Theorems · Theorem · commutative algebra
IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors
∀ {R : Type u_1} [inst : CommRing R] {K : Type u_5} [inst_1 : CommRing K] [inst_2 : Algebra R K] [IsFractionRing R K]
[Nontrivial R] {x : R}, x ∈ nonZeroDivisors R → (algebraMap R K) x ≠ 0- Cited by
- 10 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- Submonoidstatement · cited by 3,086
- Nontrivialstatement and proof · cited by 2,416
- le_rflproof · cited by 1,558
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisorsproof · cited by 3
Cited by10
Results whose statement or proof uses this declaration.
- IsDiscreteValuationRing.exists_lift_of_le_oneproof · cited by 2
- FractionalIdeal.exists_ne_zero_mem_isIntegerproof · cited by 2
- IsFractionRing.mk'_mk_eq_divproof · cited by 2
- IsFractionRing.isInteger_of_isUnit_denproof · cited by 2
- IsFractionRing.exists_reduced_fractionproof · cited by 2
- ClassGroup.mk0_integralRepproof · cited by 1
- exists_integer_of_is_root_of_monicproof · cited by 0
- FractionalIdeal.isUnit_numproof · cited by 0
- IsFractionRing.mk'_eq_one_iff_eqproof · cited by 0
- ValuationRing.le_totalproof · cited by 0