Alphabeta MathFree Modules and Exact Sequences: Examples and Counterexamples
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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.
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✓ 10 results · all verified · 7 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 3 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
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
Example
The standard basis and a universal map from R³
Counterexample
ℤ/nℤ is generated but not free as a ℤ-module for n>1
Counterexample
A regular module with bases of sizes one and two
Example
ℤ/2ℤ is projective but not free over ℤ/6ℤ
Counterexample
0→ℤ →(×2) ℤ→ℤ/2ℤ→0 does not split
Example
0→ A→ A⊕ C→ C→0 is canonically split
Counterexample
Hom_ℤ(-,ℤ) need not preserve surjections on the right
False statement
Every short exact sequence of modules splits
False statement
Every projective module is free
False statement