Theorems · Theorem · commutative algebra
Ideal.mul_mem_mul
∀ {R : Type u} [inst : Semiring R] {I J : Ideal R} {r s : R}, r ∈ I → s ∈ J → r * s ∈ I * J- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 29 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Idealstatement and proof · cited by 4,748
- Submodule.smul_mem_smulproof · cited by 32
Cited by29
Results whose statement or proof uses this declaration.
- Ideal.mul_topproof · cited by 42
- Ideal.map_mulproof · cited by 9
- PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_idealproof · cited by 4
- Ideal.sup_mul_eq_of_coprime_leftproof · cited by 4
- Ideal.IsTwoSided.mul_oneproof · cited by 3
- Ideal.cotangentIdeal_squareproof · cited by 3
- Ideal.sup_mul_eq_of_coprime_rightproof · cited by 3
- Ideal.Cotangent.smul_eq_zero_of_memproof · cited by 2
- Ideal.radical_mulproof · cited by 2
- Ideal.mul_eq_inf_of_coprimeproof · cited by 2
- Ideal.mul_mem_mul_revproof · cited by 2
- PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopenproof · cited by 2