  
  [1X1 Introduction[0X
  
  This  package  is part of the [5Xhomalg[0m project [ht09]. The role of the package
  is described in the manual of the [5Xhomalg[0m package.
  
  
  [1X1.1 Ring Constructions for Supported External Computer Algebra Systems[0X
  
  Here are some of the supported ring constructions:
  
  
  [1X1.1-1 external [5XGAP[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> ZZ := HomalgRingOfIntegersInExternalGAP( );[0X
    [4X<A homalg external ring residing in the CAS GAP>[0X
    [4X gap> Display( ZZ );[0X
    [4X Z[0X
    [4Xgap> F2 := HomalgRingOfIntegersInExternalGAP( 2, ZZ );[0X
    [4X<A homalg external ring residing in the CAS GAP>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
  [4X------------------------------------------------------------------[0X
  
  [10XF2 := HomalgRingOfIntegersInExternalGAP( 2 )[0m would launch another GAP.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Z4 := HomalgRingOfIntegersInExternalGAP( 4, ZZ );[0X
    [4X<A homalg external ring residing in the CAS GAP>[0X
    [4X gap> Display( Z4 );[0X
    [4X Z/4Z[0X
    [4Xgap> Z_4 := HomalgRingOfIntegersInExternalGAP( ZZ ) / 4;[0X
    [4X<A homalg residue class ring>[0X
    [4Xgap> Display( Z_4 );[0X
    [4XZ/( 4 )[0X
    [4Xgap> Q := HomalgFieldOfRationalsInExternalGAP( ZZ );[0X
    [4X<A homalg external ring residing in the CAS GAP>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X1.1-2 [5XSingular[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> F2 := HomalgRingOfIntegersInSingular( 2 );[0X
    [4X<A homalg external ring residing in the CAS Singular>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
    [4Xgap> Q := HomalgFieldOfRationalsInSingular( F2 );[0X
    [4X<A homalg external ring residing in the CAS Singular>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
  [4X------------------------------------------------------------------[0X
  
  [10XQ := HomalgFieldOfRationalsInSingular( )[0m would launch another Singular.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> F2xyz := F2 * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Singular>[0X
    [4X gap> Display( F2xyz );[0X
    [4X GF(2)[x,y,z][0X
    [4Xgap> Qxyz := Q * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Singular>[0X
    [4X gap> Display( Qxyz );[0X
    [4X Q[x,y,z][0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X1.1-3 [5XMAGMA[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> ZZ := HomalgRingOfIntegersInMAGMA( );[0X
    [4X<A homalg external ring residing in the CAS MAGMA>[0X
    [4X gap> Display( ZZ );[0X
    [4X Z[0X
    [4Xgap> F2 := HomalgRingOfIntegersInMAGMA( 2, ZZ );[0X
    [4X<A homalg external ring residing in the CAS MAGMA>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
  [4X------------------------------------------------------------------[0X
  
  [10XF2 := HomalgRingOfIntegersInMAGMA( 2 )[0m would launch another MAGMA.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Z_4 := HomalgRingOfIntegersInMAGMA( ZZ ) / 4;[0X
    [4X<A homalg residue class ring>[0X
    [4Xgap> Display( Z_4 );[0X
    [4XZ/( 4 )[0X
    [4Xgap> Q := HomalgFieldOfRationalsInMAGMA( ZZ );[0X
    [4X<A homalg external ring residing in the CAS MAGMA>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
    [4Xgap> F2xyz := F2 * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS MAGMA>[0X
    [4X gap> Display( F2xyz );[0X
    [4X GF(2)[x,y,z][0X
    [4Xgap> Qxyz := Q * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS MAGMA>[0X
    [4X gap> Display( Qxyz );[0X
    [4X Q[x,y,z][0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X1.1-4 [5XMacaulay2[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> ZZ := HomalgRingOfIntegersInMacaulay2( );[0X
    [4X<A homalg external ring residing in the CAS Macaulay2>[0X
    [4X gap> Display( ZZ );[0X
    [4X Z[0X
    [4Xgap> F2 := HomalgRingOfIntegersInMacaulay2( 2, ZZ );[0X
    [4X<A homalg external ring residing in the CAS Macaulay2>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
  [4X------------------------------------------------------------------[0X
  
  [10XF2 := HomalgRingOfIntegersInMacaulay2( 2 )[0m would launch another Macaulay2.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Z_4 := HomalgRingOfIntegersInMacaulay2( ZZ ) / 4;[0X
    [4X<A homalg residue class ring>[0X
    [4Xgap> Display( Z_4 );[0X
    [4XZ/( 4 )[0X
    [4Xgap> Q := HomalgFieldOfRationalsInMacaulay2( ZZ );[0X
    [4X<A homalg external ring residing in the CAS Macaulay2>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
    [4Xgap> F2xyz := F2 * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Macaulay2>[0X
    [4X gap> Display( F2xyz );[0X
    [4X GF(2)[x,y,z][0X
    [4Xgap> Qxyz := Q * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Macaulay2>[0X
    [4X gap> Display( Qxyz );[0X
    [4X Q[x,y,z][0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X1.1-5 [5XSage[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> ZZ := HomalgRingOfIntegersInSage( );[0X
    [4X<A homalg external ring residing in the CAS Sage>[0X
    [4X gap> Display( ZZ );[0X
    [4X Z[0X
    [4Xgap> F2 := HomalgRingOfIntegersInSage( 2, ZZ );[0X
    [4X<A homalg external ring residing in the CAS Sage>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
  [4X------------------------------------------------------------------[0X
  
  [10XF2 := HomalgRingOfIntegersInSage( 2 )[0m would launch another Sage.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Z_4 := HomalgRingOfIntegersInSage( ZZ ) / 4;[0X
    [4X<A homalg residue class ring>[0X
    [4Xgap> Display( Z_4 );[0X
    [4XZ/( 4 )[0X
    [4Xgap> Q := HomalgFieldOfRationalsInSage( ZZ );[0X
    [4X<A homalg external ring residing in the CAS Sage>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
    [4Xgap> F2x := F2 * "x";[0X
    [4X<A homalg external ring residing in the CAS Sage>[0X
    [4X gap> Display( F2x );[0X
    [4X GF(2)[x][0X
    [4Xgap> Qx := Q * "x";[0X
    [4X<A homalg external ring residing in the CAS Sage>[0X
    [4X gap> Display( Qx );[0X
    [4X Q[x][0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X1.1-6 [5XMaple[1X[0X
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> ZZ := HomalgRingOfIntegersInMaple( );[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( ZZ );[0X
    [4X Z[0X
    [4Xgap> F2 := HomalgRingOfIntegersInMaple( 2, ZZ );[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( F2 );[0X
    [4X GF(2)[0X
  [4X------------------------------------------------------------------[0X
  
  [10XF2 := HomalgRingOfIntegersInMaple( 2 )[0m would launch another Maple.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Z4 := HomalgRingOfIntegersInMaple( 4, ZZ );[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( Z4 );[0X
    [4X Z/4Z[0X
    [4Xgap> Z_4 := HomalgRingOfIntegersInMaple( ZZ ) / 4;[0X
    [4X<A homalg residue class ring>[0X
    [4Xgap> Display( Z_4 );[0X
    [4XZ/( 4 )[0X
    [4Xgap> Q := HomalgFieldOfRationalsInMaple( ZZ );[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( Q );[0X
    [4X Q[0X
    [4Xgap> F2xyz := F2 * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( F2xyz );[0X
    [4X GF(2)[x,y,z][0X
    [4Xgap> Qxyz := Q * "x,y,z";[0X
    [4X<A homalg external ring residing in the CAS Maple>[0X
    [4X gap> Display( Qxyz );[0X
    [4X Q[x,y,z][0X
  [4X------------------------------------------------------------------[0X
  
