Theorems · Theorem · commutative algebra
Ideal.mem_bot
∀ {R : Type u} [inst : Semiring R] {x : R}, x ∈ ⊥ ↔ x = 0- Defined in
- Mathlib.RingTheory.Ideal.Lattice
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 15 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.
- Semiringstatement and proof · cited by 13,802
- Idealstatement · cited by 4,748
- Bot.botstatement · cited by 4,720
- Submodule.mem_botproof · cited by 55
Cited by21
Results whose statement or proof uses this declaration.
- Ideal.absNorm_eq_zero_iffproof · cited by 4
- mem_pNilradicalproof · cited by 4
- IsNilpotent.isUnit_quotient_mk_iffproof · cited by 3
- Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRingproof · cited by 1
- Algebra.FormallyEtale.of_isSeparable_auxproof · cited by 1
- Algebra.FormallyUnramified.ext_of_iInfproof · cited by 1
- isStronglyTranscendental_mk_of_mem_minimalPrimesproof · cited by 1
- MvPolynomial.zeroLocus_botproof · cited by 1
- Ideal.comap_ne_bot_of_root_memproof · cited by 1
- pow_expChar_pow_inj_of_pNilradical_eq_botproof · cited by 1
- Ideal.injective_quotient_le_comap_mapproof · cited by 1
- ValuativeRel.vle_zero_iffproof · cited by 1