Theorems · Theorem · commutative algebra
Ideal.span_singleton_le_iff_mem
∀ {α : Type u} [inst : Semiring α] (I : Ideal α) {x : α}, Ideal.span {x} ≤ I ↔ x ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Span
- Cited by
- 29 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
- Assumes
- Semiring
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.
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Idealstatement and proof · cited by 4,748
- Ideal.spanstatement · cited by 948
Cited by29
Results whose statement or proof uses this declaration.
- Ideal.spanNorm_singletonproof · cited by 5
- Ideal.absNorm_eq_zero_iffproof · cited by 4
- Module.End.IsSemisimple.of_mem_adjoin_pairproof · cited by 3
- Ideal.finite_minimalPrimes_of_isNoetherianRingproof · cited by 3
- ClassGroup.mk_eq_one_of_coe_idealproof · cited by 2
- Ideal.eq_span_singleton_of_height_eq_oneproof · cited by 2
- IsDedekindDomain.exists_sup_span_eqproof · cited by 2
- ringKrullDim_succ_le_of_surjectiveproof · cited by 2
- FractionalIdeal.not_inv_le_one_of_ne_botproof · cited by 2
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- Ideal.isOka_isPrincipalproof · cited by 1
- Valuation.ideal_isPrincipalproof · cited by 1