Theorems · Theorem · ring theory
Algebra.self_mem_adjoin_singleton
∀ (R : Type uR) {A : Type uA} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (x : A), x ∈ R[x]- Cited by
- 37 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement · cited by 535
- Set.mem_singleton_iffproof · cited by 172
- Algebra.subset_adjoinproof · cited by 109
Cited by37
Results whose statement or proof uses this declaration.
- isCyclotomicExtension_singleton_iff_eq_adjoinproof · cited by 3
- RatFunc.algEquivOfTranscendental_algebraMapproof · cited by 3
- IsArtinianRing.isUnit_of_isIntegral_of_nonZeroDivisorproof · cited by 2
- LocalSubring.exists_valuationRing_of_isMaxproof · cited by 2
- MulChar.apply_mem_algebraAdjoin_of_pow_eq_oneproof · cited by 2
- isIntegral_of_isIntegralElem_of_monic_of_natDegree_ltproof · cited by 1
- LocalSubring.mem_of_isMax_of_isIntegralproof · cited by 1
- eq_of_powMul_faithfulproof · cited by 1
- IsAlmostIntegral.isIntegral_of_nonZeroDivisors_le_comapproof · cited by 1
- Polynomial.Bivariate.aeval_aeval_eq_aeval_algEquivAdjoinstatement and proof · cited by 1
- Polynomial.algEquivOfTranscendental_symm_aevalstatement and proof · cited by 1