Theorems · Theorem · commutative algebra
Ideal.finiteQuotientOfFreeOfNeBot
∀ {S : Type u_3} [inst : CommRing S] [IsDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] (I : Ideal S),
I ≠ ⊥ → Finite (S ⧸ I)A nonzero ideal over a free finite extension of ℤ has a finite quotient.
It can't be an instance because of the side condition I ≠ ⊥.
- Cited by
- 3 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.
Cites14
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
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Finitestatement · cited by 3,029
- HasQuotient.Quotientstatement · cited by 2,301
- IsDomainstatement and proof · cited by 2,196
- Module.Basisproof · cited by 1,477
- Module.Finitestatement and proof · cited by 1,032
- Module.Freestatement and proof · cited by 597
- Submodule.restrictScalarsproof · cited by 180
- Module.Free.ChooseBasisIndexproof · cited by 133
- Module.Free.chooseBasisproof · cited by 121
Cited by3
Results whose statement or proof uses this declaration.
- NumberField.HeightOneSpectrum.one_lt_absNormproof · cited by 3
- IsPrimitiveRoot.finite_quotient_span_sub_oneproof · cited by 1
- IsPrimitiveRoot.finite_quotient_toInteger_sub_oneproof · cited by 1