# Integers

The default integer type in Singular.jl is the Singular n_Z integer type.

The associated ring of integers is represented by the constant parent object which can be constructed by a call to Singular.Integers().

For convenience we define

ZZ = Singular.Integers()

so that integers can be constructed using ZZ. Note that this is the name of a specific parent object, not the name of its type.

The types of the integer ring parent objects and elements of the associated rings of integers are given in the following table according to the library providing them.

LibraryElement typeParent type
Singularn_ZSingular.Integers

All integer element types belong directly to the abstract type RingElem and all the integer ring parent object types belong to the abstract type Ring.

## Integer functionality

Singular.jl provides all the ring and possibly some parts of the Euclidean ring functionality of AbstractAlgebra.jl.

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

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

Below, we describe the functionality that is specific to the Singular integer ring.

### Constructors

ZZ(n::Integer)

Coerce a Julia integer value into the integer ring.

### Basic manipulation

Examples

a = ZZ(-12)

is_unit(a)
n = numerator(a)
d = denominator(a)
c = abs(a)

### Euclidean division

Singular.jl provides a number of Euclidean division operations. Recall that for a dividend $a$ and divisor $b$, we can write $a = bq + r$ with $0 \leq |r| < |b|$. We call $q$ the quotient and $r$ the remainder.

In the following table we list the division functions and their rounding behaviour. We also give the return value of the function, with $q$ representing return of the quotient and $r$ representing return of the remainder.

FunctionReturnRounding
divrem(a::n_Z, b::n_Z)q, rtowards zero
rem(a::n_Z, b::n_Z)rtowards zero
mod(a::n_Z, b::n_Z)rdown

Examples

a = ZZ(-12)
b = ZZ(5)

q, r = divrem(a, b)
r = mod(a, b)
c = a % b

### Comparison

Here is a list of the comparison functions implemented, with the understanding that isless provides all the usual comparison operators.

Function
isless(a::n_Z, b::n_Z)

We also provide the following ad hoc comparisons which again provide all of the comparison operators mentioned above.

Function
isless(a::n_Z, b::Integer)
isless(a::Integer, b::n_Z)

Examples

a = ZZ(12)
b = ZZ(3)

a < b
a != b
a > 4
5 <= b