Theorems · Theorem · commutative algebra
Ideal.mul_mem_right
∀ {α : Type u_1} {a : α} (b : α) [inst : Semiring α] (I : Ideal α) [I.IsTwoSided], a ∈ I → a * b ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Defs
- Cited by
- 71 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses no axioms
- Assumes
- SemiringIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Ideal.IsTwoSided.mul_mem_of_leftproof · cited by 2
Cited by71
Results whose statement or proof uses this declaration.
- Ideal.mul_topproof · cited by 42
- Ideal.radical_eq_sInfproof · cited by 21
- Ideal.mul_le_leftproof · cited by 19
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
- IsLocalization.map_underproof · cited by 12
- Ideal.mul_le_infproof · cited by 10
- Ideal.IsPrime.mul_mem_iff_mem_or_memproof · cited by 8
- Submodule.colon_univproof · cited by 7
- Ideal.mem_of_dvdproof · cited by 7
- Ideal.radical_infproof · cited by 5
- Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInfproof · cited by 5
- Ideal.sup_mul_eq_of_coprime_leftproof · cited by 4