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 $R \overset{φ}{\to} A \overset{ψ}{\to} B$ be ring homomorphisms of commutative rings. For a prime ideal $q \in Spec (A)$ define $Spec (φ) (q) = φ^{- 1} (q)$ . Then this gives a well-defined map $Spec (A) \to Spec (R)$ , and one has $Spec ({i d}_{R}) = {i d}_{Spec (R)}$ and $Spec (ψ \circ φ) = Spec (φ) \circ Spec (ψ)$ .

Facts & Assumptions

Given: Commutative rings $R, A, B$ and ring homomorphisms $φ : R \to A$ and $ψ : A \to B$ .

[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 $q \in Spec (A)$ , then $φ^{- 1} (q)$ is a proper ideal of $R$ : otherwise $1 \in φ^{- 1} (q)$ , so $1 = φ (1) \in q$ , contradicting the properness in [L1]. If $a b \in φ^{- 1} (q)$ , then $φ (a) φ (b) = φ (a b) \in q$ , so [L1] gives $a \in φ^{- 1} (q)$ or $b \in φ^{- 1} (q)$ . Therefore $φ^{- 1} (q)$ is prime.

2.1step 1.1givenalgebra

For $p \in Spec (R)$ , one has ${i d}_{R}^{- 1} (p) = p$ . For $r \in Spec (B)$ , one has $(ψ \circ φ)^{- 1} (r) = φ^{- 1} (ψ^{- 1} (r))$ , 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

Used by

A ring map induces a contraction map on prime spectra Theorem

Dependency tree · two levels

11 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

The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)