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 residue_ring 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.

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::ZZRingElem)

Coerce a Singular or Flint integer value into the ring.

Basic manipulation

Examples

julia> R, = residue_ring(ZZ, 26);

julia> a = R(5)
5

julia> is_unit(a)
true

julia> c = characteristic(R)
26