Document ideal interface (#2113)

Author: emikelsons

docs/make.jl

@@ -90,6 +90,7 @@ makedocs(
                  "field_interface.md",
                  "fraction_interface.md",
                  "module_interface.md",
+                 "ideal_interface.md",
                  "matrix_interface.md",
                  "map_interface.md",
                  "rand.md",

docs/src/ideal_interface.md

@@ -0,0 +1,107 @@
+```@meta
+CurrentModule = AbstractAlgebra
+DocTestSetup = AbstractAlgebra.doctestsetup()
+```
+
+# Ideal Interface
+
+AbstractAlgebra.jl generic code makes use of a standardised set of functions which it
+expects to be implemented by anyone implementing ideals for AbstractAlgebra rings. 
+Here we document this interface. All libraries which want to make use of the generic
+capabilities of AbstractAlgebra.jl must supply all of the required functionality for their ideals.
+There are already many helper methods in AbstractAlgebra.jl for the methods mentioned below.
+
+In addition to the required functions, there are also optional functions which can be
+provided for certain types of ideals e.g., for ideals of polynomial rings. If implemented,
+these allow the generic code to provide additional functionality for those ideals, or in
+some cases, to select more efficient algorithms.
+
+## Types and parents
+
+New ideal types should come with the following type information:
+
+```julia
+ideal_type(::Type{NewRing}) = NewIdealType 
+base_ring_type(::Type{NewIdeal}) = NewRingType
+parent_type(::Type{NewIdeal{T}}) = DefaultIdealSet{T}
+```
+
+However, new implementations of ideals needn't necessarily supply new types and could just extend
+the existing functionality for new rings as AbstractAlgebra.jl provides a generic ideal type
+based on Julia arrays which is implemented in `src/generic/Ideal.jl`. For information 
+about implementing new rings, see the [Ring interface](@ref "Ring Interface").
+
+## Required functionality for ideals
+
+In the following, we list all the functions that are required to be provided for ideals
+in AbstractAlgebra.jl or by external libraries wanting to use AbstractAlgebra.jl.
+
+We give this interface for fictitious type `NewIdeal` and `Ring` or `NewRing` for the type of the base ring
+object `R`, and `RingElem` for the type of the elements of the ring.
+We assume that the function
+
+```julia
+ideal(R::Ring, xs::Vector{U})
+```
+
+with `U === elem_type(Ring)` and `xs` a list of generators,
+is implemented by anyone implementing ideals for AbstractAlgebra rings. 
+Additionally, the following constructors are already implemented generically:
+
+```julia
+ideal(R::Ring, x::U)
+ideal(xs::Vector{U}) = ideal(parent(xs[1]), xs)
+ideal(x::U) = ideal(parent(x), x)
+*(x::RingElem, R::Ring)
+*(R::Ring, x::RingElem)
+```
+
+An implementation of an Ideal subtype should also provide the
+following methods:
+
+```julia
+base_ring(I::NewIdeal)
+```
+```julia
+gen(I::NewIdeal, k::Int)
+```
+```julia
+gens(I::NewIdeal)
+```
+```julia
+ngens(I::NewIdeal)
+```
+
+## Optional functionality for ideals
+
+Some functionality is difficult or impossible to implement for all ideals.
+If it is provided, additional functionality or performance may become available. Here
+is a list of all functions that are considered optional and can't be relied on by
+generic functions in the AbstractAlgebra Ideal interface.
+
+It may be that no algorithm, or no efficient algorithm is known to implement these
+functions. As these functions are optional, they do not need to exist. Julia will
+already inform the user that the function has not been implemented if it is called but
+doesn't exist.
+
+```julia
+in(v::RingElem, I::NewIdeal)
+```
+```julia
+issubset(I::NewIdeal, J::NewIdeal)
+```
+```julia
+iszero(I::NewIdeal)
+```
+```julia
++(I::T, J::T) where {T <: NewIdeal}
+```
+```julia
+*(I::T, J::T) where {T <: NewIdeal}
+```
+```julia
+intersect(I::T, J::T) where {T <: NewIdeal}
+```
+```julia
+==(I::T, J::T) where {T <: NewIdeal}
+```