[SONATA] 6 Nearring ideals

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# 6 Nearring ideals

### Sections

1. Construction of nearring ideals
2. Testing for ideal properties
3. Special ideal properties
4. Generators of nearring ideals
5. Near-ring ideal elements
6. Random ideal elements
7. Membership of an ideal
8. Size of ideals
9. Group reducts of ideals
10. Comparision of ideals
11. Operations with ideals
12. Commutators
13. Simple nearrings
14. Factor nearrings

For an introduction to nearring ideals we suggest [Pilz:Nearrings], [meldrum85:NATLWG], and [Clay:Nearrings].

Ideals of nearrings can either be left, right or twosided ideals. However, all of them are called ideals. Mathematicians tend to use the expression ideal also for subgroups of the group reduct of the nearring. GAP does not allow that.

Left, right or twosided ideals in GAP form their own category `IsNRI`. Whenever a left, right or twosided ideal is constructed it lies in this category. The objects in this category are what GAP considers as ideals. We will refer to them as `NRI`s.

All the functions in this chapter can be applied to all types of nearrings.

The functions described in this chapter can be found in the source files `nrid.g?`, `idlatt.g?` and `nrconstr.g?`.

## 6.1 Construction of nearring ideals

There are several ways to construct ideals in nearrings. `NearRingLeftIdealByGenerators`, `NearRingRightIdealByGenerators` and `NearRingIdealByGenerators` can be used to construct (left / right) ideals generated by a subset of the nearring. `NearRingLeftIdealBySubgroupNC`, `NearRingRightIdealBySubgroupNC` and `NearRingIdealBySubgroupNC` construct (left / right) ideals from a subgroup of the group reduct of the nearring which is an ideal. Finally `NearRingLeftIdeals`, `NearRingRightIdeals` and `NearRingIdeals` compute lists of all (left / right) ideals of a nearring.

- `NearRingIdealByGenerators(` nr`,` gens`)`

  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.

- `NearRingLeftIdealByGenerators(` nr`,` 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.

- `NearRingRightIdealByGenerators(` nr`,` 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

- `NearRingIdealBySubgroupNC(` nr`,` S`)`

  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.

- `NearRingLeftIdealBySubgroupNC(` nr`,` S`)`

  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.

- `NearRingRightIdealBySubgroupNC(` nr`,` S`)`

  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(` nr`)`

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

  For one-sided ideals the functions

- `NearRingLeftIdeals(` nr`)`

  and

- `NearRingRightIdeals(` nr`)`

  can be used.

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

  ## 6.2 Testing for ideal properties

- `IsNRI(` obj`)`

  `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.)

- `IsNearRingLeftIdeal(` I`)`

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

- `IsNearRingRightIdeal(` I`)`

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

- `IsNearRingIdeal(` I`)`

  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 >

- `IsSubgroupNearRingLeftIdeal(` nr`,` S`)`

  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.

- `IsSubgroupNearRingRightIdeal(` nr`,` S`)`

  Let (N,+,·) be a nearring. A subgroup S of the group (N,+) is a right ideal of N if S·N Í 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

- `IsPrimeNearRingIdeal(` I`)`

  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 > ]

- `IsMaximalNearRingIdeal(` I`)`

  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 > ]

  ## 6.4 Generators of nearring ideals

- `GeneratorsOfNearRingIdeal(` I`)`

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

- `GeneratorsOfNearRingLeftIdeal(` I`)`

  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.

- `GeneratorsOfNearRingRightIdeal(` I`)`

  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

- `AsList(` I`)`

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

- `AsSortedList(` I`)`

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

- `Enumerator(` I`)`

  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(` I`)`

  `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

- ``n`in` I``

  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(` I`)`

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

  ## 6.9 Group reducts of ideals

- `GroupReduct(` I`)`

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

  ## 6.10 Comparision of ideals

- ``I`=` J``

  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.

- `Intersection(` ideallist`)`

  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.

- `Intersection(` I1`, ...,` In`)`

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

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

- `ClosureNearRingLeftIdeal(` L1`,` L2`)`

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

- `ClosureNearRingRightIdeal(` R1`,` R2`)`

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

- `ClosureNearRingIdeal(` I1`,` I2`)`

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

  ## 6.12 Commutators

- `NearRingCommutator(` I`,` J`)`

  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

- `IsSimpleNearRing(` nr`)`

  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

- `FactorNearRing(` nr`,` I`)`

  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,

- ``nr`/` I``

  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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