Theorems · Theorem · linear algebra

rank_tensorProduct

∀ {R : Type u} {S : Type u'} {M : Type v} {M' : Type v'} [inst : Semiring R] [inst_1 : CommSemiring S]
  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [StrongRankCondition R]
  [StrongRankCondition S] [inst_7 : Module S M] [inst_8 : Module S M'] [Module.Free S M'] [inst_10 : Algebra S R]
  [inst_11 : IsScalarTower S R M] [Module.Free R M],
  Module.rank R (TensorProduct S M M') =
    Cardinal.lift.{v', v} (Module.rank R M) * Cardinal.lift.{v, v'} (Module.rank S M')

The S-rank of M ⊗[R] M' is (Module.rank S M).lift * (Module.rank R M').lift.

Defined in

Mathlib.LinearAlgebra.Dimension.Constructions

Cited by

4 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringCommSemiringAddCommMonoidAddCommMonoidModuleStrongRankConditionStrongRankConditionModuleModuleModule.FreeAlgebraIsScalarTowerModule.Free