1 Introduction
  
  This  package is part of the homalg project [hpa10]. The role of the package
  is described in the manual of the homalg package.
  
  
  1.1 Ring Constructions for Supported External Computer Algebra Systems
  
  Here are some of the supported ring constructions:
  
  
  1.1-1 external GAP
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInExternalGAP( );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS GAP>
    gap> F2 := HomalgRingOfIntegersInExternalGAP( 2, ZZ );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS GAP>
  
  
  F2 := HomalgRingOfIntegersInExternalGAP( 2 ) would launch another GAP.
  
    Example  
    gap> Z4 := HomalgRingOfIntegersInExternalGAP( 4, ZZ );
    Z/4Z
    gap> Display( Z4 );
    <An external ring residing in the CAS GAP>
    gap> Z_4 := HomalgRingOfIntegersInExternalGAP( ZZ ) / 4;
    Z/( 4 )
    gap> Display( Z_4 );
    <A residue class ring>
    gap> Q := HomalgFieldOfRationalsInExternalGAP( ZZ );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS GAP>
  
  
  
  1.1-2 Singular
  
    Example  
    gap> F2 := HomalgRingOfIntegersInSingular( 2 );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS Singular>
    gap> F2s := HomalgRingOfIntegersInSingular( 2, "s" ,F2 );
    GF(2)(s)
    gap> Display( F2s );
    <An external ring residing in the CAS Singular>
    gap> ZZ := HomalgRingOfIntegersInSingular( F2 );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS Singular>
    gap> Q := HomalgFieldOfRationalsInSingular( F2 );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS Singular>
    gap> Qs := HomalgFieldOfRationalsInSingular( "s", F2 );
    Q(s)
    gap> Display( Qs );
    <An external ring residing in the CAS Singular>
    gap> Qi := HomalgFieldOfRationalsInSingular( "i", "i^2+1", Q );
    Q[i]/(i^2+1)
    gap> Display( Qi );
    <An external ring residing in the CAS Singular>
  
  
  Q := HomalgFieldOfRationalsInSingular( ) would launch another Singular.
  
    Example  
    gap> F2xyz := F2 * "x,y,z";
    GF(2)[x,y,z]
    gap> Display( F2xyz );
    <An external ring residing in the CAS Singular>
    gap> F2sxyz := F2s * "x,y,z";
    GF(2)(s)[x,y,z]
    gap> Display( F2sxyz );
    <An external ring residing in the CAS Singular>
    gap> F2xyzw := F2xyz * "w";
    GF(2)[x,y,z][w]
    gap> Display( F2xyzw );
    <An external ring residing in the CAS Singular>
    gap> F2sxyzw := F2sxyz * "w";
    GF(2)(s)[x,y,z][w]
    gap> Display( F2sxyzw );
    <An external ring residing in the CAS Singular>
    gap> ZZxyz := ZZ * "x,y,z";
    Z[x,y,z]
    gap> Display( ZZxyz );
    <An external ring residing in the CAS Singular>
    gap> ZZxyzw := ZZxyz * "w";
    Z[x,y,z][w]
    gap> Display( ZZxyzw );
    <An external ring residing in the CAS Singular>
    gap> Qxyz := Q * "x,y,z";
    Q[x,y,z]
    gap> Display( Qxyz );
    <An external ring residing in the CAS Singular>
    gap> Qsxyz := Qs * "x,y,z";
    Q(s)[x,y,z]
    gap> Display( Qsxyz );
    <An external ring residing in the CAS Singular>
    gap> Qixyz := Qi * "x,y,z";
    (Q[i]/(i^2+1))[x,y,z]
    gap> Display( Qixyz );
    <An external ring residing in the CAS Singular>
    gap> Qxyzw := Qxyz * "w";
    Q[x,y,z][w]
    gap> Display( Qxyzw );
    <An external ring residing in the CAS Singular>
    gap> Qsxyzw := Qsxyz * "w";
    Q(s)[x,y,z][w]
    gap> Display( Qsxyzw );
    <An external ring residing in the CAS Singular>
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
    Q[x,y,z]<Dx,Dy,Dz>
    gap> Display( Dxyz );
    <An external ring residing in the CAS Singular>
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );
    Q{e,f,g}
    gap> Display( Exyz );
    <An external ring residing in the CAS Singular>
    gap> Dsxyz := RingOfDerivations( Qsxyz, "Dx,Dy,Dz" );
    Q(s)[x,y,z]<Dx,Dy,Dz>
    gap> Display( Dsxyz );
    <An external ring residing in the CAS Singular>
    gap> Esxyz := ExteriorRing( Qsxyz, "e,f,g" );
    Q(s){e,f,g}
    gap> Display( Esxyz );
    <An external ring residing in the CAS Singular>
    gap> Dixyz := RingOfDerivations( Qixyz, "Dx,Dy,Dz" );
    (Q[i]/(i^2+1))[x,y,z]<Dx,Dy,Dz>
    gap> Display( Dixyz );
    <An external ring residing in the CAS Singular>
    gap> Eixyz := ExteriorRing( Qixyz, "e,f,g" );
    (Q[i]/(i^2+1)){e,f,g}
    gap> Display( Eixyz );
    <An external ring residing in the CAS Singular>
  
  
  
  1.1-3 MAGMA
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInMAGMA( );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS MAGMA>
    gap> F2 := HomalgRingOfIntegersInMAGMA( 2, ZZ );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS MAGMA>
  
  
  F2 := HomalgRingOfIntegersInMAGMA( 2 ) would launch another MAGMA.
  
    Example  
    gap> Z_4 := HomalgRingOfIntegersInMAGMA( ZZ ) / 4;
    Z/( 4 )
    gap> Display( Z_4 );
    <A residue class ring>
    gap> Q := HomalgFieldOfRationalsInMAGMA( ZZ );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS MAGMA>
    gap> F2xyz := F2 * "x,y,z";
    GF(2)[x,y,z]
    gap> Display( F2xyz );
    <An external ring residing in the CAS MAGMA>
    gap> Qxyz := Q * "x,y,z";
    Q[x,y,z]
    gap> Display( Qxyz );
    <An external ring residing in the CAS MAGMA>
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );
    Q{e,f,g}
    gap> Display( Exyz );
    <An external ring residing in the CAS MAGMA>
  
  
  
  1.1-4 Macaulay2
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInMacaulay2( );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS Macaulay2>
    gap> F2 := HomalgRingOfIntegersInMacaulay2( 2, ZZ );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS Macaulay2>
  
  
  F2 := HomalgRingOfIntegersInMacaulay2( 2 ) would launch another Macaulay2.
  
    Example  
    gap> Z_4 := HomalgRingOfIntegersInMacaulay2( ZZ ) / 4;
    Z/( 4 )
    gap> Display( Z_4 );
    <A residue class ring>
    gap> Q := HomalgFieldOfRationalsInMacaulay2( ZZ );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS Macaulay2>
    gap> F2xyz := F2 * "x,y,z";
    GF(2)[x,y,z]
    gap> Display( F2xyz );
    <An external ring residing in the CAS Macaulay2>
    gap> Qxyz := Q * "x,y,z";
    Q[x,y,z]
    gap> Display( Qxyz );
    <An external ring residing in the CAS Macaulay2>
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
    Q[x,y,z]<Dx,Dy,Dz>
    gap> Display( Dxyz );
    <An external ring residing in the CAS Macaulay2>
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );
    Q{e,f,g}
    gap> Display( Exyz );
    <An external ring residing in the CAS Macaulay2>
  
  
  
  1.1-5 Sage
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInSage( );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS Sage>
    gap> F2 := HomalgRingOfIntegersInSage( 2, ZZ );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS Sage>
  
  
  F2 := HomalgRingOfIntegersInSage( 2 ) would launch another Sage.
  
    Example  
    gap> Z_4 := HomalgRingOfIntegersInSage( ZZ ) / 4;
    Z/( 4 )
    gap> Display( Z_4 );
    <A residue class ring>
    gap> Q := HomalgFieldOfRationalsInSage( ZZ );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS Sage>
    gap> F2x := F2 * "x";
    GF(2)[x]
    gap> Display( F2x );
    <An external ring residing in the CAS Sage>
    gap> Qx := Q * "x";
    Q[x]
    gap> Display( Qx );
    <An external ring residing in the CAS Sage>
  
  
  
  1.1-6 Maple
  
    Example  
    gap> ZZ := HomalgRingOfIntegersInMaple( );
    Z
    gap> Display( ZZ );
    <An external ring residing in the CAS Maple>
    gap> F2 := HomalgRingOfIntegersInMaple( 2, ZZ );
    GF(2)
    gap> Display( F2 );
    <An external ring residing in the CAS Maple>
  
  
  F2 := HomalgRingOfIntegersInMaple( 2 ) would launch another Maple.
  
    Example  
    gap> Z4 := HomalgRingOfIntegersInMaple( 4, ZZ );
    Z/4Z
    gap> Display( Z4 );
    <An external ring residing in the CAS Maple>
    gap> Z_4 := HomalgRingOfIntegersInMaple( ZZ ) / 4;
    Z/( 4 )
    gap> Display( Z_4 );
    <A residue class ring>
    gap> Q := HomalgFieldOfRationalsInMaple( ZZ );
    Q
    gap> Display( Q );
    <An external ring residing in the CAS Maple>
    gap> F2xyz := F2 * "x,y,z";
    GF(2)[x,y,z]
    gap> Display( F2xyz );
    <An external ring residing in the CAS Maple>
    gap> Qxyz := Q * "x,y,z";
    Q[x,y,z]
    gap> Display( Qxyz );
    <An external ring residing in the CAS Maple>
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
    Q[x,y,z]<Dx,Dy,Dz>
    gap> Display( Dxyz );
    <An external ring residing in the CAS Maple>
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );
    Q{e,f,g}
    gap> Display( Exyz );
    <An external ring residing in the CAS Maple>