Theorems · Theorem · commutative algebra
Subalgebra.rank_sup_eq_rank_left_mul_rank_of_free
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (A B : Subalgebra R S)
[Module.Free R ↥A] [Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)],
Module.rank R ↥(A ⊔ B) = Module.rank R ↥A * Module.rank ↥A ↥(Algebra.adjoin ↥A ↑B)- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Rank
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- SetLike.coestatement and proof · cited by 8,199
- IsScalarTowerproof · cited by 3,896
- mul_oneproof · cited by 3,885
- Cardinalstatement and proof · cited by 2,598
- Nontrivialproof · cited by 2,416
- Subalgebrastatement and proof · cited by 1,353
- Module.Freestatement and proof · cited by 597
- Algebra.adjoinstatement and proof · cited by 535
- Module.rankstatement and proof · cited by 496
- subsingleton_or_nontrivialproof · cited by 161
Cited by2
Results whose statement or proof uses this declaration.
- Subalgebra.finrank_sup_eq_finrank_left_mul_finrank_of_freeproof · cited by 2
- Subalgebra.rank_sup_eq_rank_right_mul_rank_of_freeproof · cited by 0