Theorems · Theorem · commutative algebra
Ideal.mul_mem_left
∀ {α : Type u} [inst : Semiring α] (I : Ideal α) (a : α) {b : α}, b ∈ I → a * b ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Defs
- Cited by
- 107 results in Mathlib
- Foundations
- Depth 15 from the axioms, rests on 160 definitions · uses no axioms
- 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_memproof · cited by 204
Cited by107
Results whose statement or proof uses this declaration.
- Ideal.IsMaximal.isPrimeproof · cited by 53
- Ideal.radical_eq_sInfproof · cited by 21
- Ideal.mul_le_rightproof · cited by 20
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
- Ideal.mul_le_infproof · cited by 10
- Ideal.pow_mem_of_memproof · cited by 10
- Ideal.IsPrime.mul_mem_iff_mem_or_memproof · cited by 8
- Ideal.unit_mul_mem_iff_memproof · cited by 6
- Ideal.mem_image_of_mem_map_of_surjectiveproof · cited by 6
- Ideal.mem_map_C_iffproof · cited by 6
- Ideal.radical_infproof · cited by 5
- IsLocalization.mk'_mem_map_algebraMap_iffproof · cited by 5