[Merged by Bors] - feat(RingTheory/FiniteType): generalize some results to non-commutative rings #6681
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…ve rings (#6681) I was hoping to use this in combination with #6680 to show the `TensorAlgebra` is finitely generated, where I needed to generalize `FiniteType.equiv`; but it turns out that the `FiniteType` instance on `MonoidAlgebra` also isn't generalized! The summary here is: * Move `Algebra.adjoin_algebraMap` from `Mathlib/RingTheory/Adjoin/Tower.lean` to `Mathlib/RingTheory/Adjoin/Basic.lean` and golf the proof to oblivion * Provide an alternative statement of `adjoin_union_eq_adjoin_adjoin`, `adjoin_algebraMap_image_union_eq_adjoin_adjoin`, which works in non-commutative rings, and use it along with a new `adjoin_top` lemma to prove `Algebra.fg_trans'` more generally. * Introduce a new `S` variable throughout, with the convention that `R` and `S` are commutative, `A` and `B` remain not-necessarily-commutative, and `A/S/R` is a tower of algebras. * Apply some zero-effort generalizations to semirings.
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I was hoping to use this in combination with #6680 to show the
TensorAlgebrais finitely generated, where I needed to generalizeFiniteType.equiv; but it turns out that theFiniteTypeinstance onMonoidAlgebraalso isn't generalized!The summary here is:
Algebra.adjoin_algebraMapfromMathlib/RingTheory/Adjoin/Tower.leantoMathlib/RingTheory/Adjoin/Basic.leanand golf the proof to oblivionadjoin_union_eq_adjoin_adjoin,adjoin_algebraMap_image_union_eq_adjoin_adjoin, which works in non-commutative rings, and use it along with a newadjoin_toplemma to proveAlgebra.fg_trans'more generally.Svariable throughout, with the convention thatRandSare commutative,AandBremain not-necessarily-commutative, andA/S/Ris a tower of algebras.