feat(ring_theory/ideal/basic): define right ideals and show division rings are left and right Artinian #14399
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…rings are left and right Artinian
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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| /-- A right ideal is a left ideal of the opposite (semi)ring. -/ | ||
| @[simps] def to_ideal : ideal αᵐᵒᵖ := |
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I can see this being more useful as an equiv or even order_iso between ideal αᵐᵒᵖ and right_ideal α. In particular, having things like to_ideal.injective and to_ideal.map_bot seem valuable.
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In the same vein, ideal.to_right_ideal : ideal α \equivo right_ideal αᵐᵒᵖ would go in the other direction
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| protected lemma zero_mem : (0 : α) ∈ I := I.zero_mem | ||
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| protected lemma add_mem : a ∈ I → b ∈ I → a + b ∈ I := I.add_mem |
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We should be able to safely delete these lemmas since there should already be an add_submonoid_class instance.
Suggested change
| protected lemma zero_mem : (0 : α) ∈ I := I.zero_mem | |
| protected lemma add_mem : a ∈ I → b ∈ I → a + b ∈ I := I.add_mem |
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| zero_mem' := I.zero_mem, | ||
| add_mem' := λ a b ha hb, I.add_mem ha hb, |
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Suggested change
| zero_mem' := I.zero_mem, | |
| add_mem' := λ a b ha hb, I.add_mem ha hb, | |
| zero_mem' := zero_mem I, | |
| add_mem' := λ a b ha hb, add_mem ha hb, |
| intros h1, | ||
| rw eq_bot_iff, | ||
| intros r hr, | ||
| by_cases H : r = 0, {simpa}, |
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Suggested change
| by_cases H : r = 0, {simpa}, | |
| by_cases H : r = 0, { simpa }, |
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| lemma sum_mem {ι : Type*} {t : finset ι} {f : ι → α} : | ||
| (∀c∈t, f c ∈ I) → (∑ i in t, f i) ∈ I := submodule.sum_mem I |
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This is also provided by add_submonoid_class.
Suggested change
| lemma sum_mem {ι : Type*} {t : finset ι} {f : ι → α} : | |
| (∀c∈t, f c ∈ I) → (∑ i in t, f i) ∈ I := submodule.sum_mem I |
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