Theorems · Theorem · commutative algebra
Ideal.add_mem
∀ {α : Type u} [inst : Semiring α] (I : Ideal α) {a b : α}, a ∈ I → b ∈ I → a + b ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Defs
- Cited by
- 36 results in Mathlib
- Foundations
- Depth 15 from the axioms · 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.add_memproof · cited by 75
Cited by36
Results whose statement or proof uses this declaration.
- Ideal.radical_eq_sInfproof · cited by 21
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
- Ideal.mem_image_of_mem_map_of_surjectiveproof · cited by 6
- Ideal.mem_map_C_iffproof · cited by 6
- Ideal.ideal_prod_eqproof · cited by 4
- IsLocalRing.of_unique_max_idealproof · cited by 4
- Submodule.mem_ideal_smul_span_iff_exists_sumproof · cited by 3
- Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smulproof · cited by 3
- Submodule.mem_smul_span_singletonproof · cited by 2
- PowerSeries.exist_eq_span_eq_ncard_of_X_notMemproof · cited by 2
- Ideal.mul_eq_inf_of_coprimeproof · cited by 2
- ContinuousMap.idealOfSet_ofIdeal_eq_closureproof · cited by 2