%
\Chapter{Ideals, factors, and direct products of nearrings}
%
An *ideal* of a nearring $(N,+,*)$ is a subset $I$ such that
$I$ is a normal subgroup of $(N,+)$, and
for all $i
\in I$, $n,m
\in N$, we have
$(m+i)*n - m*n
\in I$ and $n*i
\in I$. Ideals are in
one-to-one correspondence to the congruence relations
on $(N,+,*)$.
A *right ideal* of a nearring $(N,+,*)$ is a subset $I$ such that
$I$ is a normal subgroup of $(N,+)$, and
for all $i
\in I$, $n,m
\in N$, we have
$(m+i)*n - m*n
\in I$. Right ideals are in
one-to-one correspondence to the congruence relations
on $(N,+,
\{ \lambda_m | m
\in M
\} )$, where
$
\lambda_m (n) := n*m$. Hence, right ideals
describe the congruences of the $N$-group
$N_N$.
A *left ideal* of a nearring $(N,+,*)$ is a subset $I$ such that
$I$ is a normal subgroup of $(N,+)$, and
for all $i
\in I$, $n
\in N$, we have
$n*i
\in I$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Direct products}
For all sorts of nearrings direct products $A
\times B$ can be constructed. The
result is again a nearring. In the case that both <A> and <B>
are `TransformationNearRings
', the result will be a `TransformationNearRing'
acting on the direct product of the groups <A> and <B> act on. In any other
case the result is an `ExplicitMultiplicationNearRing
', even if one of the
factors is a `TransformationNearRing
'. In any case, the elements of a direct
product are *not* pairs or tuples.
%\beginexample
% gap> M1 := MapNearRing( GTW4_2 );
% TransformationNearRing(4/2)
% gap> M2 := MapNearRing( GTW8_4 );
% TransformationNearRing(8/4)
% gap> D := DirectProductNearRing( M1, M2 );
% DirectProductNearRing( TransformationNearRing(4/2),
% TransformationNearRing(8/4) )
% gap> Gamma( D );
% Group([ (1,2), (3,4), (5,6,7,8), (6,8) ])
% gap> Random( D );
% [ 24, 18, 19, 24, 17, 22, 23, 19, 8, 2, 3, 8, 1, 6, 7, 3, 24, 18, 19,
% 24, 17, 22, 23, 19, 8, 2, 3, 8, 1, 6, 7, 3 ] on Group(
% [ (1,2), (3,4), (5,6,7,8), (6,8) ])
% gap> D!.factors;
% [ TransformationNearRing(4/2), TransformationNearRing(8/4) ]
%\endexample
% The function `Gamma' returns the group on which the
% `TransformationNearRing' acts. Here the resulting nearring is
% again a nearring of `GroupTransformations'. The direct factors can
% be read from `D!.factors'.
\beginexample
gap> A := LibraryNearRing( GTW8_2,
12 );
LibraryNearRing(
8/
2,
12)
gap> B := LibraryNearRing( GTW12_4,
13 );
LibraryNearRing(
12/
4,
13)
gap> D := DirectProductNearRing( A, B );
DirectProductNearRing( LibraryNearRing(
8/
2,
12),
LibraryNearRing(
12/
4,
13) )
gap> SetName( D,
"A x B" );
gap> D;
A x B
\endexample
In this case the result is an `ExplicitMultiplicationNearRing
'.
It is a good idea to give a shorter name to the nearring <D>, because
we will investigate one of its ideals in the next section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Ideals and factors}
We go on with the last example of the previous section and try to
compute a left ideal which is generated by two elements, namely the
second and the twenty-fifth in the sorted list of elements. The {
\GAP}
function `list
\{[ poss ]
\}' constructs a list of those elements
of the list `list
' the position in the list `list' of which is
in the list `poss
'. For short, `elms\{[2,25]\}' is a list which
contains the second and the twenty-fifth element of the list `elms
'.
\beginexample
gap> elms := AsSortedList( D );;
gap> gens := elms{[
2,
25]};
[ ((
8,
9,
10)), ((
3,
5)(
4,
6)) ]
gap> L := NearRingLeftIdealByGenerators( D, gens );
< nearring left ideal >
\endexample
Now we can start investigating <I>. We can compute its size and test
if it is an ideal.
\beginexample
gap> Size( L );
24
gap> IsNearRingRightIdeal( L );
true
gap> L;
< nearring ideal of size
24 >
\endexample
So <L> is a two-sided ideal with
24 elements. Now we are getting
interested in <L>. Is it a maximal ideal, what is the factor <D/L>?
\beginexample
gap> IsMaximalNearRingIdeal( L );
false
gap> F := D/L;
FactorNearRing( A x B, < nearring ideal of size
24 > )
gap> PrintTable( F,
"am" );
+ | n0 n1 n2 n3
--------------------
n0 | n0 n1 n2 n3
n1 | n1 n0 n3 n2
n2 | n2 n3 n0 n1
n3 | n3 n2 n1 n0
* | n0 n1 n2 n3
--------------------
n0 | n0 n0 n0 n0
n1 | n0 n0 n0 n0
n2 | n0 n0 n0 n0
n3 | n0 n0 n0 n0
\endexample
Here, we use `PrintTable
' with a second argument, because we do
not want to see all the information. Here `a
' stands for addition and `m'
stands for multiplication table. For more options see the reference
manual. Obviously, <F> is a constant nearring on a group of order
4.
The additive group of the nearring is $
\Z_2 \times \Z_2$. To make this
fact more obvious, we choose other names (symbols) for the elements
of the nearring and print the addition table again.
\beginexample
gap> IsElementaryAbelian( GroupReduct( F ) );
true
gap> # this would also convince us
gap> IsCyclic( GroupReduct( F ) );
false
gap> SetSymbols( F, [
"(0,0)",
"(0,1)",
"(1,0)",
"(1,1)"] );
gap> PrintTable( F,
"m" );
* | (
0,
0) (
0,
1) (
1,
0) (
1,
1)
-----------------------------------
(
0,
0) | (
0,
0) (
0,
0) (
0,
0) (
0,
0)
(
0,
1) | (
0,
0) (
0,
0) (
0,
0) (
0,
0)
(
1,
0) | (
0,
0) (
0,
0) (
0,
0) (
0,
0)
(
1,
1) | (
0,
0) (
0,
0) (
0,
0) (
0,
0)
\endexample
So <F> is the zero-ring on $
\Z_2 \times \Z_2$, which is not simple,
but we knew that before.
Of course all this operations can be applied to all nearrings.
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