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<div><h1>Module / Module -- quotient module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>M/N</tt></div>
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<li><span>Operator: <a href="__sl.html" title="a binary operator, usually used for division">/</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
<li><span><tt>N</tt>, <span>a <a href="___Module.html">module</a></span>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___List.html">list</a></span>, <span>a <a href="___Sequence.html">sequence</a></span>, <span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Vector.html">vector</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, The quotient module M/N of M</span></li>
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<div class="single"><h2>Description</h2>
<div>If N is an ideal, ring element, or list or sequence of ring elements (in the ring of M), then the quotient is by the submodule N*M of M.<p/>
If N is a submodule of M, or a list or sequence of submodules, or a vector, then the quotient is by these elements or submodules.<table class="examples"><tr><td><pre>i1 : R = ZZ/173[a..d]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = ker matrix{{a^3-a*c*d,a*b*c-b^3,a*b*d-b*c^2}}
o2 = image {3} | 0 b3-abc bc2-abd |
{3} | -c2+ad a3-acd 0 |
{3} | b2-ac 0 a3-acd |
3
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : M/a == M/(a*M)
o3 = true</pre>
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<tr><td><pre>i4 : M/M_0
o4 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 |)
{3} | -c2+ad a3-acd 0 | {3} | -c2+ad |
{3} | b2-ac 0 a3-acd | {3} | b2-ac |
3
o4 : R-module, subquotient of R</pre>
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<tr><td><pre>i5 : M/(R*M_0 + b*M)
o5 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 0 b4-ab2c b2c2-ab2d |)
{3} | -c2+ad a3-acd 0 | {3} | -c2+ad -bc2+abd a3b-abcd 0 |
{3} | b2-ac 0 a3-acd | {3} | b2-ac b3-abc 0 a3b-abcd |
3
o5 : R-module, subquotient of R</pre>
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<tr><td><pre>i6 : M/(M_0,a*M_1+M_2)
o6 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 ab3-a2bc+bc2-abd |)
{3} | -c2+ad a3-acd 0 | {3} | -c2+ad a4-a2cd |
{3} | b2-ac 0 a3-acd | {3} | b2-ac a3-acd |
3
o6 : R-module, subquotient of R</pre>
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<tr><td><pre>i7 : presentation oo
o7 = {5} | -1 0 -a3+acd |
{6} | 0 -a -c2+ad |
{6} | 0 -1 b2-ac |
3 3
o7 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_subquotient_spmodules.html" title="the way Macaulay2 represents modules">subquotient modules</a> -- the way Macaulay2 represents modules</span></li>
<li><span><a href="_presentation.html" title="presentation of a module or ring">presentation</a> -- presentation of a module or ring</span></li>
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