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docs(linear_algebra/invariant_basis_number): Drop a TODO (#14973)
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This TODO was fixed some time ago by @riccardobrasca, reference the relevant instance in the docstring.
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urkud committed Jun 26, 2022
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14 changes: 5 additions & 9 deletions src/linear_algebra/invariant_basis_number.lean
Expand Up @@ -261,15 +261,11 @@ end
section
local attribute [instance] ideal.quotient.field

-- TODO: in fact, any nontrivial commutative ring satisfies the strong rank condition.
-- To see this, consider `f : (fin m → R) →ₗ[R] (fin n → R)`,
-- and consider the subring `A` of `R` generated by the matrix entries.
-- That subring is noetherian, and `f` induces a new linear map `f' : (fin m → A) →ₗ[R] (fin n → A)`
-- which is injective if `f` is.
-- Since we've already established the strong rank condition for noetherian rings,
-- this gives the result.

/-- Nontrivial commutative rings have the invariant basis number property. -/
/-- Nontrivial commutative rings have the invariant basis number property.
In fact, any nontrivial commutative ring satisfies the strong rank condition, see
`comm_ring_strong_rank_condition`. We prove this instance separately to avoid dependency on
`linear_algebra.charpoly.basic`. -/
@[priority 100]
instance invariant_basis_number_of_nontrivial_of_comm_ring {R : Type u} [comm_ring R]
[nontrivial R] : invariant_basis_number R :=
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