Integers mod n
Integers mod $n$ are implemented via the Singular n_Zn type for any positive modulus that can fit in a Julia Int.
The associated ring of integers mod $n$ is represented by a parent object which can be constructed by a call to the ResidueRing constructor.
The types of the parent objects and elements of the associated rings of integers modulo n are given in the following table according to the library providing them.
| Library | Element type | Parent type |
|---|---|---|
| Singular | n_Zn | Singular.N_ZnRing |
All integer mod $n$ element types belong directly to the abstract type RingElem and all the parent object types belong to the abstract type Ring.
Integer mod $n$ functionality
Singular.jl integers modulo $n$ implement the Ring and Residue Ring interfaces of AbstractAlgebra.jl.
https://nemocas.github.io/AbstractAlgebra.jl/rings.html
https://nemocas.github.io/AbstractAlgebra.jl/residue_rings.html
Parts of the Euclidean Ring interface may also be implemented, though Singular will report an error if division is meaningless (even after cancelling zero divisors).
https://nemocas.github.io/AbstractAlgebra.jl/euclidean.html
Below, we describe the functionality that is specific to the Singular integers mod $n$ ring and not already listed at the given links.
Constructors
Given a ring $R$ of integers modulo $n$, we also have the following coercions in addition to the standard ones expected.
R(n::n_Z)
R(n::fmpz)Coerce a Singular or Flint integer value into the ring.
Basic manipulation
AbstractAlgebra.Generic.isunit โ Method.isunit(n::n_Zn)Return
trueif the given value is a unit in the integers modulo $n$.
AbstractAlgebra.Generic.characteristic โ Method.characteristic(R::N_ZnRing)Return the characteristic $n$ of the ring.
Examples
R = ResidueRing(ZZ, 26)
a = R(5)
isunit(a)
c = characteristic(R)