[Merged by Bors] - [Merged by Bors] - feat(ring_theory/finiteness): prove that a surjective endomorphism of a finite module over a comm ring is injective #10944
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… a finite module over a comm ring is injective
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src/ring_theory/finiteness.lean
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| obtain ⟨y, rfl⟩ := f_surj a, | ||
| rw [← X_mul y], | ||
| exact submodule.smul_mem_smul (ideal.mem_span_singleton.mpr (dvd_refl _)) trivial, }, | ||
| haveI : is_scalar_tower R (polynomial R) M := ⟨λ x y z, by simp⟩, |
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I think it would be good to have this as a lemma after module_polynomial_ring_endo. Just to indicate it as a natural API piece. I would also move this haveI to directly after the letI that introduces the module structure.
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Yeah the issue is I'm not really sure how to state this properly as a lemma.
The type depends on the typeclases inferred, and this depends on f, I've pushed by best attempt but its quite unsatisfying as a lemma to me. Maybe its ok though.
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Thanks 🎉 bors merge |
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Using an approach of Vasconcelos, treating the module as a module over the polynomial ring, with action induced by the endomorphism.
This result was rescued from #1822.
I'm not sure of the best location for
module_polynomial_ring_endo/ whether we want more lemmata for it, I only needed the autognerated simp lemams