Rings, Subrings, Integral Domains and Fields

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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.
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✓ 15 results · all verified · 8 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 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Rings, Subrings, Integral Domains and Fields

3 · Items

Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides

Commutative ring

In any ring 0 · a = a · 0 = 0, (-a)b = a(-b) = -(ab), (-a)(-b) = ab, (-1)a = -a and a(b - c) = ab - ac

Integer multiples in a ring: (m + n)a = ma + na, m(a + b) = ma + mb, (ma)b = m(ab) = a(mb) and (ma)(nb) = (mn)(ab) for all m, n ∈ ℤ and a, b ∈ R

The units of a ring are the invertible elements of its multiplicative monoid, and R^× is a group under multiplication; 0 ∈ R^× only in the zero ring

Zero divisor, and integral domain: a commutative ring with 1 ≠ 0 and no zero divisors

Cancellation characterises domains: in a commutative ring with 1 ≠ 0, the implication ab = ac and a ≠ 0 imply b = c holds if and only if the ring has no zero divisors

Division ring: a ring with 1 ≠ 0 in which every nonzero element is a unit

Every field is a commutative ring with 1 ≠ 0; it is an integral domain, and it is a commutative division ring

Every commutative division ring is a field, so "field" and "commutative division ring" name the same structures and the published definition and the ring-theoretic one agree

The quaternions ℍ: real quadruples with componentwise addition and an explicit multiplication formula matching the table on 1, i, j, k

ℍ is a division ring that is not commutative, hence not a field: q⁻¹ = q̄ / N(q) for q ≠ 0, while ij = k and ji = -k

Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication

The order presentation and the positive-cone presentation of an ordered ring determine each other: P = { x : 0 < x } satisfies trichotomy and closure, and a < b :⟺ b - a ∈ P recovers the order

Every ordered field is an ordered ring, and its order is the one its positive cone induces

Subring: a subset containing 1_R and closed under addition, additive inverses and multiplication

Subring criterion: S ⊆ R is a subring if and only if 1_R ∈ S and a - b ∈ S and ab ∈ S for all a, b ∈ S; and an intersection of subrings is a subring

Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations

Ring homomorphism: additive, multiplicative, and required to send 1 to 1

A ring homomorphism satisfies f(0) = 0, f(-a) = -f(a) and f(ma) = m f(a) for m ∈ ℤ, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms

A ring homomorphism between fields is a field homomorphism in the published sense, and every such map is injective

The product ring R × S with componentwise operations, its identity (1_R, 1_S) and its units R^× × S^×

The ring R^X of all functions from a set X into a ring, with pointwise operations

In a field, the additive multiple n · 1_F is the canonical natural ι(n): the additive power of the group-power definition and the canonical natural are the same function, both being the unique one given by the recursion ι(0) = 0_F, ι(σ(n)) = ι(n) + 1_F

The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise

The characteristic of a ring is the additive order of 1_R, with 0 recording infinite order; n · 1_R = 0 holds exactly when char(R) ∣ n; and in an integral domain every nonzero element has the same additive order as 1_R

A finite sum in a commutative monoid indexed by an arbitrary finite set

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule