## <#GAPDoc Label="Singular">
## <Subsection Label="Singular">
## <Heading>&Singular;</Heading>
## <Example><![CDATA[
## #@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Singular ).stdout )
## 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>
## #@fi
## ]]></Example>
## <C>Q := HomalgFieldOfRationalsInSingular( )</C> would launch another Singular.
## <Example><![CDATA[
## #@if IsBound( TryLaunchCAS_IO_ForHomalg( HOMALG_IO_Singular ).stdout )
## 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>
## gap> qring := HomalgQRingInSingular( Qxyz, "x*y" );
## Q[x,y,z]/( x*y )
## gap> Display( qring );
## <An external ring residing in the CAS Singular>
## gap> "z + x*y" / qring = "z" / qring;
## true
## #@fi
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg" );
Print( "~~~~~~~~~~~\n\n" );
Print( "Singular\n\n" );
F2 := HomalgRingOfIntegersInSingular( 2 );
Display( F2 );
F2s := HomalgRingOfIntegersInSingular( 2, "s", F2 );
Display( F2s );
zz := HomalgRingOfIntegersInSingular( F2 );
Display( zz );
Q := HomalgFieldOfRationalsInSingular( F2 );
Display( Q );
Qs := HomalgFieldOfRationalsInSingular( "s", F2 );
Display( Qs );
Qi := HomalgFieldOfRationalsInSingular( "i", "i^2+1", Q );
Display( Qi );
F2xyz := F2 * "x,y,z";
Display( F2xyz );
F2sxyz := F2s * "x,y,z";
Display( F2sxyz );
F2xyzw := F2xyz * "w";
Display( F2xyzw );
F2sxyzw := F2sxyz * "w";
Display( F2sxyzw );
ZZxyz := zz * "x,y,z";
Display( zzxyz );
ZZxyzw := zzxyz * "w";
Display( zzxyzw );
Qxyz := Q * "x,y,z";
Display( Qxyz );
Qsxyz := Qs * "x,y,z";
Display( Qsxyz );
Qixyz := Qi * "x,y,z";
Display( Qixyz );
Qxyzw := Qxyz * "w";
Display( Qxyzw );
Qsxyzw := Qsxyz * "w";
Display( Qsxyzw );
Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
Display( Dxyz );
Exyz := ExteriorRing( Qxyz, "e,f,g" );
Display( Exyz );
Dsxyz := RingOfDerivations( Qsxyz, "Dx,Dy,Dz" );
Display( Dsxyz );
Esxyz := ExteriorRing( Qsxyz, "e,f,g" );
Display( Esxyz );
Dixyz := RingOfDerivations( Qixyz, "Dx,Dy,Dz" );
Display( Dixyz );
Eixyz := ExteriorRing( Qixyz, "e,f,g" );
Display( Eixyz );
