B Overview of the GradedRingForHomalg Package Source Code
  
  This  appendix  is  included in the documentation to shine some light on the
  mathematical  backgrounds of this Package. Neither is it needed to work with
  this  package  nor should the methods presented here be called directly. The
  functions documented here are entries of the so called ring table and not to
  be called directly. There are higher level methods in declared and installed
  in  MatricesForHomalg, which call this functions (--> ?MatricesForHomalg:The
  Basic Matrix Operations).
  
  
  B.1 The generic Methods
  
  We will present some methods as an example, to show the idea:
  
  B.1-1 BasisOfRowModule
  
  BasisOfRowModule( M )  function
  Returns:  a distinguished basis (i.e. a distinguished generating set) of the
            module generated by M
  
    Code  
    BasisOfRowModule :=
      function( M )
         return MatrixOverGradedRing(
                        BasisOfRowModule( UnderlyingMatrixOverNonGradedRing( M ) ),
                        HomalgRing( M ) );
      end,
  
  
  B.1-2 DecideZeroRows
  
  DecideZeroRows( A, B )  function
  Returns:  a reduced form of A with respect to B
  
    Code  
    DecideZeroRows :=
      function( A, B )
        return MatrixOverGradedRing(
                       DecideZeroRows( UnderlyingMatrixOverNonGradedRing( A ),
                               UnderlyingMatrixOverNonGradedRing( B ) ),
                       HomalgRing( A ) );
      end,
  
  
  B.1-3 SyzygiesGeneratorsOfRows
  
  SyzygiesGeneratorsOfRows( M )  function
  Returns:  a distinguished basis of the syzygies of the argument
  
    Code  
    SyzygiesGeneratorsOfRows :=
      function( M )
        return MatrixOverGradedRing(
                       SyzygiesGeneratorsOfRows( UnderlyingMatrixOverNonGradedRing( M ) ),
                       HomalgRing( M ) );
      end,
  
  
  
  B.2 Tools
  
  The  package  GradedRingForHomalg  also implements tool functions. These are
  referred  to  from  MatricesForHomalg automatically. We list the implemented
  methods  here  are  and  refer  to  the MatricesForHomalg documentation (-->
  ?MatricesForHomalg:       The      Matrix      Tool      Operations      and
  ?MatricesForHomalg:RingElement)   for  details.  All  tools  functions  from
  MatricesForHomalg not listed here are also supported by fallback tools.
  
      IsZero
  
      IsOne
  
      Minus
  
      DivideByUnit
  
      IsUnit
  
      Sum
  
      Product
  
      ShallowCopy
  
      ZeroMatrix
  
      IdentityMatrix
  
      AreEqualMatrices
  
      Involution
  
      CertainRows
  
      CertainColumns
  
      UnionOfRows
  
      UnionOfColumns
  
      DiagMat
  
      KroneckerMat
  
      MulMat
  
      AddMat
  
      SubMat
  
      Compose
  
      NrRows
  
      NrColumns
  
      IsZeroMatrix
  
      IsDiagonalMatrix
  
      ZeroRows
  
      ZeroColumns