Theorems · Theorem · commutative algebra
IsFractional.div_of_nonzero
∀ {R₁ : Type u_3} [inst : CommRing R₁] {K : Type u_4} [inst_1 : Field K] [inst_2 : Algebra R₁ K] [IsFractionRing R₁ K]
[IsDomain R₁] {I J : Submodule R₁ K},
IsFractional (nonZeroDivisors R₁) I →
IsFractional (nonZeroDivisors R₁) J → J ≠ 0 → IsFractional (nonZeroDivisors R₁) (I / J)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- CommSemiringproof · cited by 10,911
- Fieldstatement and proof · cited by 7,404
- Submodulestatement and proof · cited by 7,192
- Algebra.algebraMapproof · cited by 4,706
- mul_commproof · cited by 2,262
- IsDomainstatement and proof · cited by 2,196
- map_zeroproof · cited by 1,614
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
Cited by1
Results whose statement or proof uses this declaration.
- FractionalIdeal.isFractional_div_of_ne_zeroproof · cited by 5