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<html><head><title>[SONATA-tutorial] 6 Ideals, factors, and direct products of nearrings</title></head>
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<h1>6 Ideals, factors, and direct products of nearrings</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP006.htm#SECT001">Direct products</a>
<li> <A HREF="CHAP006.htm#SECT002">Ideals and factors</a>
</ol>

An ideal of a nearring (N,+,*) is a subset I such that
I is a normal subgroup of (N,+), and
for all i ∈ I, n,m ∈ N, we have
(m+i)*n − m*n ∈ I and n*i ∈ 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 ∈ I, n,m ∈ N, we have
(m+i)*n − m*n ∈ I. Right ideals are in
one-to-one correspondence to the congruence relations
on (N,+, { λm | m ∈ M } ), where
λm (n) : = n*m. Hence, right ideals
describe the congruences of the N-group
NN.

A left ideal of a nearring (N,+,*) is a subset I such that
I is a normal subgroup of (N,+), and
for all i ∈ I, n ∈ N, we have
n*i ∈ I.


<h2><a name="SECT001">6.1 Direct products</a></h2>

For all sorts of nearrings direct products A ×B can be constructed. The
result is again a nearring. In the case that both <var>A</var> and <var>B</var>
are <code>TransformationNearRings</code>, the result will be a <code>TransformationNearRing</code>
acting on the direct product of the groups <var>A</var> and <var>B</var> act on. In any other
case the result is an <code>ExplicitMultiplicationNearRing</code>, even if one of the
factors is a <code>TransformationNearRing</code>. In any case, the elements of a direct
product are not pairs or tuples.
<pre>
 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
</pre>
 In this case the result is an <code>ExplicitMultiplicationNearRing</code>.
 It is a good idea to give a shorter name to the nearring <var>D</var>, because
 we will investigate one of its ideals in the next section.


<h2><a name="SECT002">6.2 Ideals and factors</a></h2>

 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 <code>list{[ poss ]}</code> constructs a list of those elements
 of the list <code>list</code> the position in the list <code>list</code> of which is
 in the list <code>poss</code>. For short, <code>elms{[2,25]}</code> is a list which
 contains the second and the twenty-fifth element of the list <code>elms</code>.
<pre>
 gap> elms := AsSortedList( D );;
 gap> gens := elms{[2,25]};
 [ (( 8, 9,10)), ((3,5)(4,6)) ]
 gap> L := NearRingLeftIdealByGenerators( D, gens );
 < nearring left ideal >
</pre>
 Now we can start investigating <var>I</var>. We can compute its size and test
 if it is an ideal.
<pre>
 gap> Size( L );
 24
 gap> IsNearRingRightIdeal( L );
 true
 gap> L;
 < nearring ideal of size 24 >
</pre>
 So <var>L</var> is a two-sided ideal with 24 elements. Now we are getting
 interested in <var>L</var>. Is it a maximal ideal, what is the factor <var>D/L</var>?
<pre>
 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
</pre>
 Here, we use <code>PrintTable</code> with a second argument, because we do
 not want to see all the information. Here <code>a</code> stands for addition and <code>m</code>
 stands for multiplication table. For more options see the reference
 manual. Obviously, <var>F</var> is a constant nearring on a group of order 4.
 The additive group of the nearring is Z2 ×Z2. To make this
 fact more obvious, we choose other names (symbols) for the elements
 of the nearring and print the addition table again.
<pre>
 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)
</pre>
 So <var>F</var> is the zero-ring on Z2 ×Z2, which is not simple,
 but we knew that before.

 Of course all this operations can be applied to all nearrings.


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<address>SONATA-tutorial manual September 2025
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