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    <title>NCGroebnerBasis -- Type of a Groebner basis for an NCIdeal in an NCRing.</title>
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o1 = R

o1 : QuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : A = R {x,y,z}

o2 = A

o2 : NCPolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i3 : I = ncIdeal {a*x*y,b*z^2}

2
o3 = Two-sided ideal {axy, bz }

o3 : NCIdeal</code></pre>
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              <pre><code class="language-macaulay2">i4 : Igb = ncGroebnerBasis(I, InstallGB=>true)

2                2
o4 = bz ; Lead Term = (z , b)
     axy; Lead Term = (xy, a)

o4 : NCGroebnerBasis</code></pre>
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          <p>Note that after the InstallGB flag is set, no checking is done to ensure that the input is in fact a Groebner basis.</p>
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              <pre><code class="language-macaulay2">i5 : c*z^2 % Igb

2
o5 = cz

o5 : A</code></pre>
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              <pre><code class="language-macaulay2">i6 : b*z^2 % Igb

o6 = 0

o6 : A</code></pre>
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              <pre><code class="language-macaulay2">i7 : A = QQ{x,y,z}

o7 = A

o7 : NCPolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i8 : p = y*z + z*y - x^2

2
o8 = zy+yz-x

o8 : A</code></pre>
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              <pre><code class="language-macaulay2">i9 : q = x*z + z*x - y^2

2
o9 = zx-y +xz

o9 : A</code></pre>
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              <pre><code class="language-macaulay2">i10 : r = z^2 - x*y - y*x

2
o10 = z -yx-xy

o10 : A</code></pre>
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              <pre><code class="language-macaulay2">i11 : I = ncIdeal {p,q,r}

2      2      2
o11 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o11 : NCIdeal</code></pre>
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              <pre><code class="language-macaulay2">i12 : Igb = ncGroebnerBasis I
--Calling Bergman for NCGB calculation.
Complete!

2    2                2
o12 = y x-xy ; Lead Term = (y x, 1)
        2  2                  2
      yx -x y; Lead Term = (yx , 1)
          2
      zx-y +xz; Lead Term = (zx, 1)
             2
      zy+yz-x ; Lead Term = (zy, 1)
       2                      2
      z -yx-xy; Lead Term = (z , 1)

o12 : NCGroebnerBasis</code></pre>
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              <pre><code class="language-macaulay2">i13 : normalFormBergman(z^17,Igb)
--Calling Bergman for NF calculation for 1 elements.
Complete!
Writing bergman input file.
Writing bergman init file.

2             2    3           3     4         4     5       5     6     6     7   7     8 8
o13 = yxyxyxyxyxyxyxyxz+xyxyxyxyxyxyxyxyz+8x yxyxyxyxyxyxy z+8x yxyxyxyxyxy z+28x yxyxyxyxy z+28x yxyxyxy z+56x yxyxy z+56x yxy z+70x y z

o13 : A</code></pre>
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          <p>stuff</p>
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        <div class="waystouse">
          <h2>Methods that use an object of class NCGroebnerBasis:</h2>
          <ul>
            <li><span><a title="The list of algebra generators of an NCGroebnerBasis" href="_generators_lp__N__C__Groebner__Basis_rp.html">generators(NCGroebnerBasis)</a> -- The list of algebra generators of an NCGroebnerBasis</span></li>
            <li><span><kbd>isCentral(NCRingElement,NCGroebnerBasis)</kbd> -- see <span><a title="Determines if an element is central" href="_is__Central.html">isCentral</a> -- Determines if an element is central</span></span></li>
            <li><span><a title="Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis" href="___N__C__Matrix_sp_pc_sp__N__C__Groebner__Basis.html">NCMatrix % NCGroebnerBasis</a> -- Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis</span></li>
            <li><span><a title="Reduces a NCRingElement by a NCGroebnerBasis" href="___N__C__Ring__Element_sp_pc_sp__N__C__Groebner__Basis.html">NCRingElement % NCGroebnerBasis</a> -- Reduces a NCRingElement by a NCGroebnerBasis</span></li>
            <li><span><kbd>QQ % NCGroebnerBasis</kbd> -- see <span><a title="Reduces a NCRingElement by a NCGroebnerBasis" href="___N__C__Ring__Element_sp_pc_sp__N__C__Groebner__Basis.html">NCRingElement % NCGroebnerBasis</a> -- Reduces a NCRingElement by a NCGroebnerBasis</span></span></li>
            <li><span><kbd>ZZ % NCGroebnerBasis</kbd> -- see <span><a title="Reduces a NCRingElement by a NCGroebnerBasis" href="___N__C__Ring__Element_sp_pc_sp__N__C__Groebner__Basis.html">NCRingElement % NCGroebnerBasis</a> -- Reduces a NCRingElement by a NCGroebnerBasis</span></span></li>
            <li><span><kbd>NCRingMap NCGroebnerBasis</kbd> -- see <span><a title="Apply a ring map to the generators of an ideal" href="___N__C__Ring__Map_sp__N__C__Ideal.html">NCRingMap NCIdeal</a> -- Apply a ring map to the generators of an ideal</span></span></li>
            <li><span><kbd>normalFormBergman(List,NCGroebnerBasis)</kbd> -- see <span><a title="Calls Bergman for a normal form calculation" href="_normal__Form__Bergman.html">normalFormBergman</a> -- Calls Bergman for a normal form calculation</span></span></li>
            <li><span><kbd>normalFormBergman(NCRingElement,NCGroebnerBasis)</kbd> -- see <span><a title="Calls Bergman for a normal form calculation" href="_normal__Form__Bergman.html">normalFormBergman</a> -- Calls Bergman for a normal form calculation</span></span></li>
            <li><span><kbd>ring(NCGroebnerBasis)</kbd> -- see <span><a title="Returns the ring of an NCIdeal or NCGroebnerBasis" href="_ring_lp__N__C__Ideal_rp.html">ring(NCIdeal)</a> -- Returns the ring of an NCIdeal or NCGroebnerBasis</span></span></li>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Type of a Groebner basis for an NCIdeal in an NCRing." href="___N__C__Groebner__Basis.html">NCGroebnerBasis</a> is <span>a <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">HashTable</a> &lt; <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p>
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        <div class="waystouse">
          <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:1449:0</span>.</p>
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