Alphabeta MathThe Field of Fractions and Localisation: Examples
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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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✓ 7 results · all verified · 6 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 1 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
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
- The Field of Fractions and Localisation
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
Frac(ℤ) is canonically isomorphic to ℚ
Example
ℤ[1/6] consists exactly of rationals a/6ⁿ and inverts precisely the primes 2 and 3
Example
ℤ₍ₚ₎ consists of rationals with denominator not divisible by p, has maximal ideal pℤ₍ₚ₎, and residue field Fₚ
Counterexample
If 0∈ S, localisation collapses: S⁻¹R is the zero ring
Counterexample
Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6
Counterexample
Localising at a zero divisor need not be injective: inverting 3 in ℤ/6 kills 2
Counterexample