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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.
  • 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.

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Primes of a localization avoid the denominator set

Statement

Let R be a commutative ring, let S⊆R be a multiplicative subset, and let λ:R→S−1R be the localization map. Contraction sends each prime ideal of S−1R to a prime ideal of R disjoint from S, and extension sends each prime ideal of R disjoint from S back to a prime ideal of S−1R. These two operations are inverse and preserve strict inclusion.

Facts & Assumptions

Given: A commutative ring R, a multiplicative subset S⊆R, and the localization map λ:R→S−1R.

[L1]

Ideals of S−1R correspond to S-saturated ideals of R, and primes correspond exactly to the prime ideals disjoint from S (Ideals of S−1R correspond to S-saturated ideals of R, and prime ideals correspond to primes disjoint from S).

Proof

technique · direct
1.1L1

The prime-ideal part of [L1] says exactly that if q∈Spec⁡(S−1R), then λ−1(q) is a prime ideal of R disjoint from S, and if p∈Spec⁡(R) with p∩S=∅, then S−1p is a prime ideal of S−1R.

1.2L1

The same statement [L1] asserts that these assignments are inverse inclusion-preserving bijections. In particular they preserve strict inclusion.

2.1step 1.1step 1.2∎

Therefore primes of the localization are exactly the primes of R that avoid the denominator set.

Depends on

Dependency tree · 2 levels

Sources