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1 Introduction
1.1 Ring Constructions for Supported External Computer Algebra Systems
1.1-1 external GAP
1.1-2 Singular
1.1-3 MAGMA
1.1-4 Macaulay2
1.1-5 Sage
1.1-6 Maple

### 1 Introduction

This package is part of the homalg project [ht09]. 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

    gap> ZZ := HomalgRingOfIntegersInExternalGAP( );
    <A homalg external ring residing in the CAS GAP>
     gap> Display( ZZ );
     Z
    gap> F2 := HomalgRingOfIntegersInExternalGAP( 2, ZZ );
    <A homalg external ring residing in the CAS GAP>
     gap> Display( F2 );
     GF(2)

`F2 := HomalgRingOfIntegersInExternalGAP( 2 )` would launch another GAP.

    gap> Z4 := HomalgRingOfIntegersInExternalGAP( 4, ZZ );
    <A homalg external ring residing in the CAS GAP>
     gap> Display( Z4 );
     Z/4Z
    gap> Z_4 := HomalgRingOfIntegersInExternalGAP( ZZ ) / 4;
    <A homalg residue class ring>
    gap> Display( Z_4 );
    Z/( 4 )
    gap> Q := HomalgFieldOfRationalsInExternalGAP( ZZ );
    <A homalg external ring residing in the CAS GAP>
     gap> Display( Q );
     Q

##### 1.1-2 Singular

    gap> F2 := HomalgRingOfIntegersInSingular( 2 );
    <A homalg external ring residing in the CAS Singular>
     gap> Display( F2 );
     GF(2)
    gap> Q := HomalgFieldOfRationalsInSingular( F2 );
    <A homalg external ring residing in the CAS Singular>
     gap> Display( Q );
     Q

`Q := HomalgFieldOfRationalsInSingular( )` would launch another Singular.

    gap> F2xyz := F2 * "x,y,z";
    <A homalg external ring residing in the CAS Singular>
     gap> Display( F2xyz );
     GF(2)[x,y,z]
    gap> Qxyz := Q * "x,y,z";
    <A homalg external ring residing in the CAS Singular>
     gap> Display( Qxyz );
     Q[x,y,z]

##### 1.1-3 MAGMA

    gap> ZZ := HomalgRingOfIntegersInMAGMA( );
    <A homalg external ring residing in the CAS MAGMA>
     gap> Display( ZZ );
     Z
    gap> F2 := HomalgRingOfIntegersInMAGMA( 2, ZZ );
    <A homalg external ring residing in the CAS MAGMA>
     gap> Display( F2 );
     GF(2)

`F2 := HomalgRingOfIntegersInMAGMA( 2 )` would launch another MAGMA.

    gap> Z_4 := HomalgRingOfIntegersInMAGMA( ZZ ) / 4;
    <A homalg residue class ring>
    gap> Display( Z_4 );
    Z/( 4 )
    gap> Q := HomalgFieldOfRationalsInMAGMA( ZZ );
    <A homalg external ring residing in the CAS MAGMA>
     gap> Display( Q );
     Q
    gap> F2xyz := F2 * "x,y,z";
    <A homalg external ring residing in the CAS MAGMA>
     gap> Display( F2xyz );
     GF(2)[x,y,z]
    gap> Qxyz := Q * "x,y,z";
    <A homalg external ring residing in the CAS MAGMA>
     gap> Display( Qxyz );
     Q[x,y,z]

##### 1.1-4 Macaulay2

    gap> ZZ := HomalgRingOfIntegersInMacaulay2( );
    <A homalg external ring residing in the CAS Macaulay2>
     gap> Display( ZZ );
     Z
    gap> F2 := HomalgRingOfIntegersInMacaulay2( 2, ZZ );
    <A homalg external ring residing in the CAS Macaulay2>
     gap> Display( F2 );
     GF(2)

`F2 := HomalgRingOfIntegersInMacaulay2( 2 )` would launch another Macaulay2.

    gap> Z_4 := HomalgRingOfIntegersInMacaulay2( ZZ ) / 4;
    <A homalg residue class ring>
    gap> Display( Z_4 );
    Z/( 4 )
    gap> Q := HomalgFieldOfRationalsInMacaulay2( ZZ );
    <A homalg external ring residing in the CAS Macaulay2>
     gap> Display( Q );
     Q
    gap> F2xyz := F2 * "x,y,z";
    <A homalg external ring residing in the CAS Macaulay2>
     gap> Display( F2xyz );
     GF(2)[x,y,z]
    gap> Qxyz := Q * "x,y,z";
    <A homalg external ring residing in the CAS Macaulay2>
     gap> Display( Qxyz );
     Q[x,y,z]

##### 1.1-5 Sage

    gap> ZZ := HomalgRingOfIntegersInSage( );
    <A homalg external ring residing in the CAS Sage>
     gap> Display( ZZ );
     Z
    gap> F2 := HomalgRingOfIntegersInSage( 2, ZZ );
    <A homalg external ring residing in the CAS Sage>
     gap> Display( F2 );
     GF(2)

`F2 := HomalgRingOfIntegersInSage( 2 )` would launch another Sage.

    gap> Z_4 := HomalgRingOfIntegersInSage( ZZ ) / 4;
    <A homalg residue class ring>
    gap> Display( Z_4 );
    Z/( 4 )
    gap> Q := HomalgFieldOfRationalsInSage( ZZ );
    <A homalg external ring residing in the CAS Sage>
     gap> Display( Q );
     Q
    gap> F2x := F2 * "x";
    <A homalg external ring residing in the CAS Sage>
     gap> Display( F2x );
     GF(2)[x]
    gap> Qx := Q * "x";
    <A homalg external ring residing in the CAS Sage>
     gap> Display( Qx );
     Q[x]

##### 1.1-6 Maple

    gap> ZZ := HomalgRingOfIntegersInMaple( );
    <A homalg external ring residing in the CAS Maple>
     gap> Display( ZZ );
     Z
    gap> F2 := HomalgRingOfIntegersInMaple( 2, ZZ );
    <A homalg external ring residing in the CAS Maple>
     gap> Display( F2 );
     GF(2)

`F2 := HomalgRingOfIntegersInMaple( 2 )` would launch another Maple.

    gap> Z4 := HomalgRingOfIntegersInMaple( 4, ZZ );
    <A homalg external ring residing in the CAS Maple>
     gap> Display( Z4 );
     Z/4Z
    gap> Z_4 := HomalgRingOfIntegersInMaple( ZZ ) / 4;
    <A homalg residue class ring>
    gap> Display( Z_4 );
    Z/( 4 )
    gap> Q := HomalgFieldOfRationalsInMaple( ZZ );
    <A homalg external ring residing in the CAS Maple>
     gap> Display( Q );
     Q
    gap> F2xyz := F2 * "x,y,z";
    <A homalg external ring residing in the CAS Maple>
     gap> Display( F2xyz );
     GF(2)[x,y,z]
    gap> Qxyz := Q * "x,y,z";
    <A homalg external ring residing in the CAS Maple>
     gap> Display( Qxyz );
     Q[x,y,z]

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