Theorems · Theorem · commutative algebra
Ideal.multiset_prod_eq_bot
∀ {R : Type u_2} [inst : CommSemiring R] [IsDomain R] {s : Multiset (Ideal R)}, s.prod = ⊥ ↔ ⊥ ∈ sA product of ideals in an integral domain is zero if and only if one of the terms is zero.
- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIsDomain
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Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Bot.botstatement · cited by 4,720
- Multisetstatement and proof · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Multiset.prodstatement · cited by 528
- Multiset.prod_eq_zero_iffproof · cited by 7
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