Theorems · Theorem · commutative algebra
IsNoetherian.injective_of_surjective_endomorphism
∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsNoetherian R M]
(f : M →ₗ[R] M), Function.Surjective ⇑f → Function.Injective ⇑fAny surjective endomorphism of a Noetherian module is injective.
- Defined in
- Mathlib.RingTheory.Noetherian.Orzech
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- Ringstatement and proof · cited by 7,463
- LinearEquiv.toLinearMapproof · cited by 1,171
- IsNoetherianstatement and proof · cited by 208
- LinearEquiv.injectiveproof · cited by 162
- LinearEquiv.reflproof · cited by 143
- IsNoetherian.injective_of_surjective_of_injectiveproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- IsNoetherian.bijective_of_surjective_endomorphismproof · cited by 0