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.
The integers form a commutative ring
Statement
with the operations of Arithmetic on the integers is a commutative ring with multiplicative identity, in which every element has the additive inverse .
Facts & Assumptions
Given: with the operations of Arithmetic on the integers.
Addition on is commutative and associative, with zero as a two-sided identity (Addition is commutative, Addition is associative, Left identity for addition).
Multiplication on is commutative and associative, distributes over addition, and has the usual zero and identity laws (Multiplication is commutative, Multiplication is associative, Distributivity and the successor law for multiplication, Zero and one under multiplication).
The integer operations are independent of representatives (Integer addition and negation are well defined, Integer multiplication is well defined).
Proof
By [L3], each axiom may be verified on arbitrary fixed representatives .
Associativity of : and both equal .
Commutativity of : .
Additive identity: .
Additive inverses: , since .
Commutativity of : swapping sends to , the same pair.
Multiplicative identity: .
Associativity of : expanding, both and equal .
Distributivity: .
Steps 1.2–1.9 verify all axioms: is a commutative ring with identity and additive inverses.
Depends on
- The integers as equivalence classes of pairs of naturals
- Arithmetic on the integers
- Integer addition and negation are well defined
- Integer multiplication is well defined
- Addition is commutative
- Addition is associative
- Left identity for addition
- Multiplication is commutative
- Multiplication is associative
- Distributivity and the successor law for multiplication
- Zero and one under multiplication
Used by
- A rational root of xᵏ = m is an integer: if k ≥ 1, m ∈ ℤ, x ∈ ℚ and xᵏ is the image of m, then x is the image of an integer Corollary
- Division with remainder for any nonzero divisor: for a ∈ ℤ and b ≠ 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < |b| Corollary
- Every nonzero integer n is u ∏_i<r pᵢ with u ∈ {1,-1} and every pᵢ prime; u and r are determined by n, and the list is determined up to a permutation Corollary
- For an integer p > 1: p is prime if and only if, for all integers a and b, p ∣ ab implies p ∣ a or p ∣ b Corollary
- If a prime p divides a finite product ∏_i<n aᵢ of integers then p ∣ aᵢ for some i < n; at n = 0 the product is 1 and the hypothesis cannot hold Corollary
- If d = gcd(a,b) is nonzero then a/d and b/d are coprime Corollary
- The extended Euclidean algorithm: the same descent produces integers x, y with ax + by = gcd(a,b), so Bézout coefficients are computed and not merely shown to exist Corollary
- 2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ Counterexample
- 2ℤ is closed under addition, negation and multiplication and is not a subring of ℤ, because it does not contain 1 Counterexample
- 6 ∣ 4 · 9 while 6 ∤ 4 and 6 ∤ 9: dividing a product does not force dividing a factor, and the coprimality hypothesis is what fails Counterexample
- A nonempty subset of a group closed under the operation need not be a subgroup: the nonnegative integers inside (ℤ, +) Counterexample
- Division with a degree-small remainder can fail over ℤ when the leading coefficient of the divisor is not a unit Counterexample
- If 1 were admitted as a prime, uniqueness would fail: 6 = 2 · 3 = 1 · 2 · 3 = 1 · 1 · 2 · 3, lists of different lengths that no permutation matches Counterexample
- In the multiplicative monoid H = {1, 4, 7, 10, …} of positive integers one more than a multiple of 3, the element 100 has two genuinely different factorisations into irreducibles, 4 · 25 and 10 · 10 Counterexample
- The common divisors of (0,0) are all of ℤ and have no greatest element in the order of ℤ, so gcd(0,0) cannot be defined as a maximum and is fixed by convention Counterexample
- The doubling endomorphism of (ℤ,+) has trivial kernel but is not surjective Counterexample
- The map n ↦ (n,0) from ℤ to ℤ × ℤ preserves addition and multiplication and does not preserve 1, so the clause f(1) = 1 is not redundant Counterexample
- The zero ideal of ℤ is prime but not maximal Counterexample
- ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2 Counterexample
- Common multiple, and the least common multiple lcm(a,b), taken to be 0 when a = 0 or b = 0 Definition
- Content and primitive integer polynomials Definition
- Divisibility in ℤ: d ∣ a when a = dq for some integer q Definition
- Inversions, inversion number, the sign sgn(σ)=(-1)^inv(σ), and even and odd permutations Definition
- Prime and composite integers: p is prime when p > 1 and its only positive divisors are 1 and p Definition
- The absolute value |a| of an integer Definition
- The integer-valued Möbius function μ_P of a locally finite poset Definition
- The number-theoretic Möbius function μ(n) from prime factorisation Definition
- The p-adic valuation vₚ(a) of a nonzero integer: the greatest k ∈ ℕ with pᵏ ∣ a Definition
- The rationals as equivalence classes of pairs of integers Definition
- (ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1} Example
- ⟨ a,b∣ aba⁻¹b⁻¹⟩≅(ℤ,+)×(ℤ,+) Example
- 12ℤ + 18ℤ = 6ℤ and 12ℤ ∩ 18ℤ = 36ℤ, the arithmetic of gcd and lcm read off the subgroups of (ℤ,+) Example
- 2 is prime, and it is the only even prime: every even integer n > 2 is composite Example
- 2ℤ is an ideal of ℤ but is not a subring under the library's unital convention Example
- 360 = 2³ · 3² · 5 and 84 = 2² · 3 · 7, with gcd(360,84) = 12 and lcm(360,84) = 2520 read off the exponents Example
- A product of two rings with 1 ≠ 0 always has zero divisors: (1,0)(0,1) = (0,0) in ℤ × ℤ, so a product of integral domains is never an integral domain Example
- An integer matrix of determinant 2 is invertible over ℚ but not over ℤ Example
- ax + by = c has an integer solution exactly when gcd(a,b) ∣ c: 6x + 15y = 9 is solvable and 6x + 15y = 7 is not Example
- Bézout coefficients are not unique: 1071 · (-3) + 462 · 7 = 21 and 1071 · 19 + 462 · (-44) = 21, and for nonzero a, b every solution has the form (x₀ + t b/d, y₀ - t a/d) Example
- Consecutive Fibonacci numbers are coprime, and for every m ∈ ℕ the Euclidean algorithm on (Fₘ₊₃, Fₘ₊₂) takes exactly m+1 divisions, with quotient 1 in the first m of them and quotient 2 in the last Example
…and 77 more results.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 24 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- T. Tao, Analysis I, 3rd ed., §4.1 (standard reference, not scraped)
- E. Landau, Foundations of Analysis (standard reference, not scraped)
- Integer — construction from pairs of naturals (Wikipedia) (standard reference, not scraped)