Theorems · Theorem · commutative algebra

Submodule.finite_quotient_smul

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (I : Ideal R)
  {N : Submodule R M} [Finite (R ⧸ I)] [Finite (M ⧸ N)], N.FG → Finite (M ⧸ I • N)

Let N be a finite index f.g. R-submodule, and I be a finite index ideal. Then I • N also has finite index.

Defined in

Mathlib.RingTheory.Ideal.Quotient.Index

Cited by

2 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleFiniteFinite