6 Nearring ideals

The function NearRingIdealByGenerators takes as arguments a nearring nr and a list gens of arbitrarily many elements of nr. It returns the smallest ideal of nr containing all elements of gens.

The function NearRingLeftIdealByGenerators takes as arguments a nearring nr and a list gens of arbitrarily many elements of nr. It returns the smallest left ideal of nr containing all elements of gens.

The function NearRingRightIdealByGenerators takes as arguments a nearring nr and a list gens of arbitrarily many elements of nr. It returns the smallest right ideal of nr containing all elements of gens.

    gap> n := LibraryNearRing( GTW8_4, 12 );
    LibraryNearRing(8/4, 12)
    gap> e := AsNearRingElement( n, (1,3)(2,4) );         
    ((1,3)(2,4))
    gap> r := NearRingRightIdealByGenerators( n, [e] );
    < nearring right ideal >
    gap> l := NearRingLeftIdealByGenerators( n, [e] );
    < nearring left ideal >
    gap> i := NearRingIdealByGenerators( n, [e] );
    < nearring ideal >
    gap> r = i;
    true
    gap> l = i;
    false
    gap> l = r;
    false

From a nearring nr and a subgroup S of the group reduct of nr, NearRingIdealBySubgroupNC constructs a (GAP--) ideal of nr. It is assumed (and hence not checked) that S is an ideal of nr. See Section IsSubgroupNearRingLeftIdeal for information how to check this.

From a nearring nr and a subgroup S of the group reduct of nr, NearRingLeftIdealBySubgroupNC constructs a (GAP--) left ideal of nr. It is assumed (and hence not checked) that S is a left ideal of nr. See Section IsSubgroupNearRingLeftIdeal for information how to check this.

From a nearring nr and a subgroup S of the group reduct of nr, NearRingRightIdealBySubgroupNC constructs a (GAP--) right ideal of nr. It is assumed (and hence not checked) that S is a right ideal of nr. See Section IsSubgroupNearRingRightIdeal for information how to check this.

    gap> a := GroupReduct( n );                                          
    8/4
    gap> nsgps := NormalSubgroups( a );
    [ Group(()), Group([ (1,3)(2,4) ]), 
      Group([ (1,3)(2,4), (1,2)(3,4) ]), Group([ (1,3)(2,4), (2,4) ]), 
      Group([ (1,2,3,4), (1,3)(2,4) ]), 8/4 ]
    gap> l := Filtered( nsgps,                                             
    > s -> IsSubgroupNearRingRightIdeal( n, s ) );                      
    [ Group(()), Group([ (1,3)(2,4), (2,4) ]), 8/4 ]
    gap> l := List( l,      
    > s -> NearRingRightIdealBySubgroupNC( n, s ) );
    [ < nearring right ideal >, < nearring right ideal >, 
      < nearring right ideal > ]

NearRingIdeals computes all ideals of the nearring nr. The return value is a list of ideals of nr

For one-sided ideals the functions

and

can be used.

    gap> NearRingIdeals( n );
    [ < nearring ideal >, < nearring ideal >, < nearring ideal > ]
    gap> NearRingRightIdeals( n );
    [ < nearring right ideal >, < nearring right ideal >, 
      < nearring right ideal > ]
    gap> NearRingLeftIdeals( n );
    [ < nearring left ideal >, < nearring left ideal >, < nearring left ideal >, 
      < nearring left ideal > ]

6.2 Testing for ideal properties

IsNRI returns true if the object obj is a left ideal, a right ideal or an ideal of a nearring. (Such an object may be considered as a (one or twosided) GAP -- nearring ideal.)

The function IsNearRingLeftIdeal can be applied to any NRI. It returns true if I is a left ideal in its parent nearring.

The function IsNearRingRightIdeal can be applied to any NRI. It returns true if I is a right ideal in its parent nearring.

The function IsNearRingIdeal can be applied to any NRI. It returns true if I is an ideal in its parent nearring.

    gap> n := LibraryNearRing( GTW6_2, 39 );                      
    LibraryNearRing(6/2, 39)
    gap> e := Enumerator(n)[3];
    ((1,3,2))
    gap> l := NearRingLeftIdealByGenerators( n, [e] );
    < nearring left ideal >
    gap> IsNRI( l );
    true
    gap> IsNearRingLeftIdeal( l );
    true
    gap> IsNearRingRightIdeal( l );
    true
    gap> l;
    < nearring ideal >

Let (N,+,·) be a nearring. A subgroup S of the group (N,+) is a left ideal of N if for all a, b in N and s in S:\ a·(b+s)−a·b in S. IsSubgroupNearRingLeftIdeal takes as arguments a nearring nr and a subgroup S of the group reduct of nr and returns true if S is a nearring ideal of nr and false otherwise.

Note, that if IsSubgroupNearRingLeftIdeal returns true this means that S is a left ideal only in the mathematical sense, not in GAP--sense (it is a group, not a left ideal). You can use NearRingLeftIdealBySubgroupNC (see Section NearRingLeftIdealBySubgroupNC) to construct the corresponding left ideal.

Let (N,+,·) be a nearring. A subgroup S of the group (N,+) is a right ideal of N if S·N ⊆ S. IsSubgroupNearRingRightIdeal takes as arguments a nearring nr and a subgroup S of the group reduct of nr and returns true if S is a right ideal of nr and false otherwise.

Note, that if IsSubgroupNearRingRightIdeal returns true this means that S is a right ideal only in the mathematical sense, not in GAP--sense (it is a group, not a right ideal). You can use NearRingRightIdealBySubgroupNC (see Section NearRingRightIdealBySubgroupNC) to construct the corresponding right ideal.

    gap> n := LibraryNearRing( GTW6_2, 39 );                    
    LibraryNearRing(6/2, 39)
    gap> s := Subgroups( GroupReduct( n ) );
    [ Group(()), Group([ (2,3) ]), Group([ (1,3) ]), Group([ (1,2) ]), 
      Group([ (1,3,2) ]), Group([ (1,2,3), (1,2) ]) ]
    gap> List( s, sg -> IsSubgroupNearRingLeftIdeal( n, sg ) );
    [ true, false, false, false, true, true ]
    gap> List( s, sg -> IsSubgroupNearRingRightIdeal( n, sg ) );
    [ true, false, false, false, true, true ]

6.3 Special ideal properties

An ideal I of a nearring N is prime if for any two ideals J and K of N whenever J·K is contained in I then at least one of them is contained in I. IsPrimeNearRingIdeal returns true if I is a prime ideal in its parent nearring and false otherwise.

    gap> n := LibraryNearRingWithOne( GTW27_2, 5 );
    LibraryNearRingWithOne(27/2, 5)
    gap> Filtered( NearRingIdeals( n ), IsPrimeNearRingIdeal );
    [ < nearring ideal of size 9 >, < nearring ideal of size 27 > ]

A proper ideal I of a nearring N is maximal if there is no proper ideal containing I properly. IsMaximalNearRingIdeal( I ) returns `true if I is a maximal ideal in its parent nearring and false otherwise.

    gap> n := LibraryNearRingWithOne( GTW27_2, 5 );
    LibraryNearRingWithOne(27/2, 5)
    gap> Filtered( NearRingIdeals( n ), IsMaximalNearRingIdeal );
    [ < nearring ideal of size 9 > ]

6.4 Generators of nearring ideals

For an NRI I the function GeneratorsOfNearRingIdeal returns a set of elements of the parent nearring of I that generates I as an ideal.

For an NRI I the function GeneratorsOfNearRingLeftIdeal returns a set of elements of the parent nearring of I that generates I as a left ideal.

For an NRI I the function GeneratorsOfNearRingRightIdeal returns a set of elements of the parent nearring of I that generates I as a right ideal.

6.5 Near-ring ideal elements

The function AsList computes the elements of the (left / right) ideal I. It returns the elements as a list.

does essentially the same, but returns a set of elements.

does essentially the same as AsList, but returns an enumerator for the elements of nr.

    gap> n := LibraryNearRing( GTW8_2, 2 );
    LibraryNearRing(8/2, 2)
    gap> li := NearRingLeftIdeals( n );
    [ < nearring left ideal >, < nearring left ideal >, 
      < nearring left ideal >, < nearring left ideal >, 
      < nearring left ideal >, < nearring left ideal > ]
    gap> l := li[3];
    < nearring left ideal >
    gap> e := Enumerator( l );;
    gap> e[2];
    ((1,2)(3,6,5,4))
    gap> AsList( e ); AsList( l );
    [ (()), ((1,2)(3,6,5,4)), ((3,5)(4,6)), ((1,2)(3,4,5,6)) ]
    [ (()), ((1,2)(3,6,5,4)), ((3,5)(4,6)), ((1,2)(3,4,5,6)) ]

6.6 Random ideal elements

Random returns a random element of the (left / right) ideal I.

    gap> Random( l );
    ((3,5)(4,6))

6.7 Membership of an ideal

For a (left / right) ideal I of a nearring N and an element n of N

tests whether n is an element of I.

    gap> Random( n ) in l;
    true
    gap> Random( n ) in l;
    false

6.8 Size of ideals

Size returns the number of elements of the (left / right) ideal I.

6.9 Group reducts of ideals

GroupReduct returns the group reduct of the (left / right) ideal I.

6.10 Comparision of ideals

If I and J are (left / right) ideals of the same nearring and consist of the same elements, then true is returned. Otherwise the answer is false.

6.11 Operations with ideals

The most important operations for nearring (left / right) ideals are meet and join in the lattice. GAP offers the functions Intersection, ClosureNearRingLeftIdeal, ClosureNearRingRightIdeal and ClosureNearRingLeftIdeal for this purpose.

computes the intersection of the (left / right) ideals in the list ideallist. All of the (left / right) ideals in ideallist must be (left / right) ideals of the same nearring.

computes the intersection of the (left / right) ideals I1, ..., In.

In both cases the result is again a (left / right) ideal.

The function ClosureNearRingLeftIdeal computes the left ideal L1 + L2 of the nearRing N if both L1 and L2 are (left) ideals of N.

The function ClosureNearRingRightIdeal computes the right ideal L1 + L2 of the nearring N if both R1 and R2 are (right) ideals of N.

The function ClosureNearRingIdeal computes the ideal L1 + L2 of the nearring N if both I1 and I2 are ideals of N.

6.12 Commutators

The function NearRingCommutator returns the commutator of the two ideals I and J of a common nearring.

    gap> l := LibraryNearRing( GTW6_2, 3 );
    LibraryNearRing(6/2, 3)
    gap> i := NearRingIdeals( l );               
    [ < nearring ideal >, < nearring ideal > ]
    gap> List( i, Size );
    [ 1, 6 ]
    gap> NearRingCommutator( i[2], i[2] );
    < nearring ideal of size 6 >

The function PrintNearRingCommutatorsTable prints a complete overview over the action of the commutator operator on a group.

    gap> l := LibraryNearRing( GTW8_4, 13 );
    LibraryNearRing(8/4, 13)
    gap> NearRingIdeals( l );
    [ < nearring ideal >, < nearring ideal >, < nearring ideal > ]
    gap> PrintNearRingCommutatorsTable( l );
    [ 1, 1, 1 ]
    [ 1, 1, 2 ]
    [ 1, 2, 2 ]

6.13 Simple nearrings

The function IsSimpleNearRing returns true if the nearring nr has no proper (two-sided) ideals.

    gap> NumberLibraryNearRings( GTW4_2 );                         
    23
    gap> Filtered( AllLibraryNearRings( GTW4_2 ), IsSimpleNearRing );
    [ LibraryNearRing(4/2, 3), LibraryNearRing(4/2, 16), 
      LibraryNearRing(4/2, 17) ]

6.14 Factor nearrings

For a nearring nr and an ideal I of the nearring nr the function FactorNearRing returns the factor nearring of nr modulo the ideal I. Alternatively,

can be used and has the same effect.

The result is always an ExplicitMultiplicationNearRing, so all functions for such nearrings can be applied to the factor nearring.

    gap> n := LibraryNearRing( GTW8_2, 2 );
    LibraryNearRing(8/2, 2)
    gap> e := AsNearRingElement( n, (1,2) );
    ((1,2))
    gap> e in n;
    true
    gap> i := NearRingRightIdealByGenerators( n, [e] );
    < nearring right ideal >
    gap> Size(i);
    4
    gap> IsNearRingLeftIdeal( i );
    true
    gap> i;
    < nearring ideal of size 4 >
    gap> f := n/i;          
    FactorNearRing( LibraryNearRing(8/2, 2), < nearring ideal of size 4 > )
    gap> IdLibraryNearRing(f);
    [ 2/1, 1 ]

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