Alphabeta Math
LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-27
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 spectrum map respects composition and identities

Statement

Let R→φA→ψB be ring homomorphisms of commutative rings. For a prime ideal q∈Spec⁡(A) define Spec⁡(φ)(q)=φ−1(q). Then this gives a well-defined map Spec⁡(A)→Spec⁡(R), and one has Spec⁡(idR)=idSpec⁡(R) and Spec⁡(ψ∘φ)=Spec⁡(φ)∘Spec⁡(ψ).

Facts & Assumptions

Given: Commutative rings R,A,B and ring homomorphisms φ:R→A and ψ:A→B.

[L1]

Prime ideals are proper ideals that absorb factors of a product (Prime ideals and maximal ideals in a commutative ring).

Proof

technique · direct
1.1L1givenalgebra

If q∈Spec⁡(A), then φ−1(q) is a proper ideal of R: otherwise 1∈φ−1(q), so 1=φ(1)∈q, contradicting the properness in [L1]. If ab∈φ−1(q), then φ(a)φ(b)=φ(ab)∈q, so [L1] gives a∈φ−1(q) or b∈φ−1(q). Therefore φ−1(q) is prime.

2.1step 1.1givenalgebra

For p∈Spec⁡(R), one has idR−1(p)=p. For r∈Spec⁡(B), one has (ψ∘φ)−1(r)=φ−1(ψ−1(r)), so contraction along the composite is the composite of the contractions.

3.1step 1.1step 2.1∎

The inverse-image construction is therefore well-defined on prime spectra and respects identities and composition.

Depends on

Dependency tree · 2 levels

Sources