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<p><a id="X7FF13C7684E1122C" name="X7FF13C7684E1122C"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X7FF13C7684E1122C">4 <span class="Heading">Ideals and left ideals</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81E965A37A7EA22A">4.1 <span class="Heading">Left ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X814FEB578507E81C">4.1-1 LeftIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7AE9FAB479569BF9">4.1-2 StrongLeftIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X829DFD167A8D0D4A">4.1-3 IsLeftIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X83629803819C4A6F">4.2 <span class="Heading">Ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X879540527DA666C4">4.2-1 IsIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7EBF92377C5E417D">4.2-2 Ideals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X809F4B407D4BDE47">4.2-3 AsIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F0A2FBA87465560">4.2-4 IdealGeneratedBy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8721D11884A2CDAD">4.2-5 IntersectionOfTwoIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85A4F7FE7B627615">4.2-6 SumOfTwoIdeals</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8079CE3187FE380D">4.3 <span class="Heading">Sequences (left) ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X845E09BF86C4DD2E">4.3-1 LeftSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B9ED49481948B91">4.3-2 RightSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84C0A78F7B2845FD">4.3-3 IsLeftNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79EA70287B245D65">4.3-4 IsSimpleSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D930A7679D97788">4.3-5 IsRightNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81593E537B94350B">4.3-6 LeftNilpotentIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B6EB5A37EBFFB7D">4.3-7 RightNilpotentIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E9665EB79226E96">4.3-8 SmoktunowiczSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D503F497CB34B9D">4.3-9 Socle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FF15DAA78E08F0A">4.3-10 Annihilator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X876342AF7CF51C9B">4.4 <span class="Heading">Mutipermutation skew braces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E0053787EDFEAFB">4.4-1 SocleSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85AA85F57FF7BD73">4.4-2 MultipermutationLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X824956137F4CEF3C">4.4-3 IsMultipermutation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8616F73781699DC3">4.4-4 Fix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CE04CC57E82FD02">4.4-5 KernelOfLambda</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CE55DAF7CB85B89">4.4-6 Quotient</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7826660686D57FD6">4.5 <span class="Heading">Prime and semiprime ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82CDAD02845051FA">4.5-1 IsPrimeBrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X834AED5184F2B9AC">4.5-2 IsPrimeIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80D35A2880B39EB0">4.5-3 PrimeIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X820951168658A704">4.5-4 IsSemiprime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80961A4F7CBFBA0B">4.5-5 IsSemiprimeIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A8C53838192CEC3">4.5-6 SemiprimeIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D6E642D817352AF">4.5-7 BaerRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8571BC2F80364341">4.5-8 IsBaer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X856E8ABD7BCA81D5">4.5-9 WedderburnRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85F4D83079E1013A">4.5-10 SolvableSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86623B417F4F07FE">4.5-11 IsMinimalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X837D770278330FE0">4.5-12 MinimalIdeals</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Ideals and left ideals</span></h3>

<p>In this section we describe several functions related to ideals and left ideals of skew braces. References: <a href="chapBib_mj.html#biBMR3647970">[GV17]</a> and <a href="chapBib_mj.html#biBMR3763907">[SV18]</a>.</p>

<p><a id="X81E965A37A7EA22A" name="X81E965A37A7EA22A"></a></p>

<h4>4.1 <span class="Heading">Left ideals</span></h4>

<p>An left ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is a subgroup <span class="SimpleMath">\(I\)</span> of the additive group of <span class="SimpleMath">\(A\)</span> such that <span class="SimpleMath">\(\lambda_a(I)\subseteq I\)</span> for all <span class="SimpleMath">\(a\in A\)</span>.</p>

<p><a id="X814FEB578507E81C" name="X814FEB578507E81C"></a></p>

<h5>4.1-1 LeftIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the left ideals of the skew brace <var class="Arg">obj</var></p>

<p><a id="X7AE9FAB479569BF9" name="X7AE9FAB479569BF9"></a></p>

<h5>4.1-2 StrongLeftIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StrongLeftIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the left ideals of the skew brace <var class="Arg">obj</var> that are normal in the additive group of <span class="SimpleMath">\(A\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(24,12);</span>
<skew brace of size 24>
<span class="GAPprompt">gap></span> <span class="GAPinput">strong_left_ideals := StrongLeftIdeals(br);</span>
[ <left ideal in <skew brace of size 24>, (size 24)>,
  <left ideal in <skew brace of size 24>, (size 12)>,
  <left ideal in <skew brace of size 24>, (size 6)>,
  <left ideal in <skew brace of size 24>, (size 4)>,
  <left ideal in <skew brace of size 24>, (size 2)>,
  <left ideal in <skew brace of size 24>, (size 3)>,
  <left ideal in <skew brace of size 24>, (size 1)> ]
</pre></div>

<p><a id="X829DFD167A8D0D4A" name="X829DFD167A8D0D4A"></a></p>

<h5>4.1-3 IsLeftIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftIdeal</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the subset is a left ideal of <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(8,4);</span>
<brace of size 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">leftideals := LeftIdeals(br);</span>
[ <left ideal in <brace of size 8>, (size 1)>, <left ideal in <brace of size 8>, (size 2)>, 
<left ideal in <brace of size 8>, (size 4)>, 
<left ideal in <brace of size 8>, (size 8)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(leftideals, x->IsLeftIdeal(br, x));</span>
[ true, true, true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(leftideals, IdBrace);</span>
[ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ]
</pre></div>

<p><a id="X83629803819C4A6F" name="X83629803819C4A6F"></a></p>

<h4>4.2 <span class="Heading">Ideals</span></h4>

<p>An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is a normal subgroup <span class="SimpleMath">\(I\)</span> of the additive group of <span class="SimpleMath">\(A\)</span> such that <span class="SimpleMath">\(\lambda_a(I)\subseteq I\)</span> and <span class="SimpleMath">\(a\circ I=I\circ a\)</span> for all <span class="SimpleMath">\(a\in A\)</span>.</p>

<p><a id="X879540527DA666C4" name="X879540527DA666C4"></a></p>

<h5>4.2-1 IsIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIdeal</code>( <var class="Arg">obj</var>, <var class="Arg">subset</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the <var class="Arg">subset</var> is a left ideal of <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(8,4);</span>
<brace of size 8> 
<span class="GAPprompt">gap></span> <span class="GAPinput">leftideals := LeftIdeals(br);</span>
[ <left ideal in <brace of size 8>, (size 1)>, 
<left ideal in <brace of size 8>, (size 2)>,
<left ideal in <brace of size 8>, (size 4)>, 
<left ideal in <brace of size 8>, (size 8)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(leftideals, x->IsLeftIdeal(br, x));</span>
[ true, true, true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(leftideals, IdBrace);</span>
[ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ]
</pre></div>

<p><a id="X7EBF92377C5E417D" name="X7EBF92377C5E417D"></a></p>

<h5>4.2-2 Ideals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ideals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the ideals of the skew brace <var class="Arg">obj</var></p>

<p><a id="X809F4B407D4BDE47" name="X809F4B407D4BDE47"></a></p>

<h5>4.2-3 AsIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsIdeal</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><a id="X7F0A2FBA87465560" name="X7F0A2FBA87465560"></a></p>

<h5>4.2-4 IdealGeneratedBy</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdealGeneratedBy</code>( <var class="Arg">obj</var>, <var class="Arg">subset</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the ideal of <var class="Arg">obj</var> generated by the given <var class="Arg">subset</var></p>

<p>The ideal of a skew brace <span class="SimpleMath">\(A\)</span> generated by a subset <span class="SimpleMath">\(X\)</span> is the intersection of all the ideals of <span class="SimpleMath">\(A\)</span> containing <span class="SimpleMath">\(X\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsList(br);</span>
[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)>, <(1,4)(2,5)(3,6)>, 
  <(1,5,3,4,2,6)>, <(1,6,2,4,3,5)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdealGeneratedBy(br, [last[2]]);</span>
<ideal in <brace of size 6>, (size 3)>
</pre></div>

<p><a id="X8721D11884A2CDAD" name="X8721D11884A2CDAD"></a></p>

<h5>4.2-5 IntersectionOfTwoIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionOfTwoIdeals</code>( <var class="Arg">ideal1</var>, <var class="Arg">ideal2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the intersection of <var class="Arg">ideal1</var> and <var class="Arg">ideal2</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ideals(br);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionOfTwoIdeals(last[2],last[3]);</span>
<ideal in <brace of size 6>, (size 1)>
</pre></div>

<p><a id="X85A4F7FE7B627615" name="X85A4F7FE7B627615"></a></p>

<h5>4.2-6 SumOfTwoIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SumOfTwoIdeals</code>( <var class="Arg">ideal1</var>, <var class="Arg">ideal2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the sum of <var class="Arg">ideal1</var> and <var class="Arg">ideal2</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ideals(br);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SumOfTwoIdeals(last[2],last[3]);</span>
<ideal in <brace of size 6>, (size 6)>
</pre></div>

<p><a id="X8079CE3187FE380D" name="X8079CE3187FE380D"></a></p>

<h4>4.3 <span class="Heading">Sequences (left) ideals</span></h4>

<p><a id="X845E09BF86C4DD2E" name="X845E09BF86C4DD2E"></a></p>

<h5>4.3-1 LeftSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the left ideals of the left series of <var class="Arg">obj</var></p>

<p>The left series of a skew brace <span class="SimpleMath">\(A\)</span> is defined recursively as <span class="SimpleMath">\(A^1=A\)</span> and <span class="SimpleMath">\(A^{n+1}=A*A^n\)</span> for <span class="SimpleMath">\(n\geq1\)</span>, where <span class="SimpleMath">\(a*b=\lambda_a(b)-b\)</span>. Each <span class="SimpleMath">\(A^n\)</span> is a left ideal.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(8,20);</span>
<skew brace of size 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSeries(br);</span>
[ <skew brace of size 8>, 
<left ideal in <skew brace of size 8>, (size 2)>, 
<left ideal in <skew brace of size 8>, (size 1)> ]
</pre></div>

<p><a id="X7B9ED49481948B91" name="X7B9ED49481948B91"></a></p>

<h5>4.3-2 RightSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the ideals of the right series of <var class="Arg">obj</var></p>

<p>The right series of a skew brace 0<span class="SimpleMath">\(A\)</span> is defined recursively as <span class="SimpleMath">\(A^{(1)}=A\)</span> and <span class="SimpleMath">\(A^{(n+1)}=A*A^{(n)}\)</span> for <span class="SimpleMath">\(n\geq1\)</span>, where <span class="SimpleMath">\(a*b=\lambda_a(b)-b\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(8,20);</span>
<skew brace of size 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">RightSeries(br);</span>
[ <ideal in <skew brace of size 8>, (size 8)>, 
<ideal in <skew brace of size 8>, (size 2)>, 
<ideal in <skew brace of size 8>, (size 1)> ]
</pre></div>

<p><a id="X84C0A78F7B2845FD" name="X84C0A78F7B2845FD"></a></p>

<h5>4.3-3 IsLeftNilpotent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftNilpotent</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is left nilpotent.</p>

<p>A skew brace <span class="SimpleMath">\(A\)</span> is said to be left nilpotent if there exists <span class="SimpleMath">\(n\geq1\)</span> such that <span class="SimpleMath">\(A^n=0\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftNilpotent(SmallBrace(8,18));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftNilpotent(SmallBrace(12,2));</span>
false
</pre></div>

<p><a id="X79EA70287B245D65" name="X79EA70287B245D65"></a></p>

<h5>4.3-4 IsSimpleSkewbrace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimpleSkewbrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is simple.</p>

<p>A skew brace <span class="SimpleMath">\(A\)</span> is said to be simple if <span class="SimpleMath">\(\{0\}\)</span> and <span class="SimpleMath">\(A\)</span> are its only ideals.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimple(SmallSkewbrace(12,22));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimple(SmallSkewbrace(12,21));</span>
false
</pre></div>

<p><a id="X7D930A7679D97788" name="X7D930A7679D97788"></a></p>

<h5>4.3-5 IsRightNilpotent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightNilpotent</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is right nilpotent.</p>

<p>A skew brace <span class="SimpleMath">\(A\)</span> is said to be right nilpotent if there exists <span class="SimpleMath">\(n\geq1\)</span> such that <span class="SimpleMath">\(A^{(n)}=0\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightNilpotent(SmallBrace(8,18));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightNilpotent(SmallBrace(12,2));</span>
true
</pre></div>

<p><a id="X81593E537B94350B" name="X81593E537B94350B"></a></p>

<h5>4.3-6 LeftNilpotentIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftNilpotentIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of right or left nilpotent ideals of <var class="Arg">obj</var></p>

<p>An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is said to be left if it is left nilpotent as a skew brace.</p>

<p><a id="X7B6EB5A37EBFFB7D" name="X7B6EB5A37EBFFB7D"></a></p>

<h5>4.3-7 RightNilpotentIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightNilpotentIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of right or left nilpotent ideals of <var class="Arg">obj</var></p>

<p>An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is said to be right nilpotent if An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is said to be left if it is right nilpotent as a skew brace.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(8,18);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftNilpotent(br);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightNilpotent(br);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(LeftNilpotentIdeals(br));</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(RightNilpotentIdeals(br));</span>
2
</pre></div>

<p><a id="X7E9665EB79226E96" name="X7E9665EB79226E96"></a></p>

<h5>4.3-8 SmoktunowiczSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmoktunowiczSeries</code>( <var class="Arg">obj</var>, <var class="Arg">bound</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list of <var class="Arg">bound</var> left ideals of the Smoktunowicz's series of obj



<p>The Smoktunowicz's series of a skew brace \(A\) is defined recursively as \(A^{[1]}=A\) and \(A^{[n+1]}\) is the additive subgroup of \(A\) generated by \(A^{[i]}*A^{[n+1-i]}\) for \(1\leq i+j\leq n+1\), where \(a*b=\lambda_a(b)-b\).




<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(16,145);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmoktunowiczSeries(br,4);</span>
[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
  <brace of size 2> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SmoktunowiczSeries(br,5);</span>
[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
  <brace of size 2>, <brace of size 1> ]
</pre></div>

<p><a id="X7D503F497CB34B9D" name="X7D503F497CB34B9D"></a></p>

<h5>4.3-9 Socle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Socle</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the socle of <var class="Arg">obj</var></p>

<p>The socle of a skew brace <span class="SimpleMath">\(A\)</span> is the ideal <span class="SimpleMath">\(\ker\lambda\cap Z(A,+)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Socle(SmallSkewbrace(6,2));</span>
<ideal in <skew brace of size 6>, (size 1)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Socle(SmallBrace(8,20));</span>
<ideal in <brace of size 8>, (size 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Socle(SmallBrace(8,2));</span>
<ideal in <brace of size 8>, (size 4)>
</pre></div>

<p><a id="X7FF15DAA78E08F0A" name="X7FF15DAA78E08F0A"></a></p>

<h5>4.3-10 Annihilator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Annihilator</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the annihilator of <var class="Arg">obj</var></p>

<p>The socle of a skew brace <span class="SimpleMath">\(A\)</span> is the ideal <span class="SimpleMath">\(\ker\lambda\cap Z(A,+)\cap Z(A,\circ)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Annihilator(SmallSkewbrace(8,12));</span>
<ideal in <brace of size 8>, (size 2)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Annihilator(SmallSkewbrace(4,2));</span>
<ideal in <skew brace of size 4>, (size 2)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Annihilator(SmallSkewbrace(8,14));</span>
<ideal in <brace of size 8>, (size 4)>
</pre></div>

<p><a id="X876342AF7CF51C9B" name="X876342AF7CF51C9B"></a></p>

<h4>4.4 <span class="Heading">Mutipermutation skew braces</span></h4>

<p><a id="X7E0053787EDFEAFB" name="X7E0053787EDFEAFB"></a></p>

<h5>4.4-1 SocleSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SocleSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the socle series of <var class="Arg">obj</var></p>

<p>The socle series of a skew brace <span class="SimpleMath">\(A\)</span> is defined recursively as <span class="SimpleMath">\(A_1=A\)</span> and <span class="SimpleMath">\(A_{n+1}=A_n/\mathrm{Soc}(A_n)\)</span>, see <a href="chapBib_mj.html#biBMR3763907">[SV18]</a>.</p>

<p><a id="X85AA85F57FF7BD73" name="X85AA85F57FF7BD73"></a></p>

<h5>4.4-2 MultipermutationLevel</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultipermutationLevel</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the multipermutation level of the skew brace <var class="Arg">obj</var></p>

<p>The multipermutation level of a skew brace <span class="SimpleMath">\(A\)</span> is defined as the smallest positive integer <span class="SimpleMath">\(n\)</span> such that the <span class="SimpleMath">\(n\)</span>-th term <span class="SimpleMath">\(A_n\)</span> of the socle series has only one element, see Definition 5.17 of <a href="chapBib_mj.html#biBMR3763907">[SV18]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(8,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SocleSeries(br);</span>
[ <brace of size 8>, <brace of size 1> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MultipermutationLevel(br);</span>
2
</pre></div>

<p><a id="X824956137F4CEF3C" name="X824956137F4CEF3C"></a></p>

<h5>4.4-3 IsMultipermutation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMultipermutation</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> has finite multipermutation level and <var class="Arg">false</var> otherwise</p>

<p><a id="X8616F73781699DC3" name="X8616F73781699DC3"></a></p>

<h5>4.4-4 Fix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fix</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the left ideal <span class="SimpleMath">\(\{x\in A:\lambda_a(x)=x\;\forall a\in A\}\)</span> of the skew brace <span class="SimpleMath">\(A\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTrivialSkewbrace(br);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Fix(br);</span>
[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)>, <(1,4)(2,6)(3,5)>,
  <(1,5)(2,4)(3,6)>, <(1,6)(2,5)(3,4)> ]
</pre></div>

<p><a id="X7CE04CC57E82FD02" name="X7CE04CC57E82FD02"></a></p>

<h5>4.4-5 KernelOfLambda</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelOfLambda</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the kernel of the map <span class="SimpleMath">\(\lambda\)</span> as a subset of elements of the skew brace <var class="Arg">obj</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(6,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">KernelOfLambda(br);</span>
[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)> ]
</pre></div>

<p><a id="X7CE55DAF7CB85B89" name="X7CE55DAF7CB85B89"></a></p>

<h5>4.4-6 Quotient</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Quotient</code>( <var class="Arg">obj</var>, <var class="Arg">ideal</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the quotient <var class="Arg">obj</var> by <var class="Arg">ideal</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(8,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ideals := Ideals(br);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Quotient(br, ideals[3]);</span>
<brace of size 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">br/ideals[3];</span>
<brace of size 4>
</pre></div>

<p><a id="X7826660686D57FD6" name="X7826660686D57FD6"></a></p>

<h4>4.5 <span class="Heading">Prime and semiprime ideals</span></h4>

<p><a id="X82CDAD02845051FA" name="X82CDAD02845051FA"></a></p>

<h5>4.5-1 IsPrimeBrace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimeBrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is prime</p>

<p>A skew brace <span class="SimpleMath">\(A\)</span> is said to be prime if for all non-zero ideals <span class="SimpleMath">\(I\)</span> and <span class="SimpleMath">\(J\)</span> one has <span class="SimpleMath">\(I*J\ne 0\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeBrace(SmallBrace(24,12));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeBrace(SmallBrace(24,94));</span>
true
</pre></div>

<p><a id="X834AED5184F2B9AC" name="X834AED5184F2B9AC"></a></p>

<h5>4.5-2 IsPrimeIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimeIdeal</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the ideal <var class="Arg">obj</var> is prime</p>

<p>An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is said to be prime if <span class="SimpleMath">\(A/I\)</span> is a prime skew brace.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(24,94);</span>
<brace of size 24>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeBrace(br);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Ideals(br);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeIdeal(last[2]);</span>
true
</pre></div>

<p><a id="X80D35A2880B39EB0" name="X80D35A2880B39EB0"></a></p>

<h5>4.5-3 PrimeIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimeIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of prime ideals of the skew brace <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(PrimeIdeals(SmallBrace(24,94)));</span>
2
</pre></div>

<p><a id="X820951168658A704" name="X820951168658A704"></a></p>

<h5>4.5-4 IsSemiprime</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiprime</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is semiprime</p>

<p>An ideal <span class="SimpleMath">\(I\)</span> of a skew brace <span class="SimpleMath">\(A\)</span> is said to be semiprime if <span class="SimpleMath">\(A/I\)</span> is a semiprime skew brace.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := DirectProductSkewbraces(SmallSkewbrace(12,22),SmallSkewbrace(12,22));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemiprime(br);</span>
true
</pre></div>

<p><a id="X80961A4F7CBFBA0B" name="X80961A4F7CBFBA0B"></a></p>

<h5>4.5-5 IsSemiprimeIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiprimeIdeal</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the ideal <var class="Arg">obj</var> is semiprime</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemiprimeIdeals(SmallSkewbrace(12,24));</span>
[ <ideal in <skew brace of size 12>, (size 12)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemiprimeIdeal(last[1]);</span>
true
</pre></div>

<p><a id="X7A8C53838192CEC3" name="X7A8C53838192CEC3"></a></p>

<h5>4.5-6 SemiprimeIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemiprimeIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of semiprime ideals of the skew brace <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemiprimeIdeals(SmallSkewbrace(12,24));</span>
[ <ideal in <skew brace of size 12>, (size 12)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(SemiprimeIdeals(SmallSkewbrace(12,22)));</span>
2
</pre></div>

<p><a id="X7D6E642D817352AF" name="X7D6E642D817352AF"></a></p>

<h5>4.5-7 BaerRadical</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaerRadical</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the Baer radical of the skew brace <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BaerRadical(br);</span>
<ideal in <skew brace of size 6>, (size 6)>
</pre></div>

<p><a id="X8571BC2F80364341" name="X8571BC2F80364341"></a></p>

<h5>4.5-8 IsBaer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBaer</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is ia Baer radical skew brace.</p>

<p>A skew brace <span class="SimpleMath">\(A\)</span> is said to be Baer radical if <span class="SimpleMath">\(A=B(A)\)</span>, where <span class="SimpleMath">\(B(A)\)</span> is the Baer radical of <span class="SimpleMath">\(A\)</span> (i.e., the intersection of all prime ideals of <span class="SimpleMath">\(A\)</span>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBaer(br);</span>
true
</pre></div>

<p><a id="X856E8ABD7BCA81D5" name="X856E8ABD7BCA81D5"></a></p>

<h5>4.5-9 WedderburnRadical</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WedderburnRadical</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the Wedderburn radical of the skew brace <var class="Arg">obj</var></p>

<p>The Wedderburn radical of a skew brace is the intersection of all its prime ideals</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(6,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnRadical(br);</span>
<ideal in <skew brace of size 6>, (size 3)>
</pre></div>

<p><a id="X85F4D83079E1013A" name="X85F4D83079E1013A"></a></p>

<h5>4.5-10 SolvableSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SolvableSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the solvable series of the skew brace <var class="Arg">obj</var></p>

<p>The solvable series of a skew brace <span class="SimpleMath">\(A\)</span> is defined recursively as <span class="SimpleMath">\(A_{1}=A\)</span> and <span class="SimpleMath">\(A_{n+1}=A_{n}*A_{n}\)</span> for <span class="SimpleMath">\(n\geq1\)</span>, where <span class="SimpleMath">\(a*b=\lambda_a(b)-b\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(8,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSolvable(br);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SolvableSeries(br);</span>
[ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(12,23);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSolvable(br);</span>
false
</pre></div>

<p><a id="X86623B417F4F07FE" name="X86623B417F4F07FE"></a></p>

<h5>4.5-11 IsMinimalIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMinimalIdeal</code>( <var class="Arg">obj</var>, <var class="Arg">ideal</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if <var class="Arg">ideal</var> is a minimal ideal of <var class="Arg">obj</var> An ideal <span class="SimpleMath">\(I\)</span> of <span class="SimpleMath">\(A\)</span> is said to be <em>minimal</em> if does not contain any other ideal of <span class="SimpleMath">\(A\)</span>. To check if an ideal <span class="SimpleMath">\(I\)</span> of <span class="SimpleMath">\(A\)</span> is minimal, one computes the ideals of <span class="SimpleMath">\(I\)</span> and keep only those that are simple as a skew brace.</p>

<p><a id="X837D770278330FE0" name="X837D770278330FE0"></a></p>

<h5>4.5-12 MinimalIdeals</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of minimal ideals of the skew brace <var class="Arg">obj</var></p>


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