Theorems · Theorem · commutative algebra

IsNoetherian.subsingleton_of_prod_injective

∀ {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 → Subsingleton N

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

Defined in

Mathlib.RingTheory.Noetherian.Orzech

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModuleIsNoetherian