive along a tower of algebras Lemma
• Over a commutative ring the homomorphism group Hom_R(M,N) is an R-module Lemma
• The one-step submodule criterion; intersections and sums of submodules are submodules 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
• Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod Proposition
• Lie representations as actions before enveloping 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
• Additive functors from a ring to abelian groups are left modules Theorem
• End_R(_R R)≅ Rᵒᵖ Theorem
• For a commutative ring R, R-linear G-actions are exactly the compatible left R[G]-module structures Theorem
• For a unital ring R, the free-R-module monad on sets has left R-modules as its Eilenberg–Moore algebras Theorem
• For every ring R, the category R-Mod is complete and cocomplete Theorem
• Lie representations are U(g)-modules Theorem
• The quotient action is well defined and makes M/N a module Theorem

Dependency tree · two levels

8 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)