Theorems · Theorem · commutative algebra
IsNoetherian.subsingleton_of_injective
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [IsNoetherian R M]
{P : Type u_5} [inst_4 : AddCommMonoid P] [inst_5 : Module R P] {f : P × M →ₗ[R] M},
Function.Injective ⇑f → Subsingleton PIf P × N embeds into N for some nontrivial module P, then N cannot be a Noetherian
module. Lemma 1.36 of Chapter 1 in [lam_1999].
- Defined in
- Mathlib.RingTheory.Noetherian.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Finsuppproof · cited by 5,255
- LinearMap.rangeproof · cited by 893
- Submodule.mapproof · cited by 614
- IsNoetherianstatement and proof · cited by 208
- Finsupp.lsingleproof · cited by 75
- by_contraproof · cited by 60
Cited by1
Results whose statement or proof uses this declaration.
- StrongRankCondition.of_isNoetherianproof · cited by 0