[Merged by Bors] - feat(Algebra): generalize Basis.smul
#9382
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Basis.smul
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Add various `LinearMap.CompatibleSMul` instances that ultimately lead to generalization of `Basis.smul` to allow a noncommutative base ring. The key observations that allows the generalization are `IsScalarTower.smulHomClass` and `isScalarTower_of_injective`. Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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Generalize the conditions of the tower law [FiniteDimensional.finrank_mul_finrank'](https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Tower.html#FiniteDimensional.finrank_mul_finrank') in FieldTheory/Tower from `[CommRing F] [Algebra F K]` to `[Ring F] [Module F K]`, and remove the `[Module.Finite F K]` and `[Module.Finite K A]` conditions. The generalized version applies to situations when we have a tower C/B/A where the A-module structure on C is induced from the B-module structure via a RingHom from A to B, and the A-module structure on B is induced by the same RingHom. In particular, it applies when the A-module structure on B and the B-module structure on C come from two RingHoms, and the A-module structure on C comes from the composition of them, regardless of whether A and B are commutative or not. As prerequisites, I also generalized lemmas originally introduced by @kckennylau in [mathlib3#3355](https://github.com/leanprover-community/mathlib/pull/3355/files) to prove the tower law. They were split into three PRs: + LinearAlgebra/Span #9380: add `span_eq_closure` and `closure_induction` which say that `Submodule.span R s` is generated by `R • s` as an AddSubmonoid. I feel that the existing `span_induction` should be replaced by `closure_induction` as the latter is stronger, and allow us to remove the commutativity condition in `span_smul_of_span_eq_top` in Algebra/Tower. + Algebra/Tower #9382: switching from CommSemiring/Algebra to Semiring/Module here requires proving the curious lemma `IsScalarTower.isLinearMap` which states that for a tower of modules A/S/R, any S-linear map from S to A is also R-linear. If the map is injective, we can deduce that S/S/R also form a tower. (By `ringHomEquivModuleIsScalarTower` in #9381, there is therefore a canonical RingHom from R to S.) + Lemmas for free modules over rings including `finrank_mul_finrank'` are moved from FieldTheory/Tower to LinearAlgebra/Dimension/Free [Zulip](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Puzzle.3A.20noncommutative.20RingHom.20as.20typeclasses/near/407347711) Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Add various
LinearMap.CompatibleSMulinstances that ultimately lead to generalization ofBasis.smulto allow a noncommutative base ring. The key observations that allows the generalization areIsScalarTower.smulHomClassandisScalarTower_of_injective.