Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Modules over a ring form an abelian category
Statement
For every ring , the category of left -modules is an abelian category.
Facts & Assumptions
Given: A ring .
Left -modules and their homomorphisms form a category (Left modules over a fixed ring and module homomorphisms form the large locally small category ).
The category is complete and cocomplete (For every ring R, the category R-Mod is complete and cocomplete).
Module kernels, images, and cokernels are the usual ones (Module homomorphism and isomorphism, kernel, image and cokernel).
The first isomorphism theorem for modules identifies the coimage with the image (First isomorphism theorem for modules: ).
Proof
The hom-set of module homomorphisms is an abelian group under pointwise addition, and [L2] gives finite products and coproducts, namely the usual direct sums. So is additive.
Every module homomorphism has the kernel and cokernel from [L3], and [L4] identifies with . Thus satisfies the defining AB1 and AB2 clauses of Abelian category, so it is abelian.
Depends on
- Abelian category
- Left modules over a fixed ring and module homomorphisms form the large locally small category $R\text{-}\mathbf{Mod}$
- Module homomorphism and isomorphism, kernel, image and cokernel
- First isomorphism theorem for modules: $M/\ker f\cong\operatorname{im}f$
- For every ring R, the category R-Mod is complete and cocomplete
Used by
- A module homomorphism factors as quotient by its kernel followed by inclusion of its image Example
- Localization of modules gives an exact functor between module categories Example
- Vector spaces over a field form an abelian category Example
- FALSE: every abelian category is equivalent to a module category False statement
Dependency tree · two levels
19 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
- Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)