Alphabeta MathFree Modules, Exact Sequences, Projective and Injective Modules
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How 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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 20 results · all verified · 11 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
The direct sum of an indexed family of modules
Theorem
Universal property of a direct sum of modules
Definition
The free module on a set and its standard basis
Theorem
Universal property of the free module on a set
Corollary
Every module is a quotient of a free module
Definition
Invariant basis number and the rank of a free module
Theorem
Every nonzero commutative ring has invariant basis number for finite bases
Definition
Exact sequences and short exact sequences of modules
Lemma
The endpoints of a short exact sequence encode injectivity and surjectivity
Definition
Split short exact sequences, sections, and retractions
Theorem
The splitting lemma for short exact sequences of modules
Definition
The abelian group Hom_R(M,N) and maps induced by pre- and postcomposition
Theorem
Covariant and contravariant Hom are left exact
Lemma
The injective and surjective Four Lemmas
Theorem
The Five Lemma for modules
Theorem
The Snake Lemma for modules
Definition
Projective modules and the lifting property
Theorem
Free modules are projective, with the exact choice boundary
Theorem
Equivalent characterizations of projective modules
Theorem
Direct sums of projectives are projective, and module categories have enough projectives
Definition
Injective modules and the extension property
Theorem
Equivalent characterizations of injective modules
Theorem
Baer's criterion for injective modules
Definition
Divisible modules over an integral domain
Theorem
Over a PID, injective modules are exactly divisible modules
Lemma
Every abelian group embeds in a divisible abelian group
Theorem
Products of injective modules are injective, with the exact choice boundary
Lemma
Coinduction sends injective abelian groups to injective modules
Theorem