[Merged by Bors] - feat(ring_theory/fractional_ideal): pushforward of fractional ideals #2984
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Extend `submodule.map` to fractional ideals by showing that the pushforward is also fractional. For this, we need a slightly tweaked definition of fractional ideal: if we localize `R` at the submonoid `S`, the old definition required a nonzero `x : R` such that `xI ≤ R`, and the new definition requires `x ∈ S` instead. In the most common case, `S = non_zero_divisors R`, the results are exactly the same, and all operations are still the same. As a bonus, a proof that the fractional ideals don't depend on choice of localization map.
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Co-Authored-By: Johan Commelin <johan@commelin.net>
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…2984) Extend `submodule.map` to fractional ideals by showing that the pushforward is also fractional. For this, we need a slightly tweaked definition of fractional ideal: if we localize `R` at the submonoid `S`, the old definition required a nonzero `x : R` such that `xI ≤ R`, and the new definition requires `x ∈ S` instead. In the most common case, `S = non_zero_divisors R`, the results are exactly the same, and all operations are still the same. A practical use of these pushforwards is included: `canonical_equiv` states fractional ideals don't depend on choice of localization map. Co-authored-by: Vierkantor <Vierkantor@users.noreply.github.com>
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…eanprover-community#2984) Extend `submodule.map` to fractional ideals by showing that the pushforward is also fractional. For this, we need a slightly tweaked definition of fractional ideal: if we localize `R` at the submonoid `S`, the old definition required a nonzero `x : R` such that `xI ≤ R`, and the new definition requires `x ∈ S` instead. In the most common case, `S = non_zero_divisors R`, the results are exactly the same, and all operations are still the same. A practical use of these pushforwards is included: `canonical_equiv` states fractional ideals don't depend on choice of localization map. Co-authored-by: Vierkantor <Vierkantor@users.noreply.github.com>
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Extend
submodule.mapto fractional ideals by showing that the pushforward is also fractional.For this, we need a slightly tweaked definition of fractional ideal: if we localize
Rat the submonoidS, the old definition required a nonzerox : Rsuch thatxI ≤ R, and the new definition requiresx ∈ Sinstead. In the most common case,S = non_zero_divisors R, the results are exactly the same, and all operations are still the same.A practical use of these pushforwards is included:
canonical_equivstates fractional ideals don't depend on choice of localization map.