Theorems · Theorem · commutative algebra

Ideal.mem_minimalPrimes_sup

∀ {R : Type u_2} [inst : CommRing R] {p I J : Ideal R} [p.IsPrime],
  I ≤ p →
    Ideal.map (Ideal.Quotient.mk I) p ∈ (Ideal.map (Ideal.Quotient.mk I) J).minimalPrimes → p ∈ (I ⊔ J).minimalPrimes

Defined in: Mathlib.RingTheory.Ideal.MinimalPrime.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing Ideal.IsPrime

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Cites16

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
CommRingstatement and proof · cited by 17,173
RingHomstatement · cited by 10,189
Idealstatement and proof · cited by 4,748
HasQuotient.Quotientstatement · cited by 2,301
Ideal.IsPrimestatement and proof · cited by 827
Ideal.mapstatement and proof · cited by 692
Ideal.Quotient.mkstatement and proof · cited by 610
sup_of_le_rightproof · cited by 143
Ideal.minimalPrimesstatement and proof · cited by 74
sup_le_iffproof · cited by 58
Ideal.map_monoproof · cited by 29

Cited by1

Results whose statement or proof uses this declaration.

Ideal.height_le_height_add_spanFinrank_of_leproof · cited by 2