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.

LibraryElement typeParent type
Singularn_ZnSingular.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 $n$ provide all the AbstractAlgebra ring and residue ring functionality.

https://nemocas.github.io/AbstractAlgebra.jl/latest/ring

https://nemocas.github.io/AbstractAlgebra.jl/latest/residue

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/latest/euclidean_interface

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

Examples

R = ResidueRing(ZZ, 26)
a = R(5)

is_unit(a)
c = characteristic(R)