  GNU-Darwin Web
PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

[index]

# Algebra::ResidueClassRing

(Class of Residue Class Ring)

This class represents a residue class ring. To create concrete class, use the class method ::create or the function Algebra.ResidueClassRing() designating the base ring and the element of it.

## File Name:

• residue-class-ring.rb

• Object

none.

## Associated Functions:

`Algebra.ResidueClassRing(ring, mod)`

Same as ::create(ring, mod).

## Class Methods:

`::create(ring, mod)`

Returns the class of the residue class ring of the ring and the modulus mod.

This class is a subclass of ResidueClassRing and has the class methods ::ground, ::modulus and ` [x] ` , which return the fundamental ring ring, the modulus mod and the representing residue class of x, respectively.

Example: divide the polynomial ring by the modulus `x**2 + x + 1`.

```require "rational"
require "polynomial"
require "residue-class-ring"
Px = Algebra.Polynomial(Rational, "x")
x = Px.var
F = ResidueClassRing(Px, x**2 + x + 1)
p F[x + 1]**100     #=> -x - 1
```

When ring is Integer, all inverse elements are calculated in advance. And we can obtain the residue classes of `0, 1, ... , mod-1` by to_ary.

Example: the prime field of modulo 7

```require "residue-class-ring"
F7 = Algebra::ResidueClassRing.create(Integer, 7)
a, b, c, d, e, f, g = F7
p [e + c, e - c, e * c, e * 2001, 3 + c, 1/c, 1/c * c]
#=> [6, 2, 1, 3, 5, 4, 1]
p( (1...7).collect{|i| F7[i]**6} )
#=> [1, 1, 1, 1, 1, 1]
```
`::[x]`

Returns the residue class represented bye x.

`::zero`

Returns zero.

`::unity`

Returns unity.

## Methods:

`lift`

Returns the representative of self.

`zero?`

Returns true if self is zero.

`zero`

Returns zero.

`unity`

Returns unity.

`==(other)`

Returns true if self equals other.

`+(other)`

Returns the sum of self and other.

`-(other)`

Returns the difference of self from other.

`*(other)`

Returns the product of self and other.

`**(n)`

Returns the n-th power of self.

`/(other)`

Returns the quotient of self by other using inverse.

`inverse`

Returns the inverse element, assuming the fundamental ring is Euclidian. When it does not exist, this returns nil.