Theorems · Definition · commutative algebra

IsNoetherian.equivPUnitOfProdInjective

{R : Type u_1} →
  {M : Type u_2} →
    {N : Type w} →
      [inst : Ring R] →
        [inst_1 : AddCommGroup M] →
          [inst_2 : Module R M] →
            [inst_3 : AddCommGroup N] →
              [inst_4 : Module R N] →
                [IsNoetherian R M] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → N ≃ₗ[R] PUnit.{w + 1}

If M ⊕ N embeds into M, for M Noetherian over R, then N is trivial.

Defined in

Mathlib.RingTheory.Noetherian.Orzech

Cited by

2 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModuleIsNoetherian