The integers form a commutative ring

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
• Connected coverings of the circle are classified by the subgroups nℤ for n≥0 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 connected covering of the circle is regular 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
• The fundamental theorem of finitely generated abelian groups from PID modules Corollary
• The index of a cycle is locally constant off its trace and vanishes far from it Corollary
• The index of a full-rank subgroup of ℤⁿ is the absolute determinant of a generating matrix Corollary
• The winding number is the increment of a continuous argument divided by 2π 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
• Complex chains, their traces, and cycles Definition
• Content and primitive integer polynomials Definition
• Cyclic shifts of an integer word and its periodic partial-sum function 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
• Lattice paths, step sets and step words 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 circle as S¹=ℝ/ℤ with basepoint [0] 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

…and 114 more results.

Dependency tree · two levels

16 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

• 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)