[Merged by Bors] - feat(Algebra/Colimit): the directed system of finitely generated submodules#20264
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[Merged by Bors] - feat(Algebra/Colimit): the directed system of finitely generated submodules#20264alreadydone wants to merge 2 commits intomasterfrom
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PR summary 4d36c01dcdImport changes for modified filesNo significant changes to the import graph Import changes for all files
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jcommelin
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…odules (#20264) We show that every module is the direct limit of its finitely generated submodules. As a consequence of this and the fact that tensor products preserves colimits, we show that if `M` and `P` are arbitrary modules and `N` is a finitely generated submodule of a module `P`, then two elements of `N ⊗ M` have the same image in `P ⊗ M` if and only if they already have the same image in `N' ⊗ M` for some finitely generated submodule `N' ≥ N`. This is the theorem `Submodule.FG.exists_rTensor_fg_inclusion_eq`.
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We show that every module is the direct limit of its finitely generated submodules.
As a consequence of this and the fact that tensor products preserves colimits, we show that if
MandPare arbitrary modules andNis a finitely generated submodule of a moduleP, then two elements ofN ⊗ Mhave the same image inP ⊗ Mif and only if they already have the same image inN' ⊗ Mfor some finitely generated submoduleN' ≥ N. This is the theoremSubmodule.FG.exists_rTensor_fg_inclusion_eq.