Alphabeta Math

Rings, Domains and Fields: 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.
  • 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.

✓ 10 results · all verified · 2 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

ℤ is a commutative ring and an ordered ring, the published construction being an instance of the general definitions

Example

ℤ is an integral domain of characteristic 0 whose group of units is {1,-1}, so it is not a field: 2 is nonzero and not invertible

Example

ℚ and ℝ are fields, hence commutative rings, integral domains and ordered rings, all of characteristic 0

Example

The Cauchy sequences of rationals form a commutative ring that is not an integral domain: two eventually-constant sequences with disjoint supports multiply to zero

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

The zero ring {0}, in which 1 = 0: a commutative ring of characteristic 1 that is not a domain, not a division ring and not a field

Example

ℤ sits inside ℚ as a subring that is not a subfield, so the inverse-closure clause of the subfield definition is doing work

Counterexample

2ℤ is closed under addition, negation and multiplication and is not a subring of ℤ, because it does not contain 1

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

False statement

FALSE: In every commutative ring, each nonzero element is either a unit or a zero divisor

Sources