Theorems · Theorem · commutative algebra
IsArtinian.surjective_of_injective_endomorphism
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [IsArtinian R M]
(f : M →ₗ[R] M), Function.Injective ⇑f → Function.Surjective ⇑fAny injective endomorphism of an Artinian module is surjective.
- Defined in
- Mathlib.RingTheory.Artinian.Module
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- Top.topproof · cited by 9,680
- Submoduleproof · cited by 7,192
- LinearMap.rangeproof · cited by 893
- Submodule.mapproof · cited by 614
- pow_succproof · cited by 374
- Ne.lt_topproof · cited by 161
Cited by4
Results whose statement or proof uses this declaration.
- IsArtinianRing.isUnit_iff_isRightRegularproof · cited by 3
- IsArtinian.subsingleton_of_injectiveproof · cited by 1
- RingTheory.Sequence.eq_nil_of_isRegular_on_artinianproof · cited by 0
- IsArtinian.bijective_of_injective_endomorphismproof · cited by 0