Theorems · Theorem · commutative algebra
IsIntegral.of_mem_of_fg
∀ {R : Type u_1} {B : Type u_3} [inst : CommRing R] [inst_1 : Ring B] [inst_2 : Algebra R B] (S : Subalgebra R B),
(Subalgebra.toSubmodule S).FG → ∀ x ∈ S, IsIntegral R xIf S is a sub-R-algebra of A and S is finitely-generated as an R-module,
then all elements of S are integral over R.
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Submodulestatement · cited by 7,192
- Subalgebrastatement and proof · cited by 1,353
- Module.Finiteproof · cited by 1,032
- OrderEmbeddingstatement · cited by 619
- IsIntegralstatement · cited by 427
- Subtype.val_injectiveproof · cited by 232
- Submodule.FGstatement and proof · cited by 230
- Subalgebra.toSubmodulestatement and proof · cited by 141
Cited by13
Results whose statement or proof uses this declaration.
- isIntegral_transproof · cited by 15
- IsIntegral.smulproof · cited by 13
- IsIntegral.powproof · cited by 12
- IsIntegral.negproof · cited by 4
- RingHom.IsIntegralElem.of_mem_closureproof · cited by 4
- RingHom.Finite.to_isIntegralproof · cited by 3
- IsIntegral.invproof · cited by 2
- IntermediateField.AdjoinSimple.norm_gen_eq_oneproof · cited by 1
- IntermediateField.AdjoinSimple.trace_gen_eq_zeroproof · cited by 1
- IsAlmostIntegral.isIntegral_of_nonZeroDivisors_le_comapproof · cited by 1
- mem_integralClosure_iff_mem_fgproof · cited by 1
- isIntegral_of_submodule_noetherianproof · cited by 0