Theorems · Theorem · commutative algebra
IsLocalization.coeSubmodule_le_coeSubmodule
∀ {R : Type u_3} [inst : CommRing R] {M : Submonoid R} {S : Type u_4} [inst_1 : CommRing S] [inst_2 : Algebra R S]
[IsLocalization M S],
M ≤ nonZeroDivisors R → ∀ {I J : Ideal R}, IsLocalization.coeSubmodule S I ≤ IsLocalization.coeSubmodule S J ↔ I ≤ J- Cited by
- 4 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Submonoidstatement and proof · cited by 3,086
- nonZeroDivisorsstatement and proof · cited by 895
- IsLocalizationstatement and proof · cited by 636
- IsLocalization.coeSubmodulestatement · cited by 30
- IsLocalization.injectiveproof · cited by 22
- Submodule.map_le_map_iff_of_injectiveproof · cited by 8
Cited by4
Results whose statement or proof uses this declaration.
- IsFractionRing.coeSubmodule_le_coeSubmoduleproof · cited by 3
- FractionalIdeal.coeIdeal_le_coeIdeal'proof · cited by 2
- IsLocalization.coeSubmodule_injectiveproof · cited by 1
- IsLocalization.coeSubmodule_strictMonoproof · cited by 1