
[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.
   

