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      <h1>intclMonIdeal(Ideal) -- normalization of Rees algebra</h1>
      <ul>
        <li><span>Function: <a title="normalization of Rees algebra" href="_intcl__Mon__Ideal.html">intclMonIdeal</a></span></li>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">intclMonIdeal I</code></dd>
          </dl>
        </li>
        <li>Inputs:          <ul>
            <li><span><span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the leading monomials of the elements of the ideal are considered as generators of a monomial ideal</span></li>
          </ul>
        </li>
        <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:          <ul>
            <li><span><span class="tt">allComputations</span> (missing documentation)
<span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span></span></span></li>
            <li><span><span class="tt">grading</span> (missing documentation)
<span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the integral closure of the input ideal</span></li>
            <li><span><span>a <a title="class of monomial subalgebras" href="___Monomial__Subalgebra.html">monomial subalgebra</a></span>, the normalization of the Rees algebra of I</span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <p></p>
The leading monomials of the elements of I are considered as generators of a monomial ideal. This function computes the integral closure of I\subset R in the polynomial ring R[t] and the normalization of its Rees algebra. If f_1,\ldots,f_m are the monomial generators of I, then the normalization of the Rees algebra is the integral closure of K[f_1t,\ldots,f_nt] in R[t]. For a definition of the Rees algebra (or Rees ring) see Bruns and Herzog, Cohen-Macaulay rings, Cambridge University Press 1998, p. 182. The function returns the integral closure of the ideal I and the normalization of its Rees algebra. Since the Rees algebra is defined in a polynomial ring with an additional variable, the function creates a new polynomial ring with an additional variable.        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : R=ZZ/37[x,y];</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);

o2 : Ideal of R</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : (intCl,normRees)=intclMonIdeal I;</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : intCl

3     2   2    3
o4 = ideal (y , x*y , x y, x )

ZZ
o4 : Ideal of --[x..y, a]
              37</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i5 : normRees

o5 = MonomialSubalgebra{cache => CacheTable{...1...}                }
                                           3         2    2      3
                        generators => {y, y a, x, x*y a, x y*a, x a}
                                ZZ
                        ring => --[x..y, a]
                                37

ZZ
o5 : MonomialSubalgebra of --[x..y, a]
                           37</code></pre>
            </td>
          </tr>
        </table>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use this method:</h2>
          <ul>
            <li><span><a title="normalization of Rees algebra" href="_intcl__Mon__Ideal_lp__Ideal_rp.html">intclMonIdeal(Ideal)</a> -- normalization of Rees algebra</span></li>
          </ul>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Normaliz.m2:1851:0</span>.</p>
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