<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>Ideal / Ideal -- quotient module</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Ideal_sp^_sp__Z__Z.html">next</a> | <a href="___Ideal_sp_sl_sp__Function.html">previous</a> | <a href="___Ideal_sp^_sp__Z__Z.html">forward</a> | <a href="___Ideal_sp_sl_sp__Function.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>Ideal / Ideal -- 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>I/J</tt></div> </dd></dl> </div> </li> <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>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span></span></li> <li><span><tt>J</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, in the same ring as <tt>I</tt></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, The quotient module <tt>(I+J)/J</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : I = ideal vars R o2 = ideal (a, b, c) o2 : Ideal of R</pre> </td></tr> <tr><td><pre>i3 : M = I / I^2 o3 = subquotient (| a b c |, | a2 ab ac b2 bc c2 |) 1 o3 : R-module, subquotient of R</pre> </td></tr> </table> There is a diffference between typing I/J and (I+J)/J in Macaulay2, although conceptually they are the same module. The former has as its generating set the generators of I, while the latter has as its (redundant) generators the generators of I and J. Generally, the former method is preferable.<table class="examples"><tr><td><pre>i4 : gens M o4 = | a b c | 1 3 o4 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i5 : N = (I + I^2)/I^2 o5 = subquotient (| a b c a2 ab ac b2 bc c2 |, | a2 ab ac b2 bc c2 |) 1 o5 : R-module, subquotient of R</pre> </td></tr> <tr><td><pre>i6 : gens N o6 = | a b c a2 ab ac b2 bc c2 | 1 9 o6 : Matrix R <--- R</pre> </td></tr> </table> </div> </div> <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="_generators.html" title="provide matrix or list of generators">generators</a> -- provide matrix or list of generators</span></li> </ul> </div> </div> </body> </html>