Theorems · Theorem · commutative algebra

Ideal.ideal_prod_prime

∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (I : Ideal (R × S)),
  I.IsPrime ↔ (∃ p, p.IsPrime ∧ I = p.prod ⊤) ∨ ∃ p, p.IsPrime ∧ I = ⊤.prod p

Classification of prime ideals in product rings: the prime ideals of R × S are precisely the ideals of the form p × S or R × p, where p is a prime ideal of R or S.

Defined in

Mathlib.RingTheory.Ideal.Prod

Cited by

2 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiring