Alphabeta MathTensor Products of Modules — Examples
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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.
✓ 11 results · all verified · 2 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
- 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
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Tensor Products of Modules
Linear Algebra
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
False statement
False: ℤ/m⊗_ℤℤ/n is nonzero for all positive m,n
Example
ℚ⊗_ℤℤ/n=0 for every positive n
Example
(R/I)⊗_R(R/J)≅ R/(I+J) for ideals of a commutative ring
Example
S⊗_RR[x]≅ S[x] as S-algebras
Example
For a field extension K/F, one has K⊗_FMₙ(F)≅ Mₙ(K) as K-algebras
Example
For a field extension K/F, one has K⊗_FFⁿ≅ Kⁿ
Example
ℂ⊗_ℝℂ≅ℂ×ℂ as ℝ-algebras
Example
Tensoring the injection k[x] →(· x) k[x] with k[x]/(x) gives the zero map
False statement
False: m⊗ n=0 implies m=0 or n=0
False statement
False: every element of M⊗_RN is an elementary tensor
False statement