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<p>&nbsp;<p>
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<h2><a name="Module"></a>
Module
</h2>

Creates a module over an order.

<p><p>

<h3>
Syntax:
</h3>
<pre>M1 := Module([IL,] M);
M1 := Module(EL);</pre>

<p>

<table>
<tr>
<td>module</td>
<td><pre>  M1  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  IL  </pre></td>
<td>of ideals over a maximal order O</td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td>of algebraic elements over O</td>
</tr>
<tr>
<td>list</td>
<td><pre>  EL  </pre></td>
<td>of relative algebraic elements of o</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#ModuleIdeals">ModuleIdeals</a>, <a href="kashref.html#ModuleMatrix">ModuleMatrix</a>, <a href="kashref.html#ModuleOrder">ModuleOrder</a>

<p>

<h3>
Description:
</h3>
Let \a|1,  &hellip;  ,\a|m be fractional O-ideals (the
list <tt> IL</tt>, they are called the coefficient ideals) and
A|1,  &hellip;  ,A|m be the columns of the matrix M
of dimension n over the
quotient field of O. This function creates a pseudomatrix
which is a representation
of the O-module
M|1 = A|1 \a|1 + cdots + A|m \a|m.

If the list of coefficient ideals (<tt> IL</tt>) is omitted, the
module with trivial coefficient ideals ( 1O) will be created.

If <tt> Module</tt> is invoked with a list of relative algebraic elements,
the representations of those in a relative order
o as column vectors over O, the coefficient order of
o. Again the ideals are assumed to be trivial.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> O:=OrderMaximal(Poly(Zx,[1, 5, -6, -53, 3, 206, 244]));;
kash> IL:=List(Factor(2*O),f->f[1]);;
kash> List(IL,IdealGenerators);
[ [ 2, [0, 2, 0, 3, 0, 2] ], [ 2, [0, 0, 3, 0, 1, 2] ], 
  [ 2, [3, 3, 1, 2, 0, 0] ] ]
kash> EL1:=List(Factor(5*O),f->IdealGen(f[1],2));
[ [12, 17, 15, 7, 9, 5], [19, 18, 15, 1, 10, 24], [4, 22, 4, 10, 23, 4] ]
kash> EL2:=List(Factor(13*O),f->IdealGen(f[1],2));
[ [10, 8, 0, 1, 0, 0], [1, 12, 5, 1, 0, 0] ]
kash> EL2[3]:=IdealGen(Factor(7*O)[1][1],2);
[1, 1, 0, 0, 0, 0]
kash> EL3:=List(Factor(3*O),f->IdealGen(f[1],2));
[ [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0] ]
kash> EL3[3]:=IdealGen(Factor(7*O)[2][1],2);
[3, 1, 0, 0, 0, 0]
kash> M:=Module(IL,Mat(O,[EL1,EL2,EL3]));
> {<2, [0, 2, 0, 3, 0, 2]><2, [0, 0, 3, 0, 1, 2]><2, [3, 3, 1, 2, 0, 0]>
[[12, 17, 15, 7, 9, 5] [19, 18, 15, 1, 10, 24] [4, 22, 4, 10, 23, 4]]
[[10, 8, 0, 1, 0, 0] [1, 12, 5, 1, 0, 0] [1, 1, 0, 0, 0, 0]]
[[1, 1, 0, 0, 0, 0] [1, 0, 1, 0, 0, 0] [3, 1, 0, 0, 0, 0]]
}


</pre>

<p>


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