Theorems · Theorem · linear algebra

Submodule.finrank_quotient

∀ {R : Type u_2} {M : Type u} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [HasRankNullity.{u, u_2} R] [StrongRankCondition R] [Module.Finite R M] {S : Type u_1} [inst_6 : Ring S]
  [inst_7 : SMul R S] [inst_8 : Module S M] [inst_9 : IsScalarTower R S M] (N : Submodule S M),
  Module.finrank R (M ⧸ N) = Module.finrank R M - Module.finrank R ↥N

Rank-nullity theorem using finrank and subtraction.

Defined in

Mathlib.LinearAlgebra.Dimension.RankNullity

Cited by

1 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleHasRankNullityStrongRankConditionModule.FiniteRingSMulModuleIsScalarTower