Theorems · Theorem · commutative algebra
Algebra.norm_eq_zero_iff
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] [IsDomain R] [IsDomain S]
[Module.Free R S] [Module.Finite R S] {x : S}, (Algebra.norm R) x = 0 ↔ x = 0- Defined in
- Mathlib.RingTheory.Norm.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Finset.sumproof · cited by 5,195
- MonoidHomstatement · cited by 3,629
- Finset.univproof · cited by 3,473
- IsDomainstatement and proof · cited by 2,196
- map_zeroproof · cited by 1,614
- Module.Basisproof · cited by 1,477
- LinearEquiv.symmproof · cited by 1,461
- Module.Finitestatement and proof · cited by 1,032
Cited by6
Results whose statement or proof uses this declaration.
- Ideal.absNorm_eq_zero_iffproof · cited by 4
- Algebra.norm_ne_zero_iffproof · cited by 4
- Algebra.norm_eq_zero_iff_of_basisproof · cited by 2
- FractionalIdeal.absNorm_span_singletonproof · cited by 1
- NumberField.mixedEmbedding.norm_eq_zero_iff'proof · cited by 1
- Algebra.norm_eq_zero_iff'proof · cited by 0