[Merged by Bors] - feat(algebra/direct_sum_graded): A 0 is a ring, A i is an A 0-module, and direct_sum.of A 0 is a ring_hom
#6851
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… 0` is a ring_hom
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A 0 is a ring and direct_sum.of A 0 is a ring_homA 0 is a ring, A i is an A 0-module, and direct_sum.of A 0 is a ring_hom
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Co-authored-by: Adam Topaz <adamtopaz@users.noreply.github.com>
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…odule, and `direct_sum.of A 0` is a ring_hom (#6851) In a graded ring, grade 0 is itself a ring, and grade `i` (and therefore, the overall direct_sum) is a module over elements of grade 0. This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR. The main results here are `direct_sum.grade_zero.comm_ring`, `direct_sum.grade_zero.semimodule`, and `direct_sum.of_zero_ring_hom`. Note that we have no way to let the user provide their own ring structure on `A 0`, as `[Π i, add_comm_monoid (A i)] [semiring (A 0)]` provides `add_comm_monoid (A 0)` twice.
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…odule, and `direct_sum.of A 0` is a ring_hom (#6851) In a graded ring, grade 0 is itself a ring, and grade `i` (and therefore, the overall direct_sum) is a module over elements of grade 0. This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR. The main results here are `direct_sum.grade_zero.comm_ring`, `direct_sum.grade_zero.semimodule`, and `direct_sum.of_zero_ring_hom`. Note that we have no way to let the user provide their own ring structure on `A 0`, as `[Π i, add_comm_monoid (A i)] [semiring (A 0)]` provides `add_comm_monoid (A 0)` twice.
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bors r+ Thanks for cleaning up the conflicts, @urkud |
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…odule, and `direct_sum.of A 0` is a ring_hom (#6851) In a graded ring, grade 0 is itself a ring, and grade `i` (and therefore, the overall direct_sum) is a module over elements of grade 0. This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR. The main results here are `direct_sum.grade_zero.comm_ring`, `direct_sum.grade_zero.semimodule`, and `direct_sum.of_zero_ring_hom`. Note that we have no way to let the user provide their own ring structure on `A 0`, as `[Π i, add_comm_monoid (A i)] [semiring (A 0)]` provides `add_comm_monoid (A 0)` twice. Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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…odule, and `direct_sum.of A 0` is a ring_hom (#6851) In a graded ring, grade 0 is itself a ring, and grade `i` (and therefore, the overall direct_sum) is a module over elements of grade 0. This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR. The main results here are `direct_sum.grade_zero.comm_ring`, `direct_sum.grade_zero.semimodule`, and `direct_sum.of_zero_ring_hom`. Note that we have no way to let the user provide their own ring structure on `A 0`, as `[Π i, add_comm_monoid (A i)] [semiring (A 0)]` provides `add_comm_monoid (A 0)` twice. Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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…odule, and `direct_sum.of A 0` is a ring_hom (#6851) In a graded ring, grade 0 is itself a ring, and grade `i` (and therefore, the overall direct_sum) is a module over elements of grade 0. This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR. The main results here are `direct_sum.grade_zero.comm_ring`, `direct_sum.grade_zero.semimodule`, and `direct_sum.of_zero_ring_hom`. Note that we have no way to let the user provide their own ring structure on `A 0`, as `[Π i, add_comm_monoid (A i)] [semiring (A 0)]` provides `add_comm_monoid (A 0)` twice. Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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A 0 is a ring, A i is an A 0-module, and direct_sum.of A 0 is a ring_homA 0 is a ring, A i is an A 0-module, and direct_sum.of A 0 is a ring_hom
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In a graded ring, grade 0 is itself a ring, and grade
i(and therefore, the overall direct_sum) is a module over elements of grade 0.This stops just short of the structure necessary to make a graded ring a graded algebra over elements of grade 0; this requires some extra assumptions and probably a new typeclass, so is best left to its own PR.
The main results here are
direct_sum.grade_zero.comm_ring,direct_sum.grade_zero.semimodule, anddirect_sum.of_zero_ring_hom.Note that we have no way to let the user provide their own ring structure on
A 0, as[Π i, add_comm_monoid (A i)] [semiring (A 0)]providesadd_comm_monoid (A 0)twice.This is a rework of #6837 that avoids having to prove anything interesting.