localized-ring.rd

[index]

# Algebra::LocalizedRing

(Class of Localization of Ring)

This class creates the fraction ring of the given ring. To make a concrete class, use the class method ::create or the function Algebra.LocalizedRing().

## File Name:

- localized-ring.rb

## SuperClass:

- Object

## Included Ring

none.

## Associated Functions:

`Algebra.LocalizedRing(ring)`

  Same as ::create(ring).

`Algebra.RationalFunctionField(ring, obj)`

  Creates the rational function field over ring with the variable expressed by obj. This class is equipped with the class method ::var which returns the variable.

  Example: the quotient field over the polynomial ring over Integer

      require "localized-ring"
      F = Algebra.RationalFunctionField(Integer, "x")
      x = F.var
      p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) )
        #=> (x^3 - x^2)/(x^5 - x^3 - x^2 + 1)

## Class Method:

`::create(ring)`

  Returns the fraction ring of which the numerator and the denominator are the elements of the ring.

  This returns the subclass of Algebra::LocalizedRing. The subclass has the class method ::ground and `::[]` which return ring and `x/1` respectively.

  Example: Yet Another Rational

      require "localized-ring"
      F = Algebra.LocalizedRing(Integer)
      p F.new(1, 2) + F.new(2, 3) #=> 7/6

  Example: rational function field over Integer

      require "polynomial"
      require "localized-ring"
      P = Algebra.Polynomial(Integer, "x")
      F = Algebra.LocalizedRing(P)
      x = F[P.var]
      p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) )
        #=> (x^3 - x^2)/(x^5 - x^3 - x^2 + 1)

`::zero`

  Returns zero.

`::unity`

  Returns unity.

## Methods:

`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.

`/(other)`

  Returns the quotient of self by other using inverse.

`**(n)`

  Returns the n-th power of self.