% \Chapter{Nearring ideals} % For an introduction to nearring ideals we suggest \cite{Pilz:Nearrings}, \cite{meldrum85:NATLWG}, and \cite{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?'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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>. \beginexample 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 \endexample \>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. \beginexample 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 > ] \endexample \>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. \beginexample 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 > ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. \beginexample 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 > \endexample \>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 $S.N \subseteq S$. `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 `NearRingRight\-Ideal\-BySubgroupNC' (see Section "NearRingRightIdealBySubgroupNC") to construct the corresponding right ideal. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. \beginexample 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 > ] \endexample \>IsMaximalNearRingIdeal( <I> ) A proper ideal $I$ of a nearring $N$ is *maximal* if there is no proper ideal containing $I$ properly. `IsMaximal\-NearRingIdeal( <I> ) returns `true' if <I> is a maximal ideal in its parent nearring and `false' otherwise. \beginexample gap> n := LibraryNearRingWithOne( GTW27_2, 5 ); LibraryNearRingWithOne(27/2, 5) gap> Filtered( NearRingIdeals( n ), IsMaximalNearRingIdeal ); [ < nearring ideal of size 9 > ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Near-ring ideal elements} \>AsList( <I> )!{near ring ideals} The function `AsList' computes the elements of the (left / right) ideal <I>. It returns the elements as a list. \>AsSortedList( <I> )!{near ring ideals} does essentially the same, but returns a set of elements. \>Enumerator( <I> )!{near ring ideals} does essentially the same as `AsList', but returns an enumerator for the elements of <nr>. \beginexample 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)) ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Random ideal elements} \>Random( <I> )!{near ring ideal element} `Random' returns a random element of the (left / right) ideal <I>. \beginexample gap> Random( l ); ((3,5)(4,6)) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Membership of an ideal} For a (left / right) ideal <I> of a nearring $N$ and an element <n> of $N$ \>`<n> in <I>'{in} tests whether <n> is an element of <I>. \beginexample gap> Random( n ) in l; true gap> Random( n ) in l; false \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Size of ideals} \>Size( <I> )!{near ring ideals} `Size' returns the number of elements of the (left / right) ideal <I>. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Group reducts of ideals} \>GroupReduct( <I> )!{near ring ideals} `GroupReduct' returns the group reduct of the (left / right) ideal <I>. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations with ideals} The most important operations for nearring (left / right) ideals are *meet* and *join* in the lattice. {\GAP} offers the functions `Intersection', `Closure\-NearRing\-LeftIdeal', `Closure\-NearRing\-RightIdeal' and `Closure\-NearRing\-LeftIdeal' for this purpose. \>Intersection( <ideallist> )!{for nearring ideals} 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>, \dots, <In> ) computes the intersection of the (left / right) ideals <I1>, \dots, <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 $N$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Commutators} \>NearRingCommutator( <I>, <J> ) The function `NearRingCommutator' returns the commutator of the two ideals <I> and <J> of a common nearring. \beginexample 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 > \endexample The function `PrintNearRingCommutatorsTable' prints a complete overview over the action of the commutator operator on a group. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Simple nearrings} \>IsSimpleNearRing( <nr> ) The function `IsSimpleNearRing' returns `true' if the nearring <nr> has no proper (two-sided) ideals. \beginexample gap> NumberLibraryNearRings( GTW4_2 ); 23 gap> Filtered( AllLibraryNearRings( GTW4_2 ), IsSimpleNearRing ); [ LibraryNearRing(4/2, 3), LibraryNearRing(4/2, 16), LibraryNearRing(4/2, 17) ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. \beginexample 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 ] \endexample %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: