Theorems · Theorem · commutative algebra
Submodule.range_fg_iff_ker_cofg
∀ {R : Type u_1} [inst : Ring R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type u_3}
[inst_3 : AddCommGroup N] [inst_4 : Module R N] {f : M →ₗ[R] N}, f.range.FG ↔ f.ker.CoFGThe range of a linear map is FG if and only if the kernel is CoFG.
- Defined in
- Mathlib.RingTheory.Finiteness.Cofinite
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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.symmproof · cited by 1,461
- LinearMap.rangestatement · cited by 893
- LinearMap.kerstatement and proof · cited by 848
- Submodule.FGstatement · cited by 230
- Submodule.CoFGstatement and proof · cited by 28
- LinearMap.quotKerEquivRangeproof · cited by 23
- Module.Finite.iff_fgproof · cited by 21
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.ker_coFG_iff_hasFiniteRangeproof · cited by 2
- Submodule.CoFG.kerproof · cited by 1