<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><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP006.htm#SECT001">Direct products</a>
<li> <A HREF="CHAP006.htm#SECT002">Ideals and factors</a>
</ol><p>
<p>
An <strong>ideal</strong> of a nearring (<i>N</i>,+,*) is a subset <i>I</i> such that
<i>I</i> is a normal subgroup of (<i>N</i>,+), and
for all <i>i</i> ∈ <i>I</i>, <i>n</i>,<i>m</i> ∈ <i>N</i>, we have
(<i>m</i>+<i>i</i>)*<i>n</i> − <i>m</i>*<i>n</i> ∈ <i>I</i> and <i>n</i>*<i>i</i> ∈ <i>I</i>. Ideals are in
one-to-one correspondence to the congruence relations
on (<i>N</i>,+,*).
<p>
A <strong>right ideal</strong> of a nearring (<i>N</i>,+,*) is a subset <i>I</i> such that
<i>I</i> is a normal subgroup of (<i>N</i>,+), and
for all <i>i</i> ∈ <i>I</i>, <i>n</i>,<i>m</i> ∈ <i>N</i>, we have
(<i>m</i>+<i>i</i>)*<i>n</i> − <i>m</i>*<i>n</i> ∈ <i>I</i>. Right ideals are in
one-to-one correspondence to the congruence relations
on (<i>N</i>,+, { λ<sub><i>m</i></sub> | <i>m</i> ∈ <i>M</i> } ), where
λ<sub><i>m</i></sub> (<i>n</i>) : = <i>n</i>*<i>m</i>. Hence, right ideals
describe the congruences of the <i>N</i>-group
<i>N</i><sub><i>N</i></sub>.
<p>
A <strong>left ideal</strong> of a nearring (<i>N</i>,+,*) is a subset <i>I</i> such that
<i>I</i> is a normal subgroup of (<i>N</i>,+), and
for all <i>i</i> ∈ <i>I</i>, <i>n</i> ∈ <i>N</i>, we have
<i>n</i>*<i>i</i> ∈ <i>I</i>.
<p>
<p>
<h2><a name="SECT001">6.1 Direct products</a></h2>
<p><p>
For all sorts of nearrings direct products <i>A</i> ×<i>B</i> 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 <strong>not</strong> 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.
<p>
<p>
<h2><a name="SECT002">6.2 Ideals and factors</a></h2>
<p><p>
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 <font face="Gill Sans,Helvetica,Arial">GAP</font>
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 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 <b>Z</b><sub>2</sub> ×<b>Z</b><sub>2</sub>. 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 <b>Z</b><sub>2</sub> ×<b>Z</b><sub>2</sub>, which is not simple,
but we knew that before.
<p>
Of course all this operations can be applied to all nearrings.
<p>
<p>
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<P>
<address>SONATA-tutorial manual<br>September 2025
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