Residue rings
Nemo allows the creation of residue rings of the form $R/(a)$ for an element $a$ of a ring $R$.
We don't require $(a)$ to be a prime or maximal ideal. Instead, we allow the creation of the residue ring $R/(a)$ for any nonzero $a$ and simply raise an exception if an impossible inverse is encountered during computations involving elements of $R/(a)$. Of course, a GCD function must be available for the base ring $R$.
There is a generic implementation of residue rings of this form in AbstractAlgebra.jl, which accepts any ring $R$ as base ring.
The associated types of parent object and elements for each kind of residue rings in Nemo are given in the following table.
| Base ring | Library | Element type | Parent type |
|---|---|---|---|
| Generic ring $R$ | AbstractAlgebra.jl | EuclideanRingResidueRingElem{T} | EuclideanRingResidueRing{T} |
| $\mathbb{Z}$ (Int modulus) | Flint | zzModRingElem | zzModRing |
| $\mathbb{Z}$ (ZZ modulus) | Flint | ZZModRingElem | ZZModRing |
The modulus $a$ of a residue ring is stored in its parent object.
All residue element types belong to the abstract type ResElem and all the residue ring parent object types belong to the abstract type ResidueRing. This enables one to write generic functions that accept any Nemo residue type.
Residue functionality
All the residue rings in Nemo provide the functionality described in AbstractAlgebra for residue rings:
https://nemocas.github.io/AbstractAlgebra.jl/stable/residue
In addition, generic residue rings are available.
We describe Nemo specific residue ring functionality below.
GCD
Base.gcdx — Methodgcdx(a::zzModRingElem, b::zzModRingElem)Compute the extended gcd with the Euclidean structure inherited from $\mathbb{Z}$.
Base.gcdx — Methodgcdx(a::ZZModRingElem, b::ZZModRingElem)Compute the extended gcd with the Euclidean structure inherited from $\mathbb{Z}$.
Examples
julia> R, = residue_ring(ZZ, 123456789012345678949);
julia> g, s, t = gcdx(R(123), R(456))
(1, 123456789012345678928, 41152263004115226322)