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[Merged by Bors] - feat(ring_theory/noetherian): add two lemmas on products of prime ideals #5013
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faenuccio
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Merge branch 'master' into prime_prod
Vierkantor
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Nov 16, 2020
src/ring_theory/noetherian.lean
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| obtain ⟨x, y, hx, hy, h_xy⟩ : ∃ (x y : R), x ∉ M ∧ y ∉ M ∧ x * y ∈ M, | ||
| { rw [ideal.is_prime, not_and] at h_not_prM, | ||
| specialize h_not_prM h_not_topM, | ||
| rw not_forall at h_not_prM, | ||
| cases h_not_prM with x hx, | ||
| rw not_forall at hx, | ||
| cases hx with y hxy, | ||
| use [x, y], finish }, |
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Golfing:
obtain ⟨x, y, hxy, hx, hy⟩ : ∃ (x y : R), x * y ∈ M ∧ x ∉ M ∧ y ∉ M,
{ rw [ideal.is_prime, not_and] at h_not_prM,
specialize h_not_prM h_not_topM,
push_neg at h_not_prM,
assumption },
src/ring_theory/noetherian.lean
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| begin | ||
| have hA_nont : nontrivial A, | ||
| apply is_integral_domain.to_nontrivial (integral_domain.to_is_integral_domain A), | ||
| by_cases h_topI : I < ⊤, |
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I prefer to deal with the easy case first:
by_cases h_topI : I = ⊤,
{ rcases h_topI with rfl,
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ (p : ideal A), p ≠ 0 ∧ p.is_prime,
{ apply ring.not_is_field_iff_exists_prime.mp h_fA },
let p : prime_spectrum A := ⟨p_id, h_pp⟩,
use [({p} : multiset (prime_spectrum A)), le_top],
rwa [multiset.map_singleton, multiset.singleton_eq_singleton, multiset.prod_singleton] },
...Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
Vierkantor
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faenuccio
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I have implemented all requested changes with two minor modifications:
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Vierkantor
approved these changes
Nov 17, 2020
Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
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Great, I have implemented these two changes. |
jcommelin
reviewed
Nov 17, 2020
Co-authored-by: Johan Commelin <johan@commelin.net>
Co-authored-by: Johan Commelin <johan@commelin.net>
Co-authored-by: Johan Commelin <johan@commelin.net>
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Nov 17, 2020
…als (#5013) Add two lemmas saying that in a noetherian ring (resp. _integral domain)_ every (_nonzero_) ideal contains a (_nonzero_) product of prime ideals. Co-authored-by: faenuccio <65080144+faenuccio@users.noreply.github.com>
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Pull request successfully merged into master. Build succeeded: |
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Add two lemmas saying that in a noetherian ring (resp. integral domain) every (nonzero) ideal contains a (nonzero) product of prime ideals.