Theorems · Theorem · commutative algebra
Ideal.IsMaximal.mul_mem_pow
∀ {R : Type u} [inst : CommSemiring R] (I : Ideal R) [I.IsMaximal] {a b : R} {n : ℕ}, a * b ∈ I ^ n → a ∈ I ∨ b ∈ I ^ nSee also Ideal.IsPrime.mul_mem_pow for prime ideal in Dedekind domain.
- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIdeal.IsMaximal
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- one_mulproof · cited by 2,841
- mul_assocproof · cited by 1,667
- Ideal.IsMaximalstatement and proof · cited by 452
- add_mulproof · cited by 363
- AddMemClass.add_memproof · cited by 229
- Ideal.mul_mem_leftproof · cited by 107
- Ideal.mul_mem_rightproof · cited by 71
- Ideal.IsMaximal.exists_inv_powproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Ideal.IsPrime.mul_mem_powproof · cited by 1
- IsLocalization.AtPrime.under_maximalIdeal_powproof · cited by 1
- Ideal.IsMaximal.mem_pow_mulproof · cited by 0