Residue Ring Interface

Residue rings (currently a quotient ring modulo a principal ideal) are supported in AbstractAlgebra.jl, at least for Euclidean base rings. There is also partial support for residue rings of polynomial rings where the modulus has invertible leading coefficient.

In addition to the standard Ring interface, some additional functions are required to be present for residue rings.

Types and parents

AbstractAlgebra provides four abstract types for residue rings and their elements:

  • ResRing{T} is the abstract type for residue ring parent types
  • ResField{T} is the abstract type for residue rings known to be fields
  • ResElem{T} is the abstract type for types of elements of residue rings (residues)
  • ResFieldElem{T} is the abstract type for types of elements of residue fields

We have that ResRing{T} <: AbstractAlgebra.Ring and ResElem{T} <: AbstractAlgebra.RingElem.

Note that these abstract types are parameterised. The type T should usually be the type of elements of the base ring of the residue ring/field.

If the parent object for a residue ring has type MyResRing and residues in that ring have type MyRes then one would have:

  • MyResRing <: ResRing{BigInt}
  • MyRes <: ResElem{BigInt}

Residue rings should be made unique on the system by caching parent objects (unless an optional cache parameter is set to false). Residue rings should at least be distinguished based on their base ring and modulus (the principal ideal one is taking a quotient of the base ring by).

See src/generic/GenericTypes.jl for an example of how to implement such a cache (which usually makes use of a dictionary).

Required functionality for residue rings

In addition to the required functionality for the Ring interface the Residue Ring interface has the following required functions.

We suppose that R is a fictitious base ring, $m$ is an element of that ring, and that S is the residue ring (quotient ring) $R/(m)$ with parent object S of type MyResRing{T}. We also assume the residues $r \pmod{m}$ in the residue ring have type MyRes{T}, where T is the type of elements of the base ring.

Of course, in practice these types may not be parameterised, but we use parameterised types here to make the interface clearer.

Note that the type T must (transitively) belong to the abstract type RingElem.

Data type and parent object methods

modulus(S::MyResRing{T}) where T <: AbstractAlgebra.RingElem

Return the modulus of the given residue ring, i.e. if the residue ring $S$ was specified to be $R/(m)$, return $m$.

Basic manipulation of rings and elements

data(f::MyRes{T}) where T <: RingElem
lift(f::MyRes{T}) where T <: RingElem

Given a residue $r \pmod{m}$, represented as such, return $r$. In the special case where machine integers are used to represent the residue, data will return the machine integer, whereas lift will return a multiprecision integer. Otherwise lift falls back to data by default.