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    <title>rightKernel -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</title>
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o1 = B

o1 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : sigma = ncMap(B,B,{y,z,x})

o2 = NCRingMap B &lt;--- B

o2 : NCRingMap</code></pre>
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              <pre><code class="language-macaulay2">i3 : C = oreExtension(B,sigma,w)
--Calling Bergman for NCGB calculation.
Complete!

o3 = C

o3 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i4 : D = (ambient C)/(ideal C + ncIdeal{promote(w^2,ambient C)})
--Calling Bergman for NCGB calculation.
Complete!

o4 = D

o4 : NCQuotientRing</code></pre>
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          <p>This algebra is an Ore extension of a 3-dimensional Sklyanin algebra, factored by the normal regular element w^2. This algebra is Koszul, hence it has a linear free resolution. The rightKernel method is significantly faster than rightKernelBergman in this case. Also note the two methods return different generating sets for the kernel.</p>
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              <pre><code class="language-macaulay2">i5 : M1 = ncMatrix {{x,y,z,w}}

o5 = | x y z w |

o5 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i6 : M2 = rightKernel(M1,1)

o6 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i7 : M3 = rightKernel(M2,1)

o7 = | -y 0  w  0  0  0  0  0 |
     | -x 0  0  -w 0  0  0  0 |
     | z  -w 0  0  0  0  0  0 |
     | 0  z  -x -y 0  0  -w 0 |
     | 0  -y z  -x -w 0  0  0 |
     | 0  x  y  z  0  -w 0  0 |
     | 0  0  0  0  x  y  z  w |

o7 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i8 : rightKernelBergman(M2)
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!

o8 = | 0 0  0  0  y  0  -w 0  |
     | 0 0  0  0  x  w  0  0  |
     | 0 0  0  0  -z 0  0  w  |
     | 0 w  0  0  0  y  x  -z |
     | 0 0  0  w  0  x  -z y  |
     | 0 0  w  0  0  -z -y -x |
     | w -z -y -x 0  0  0  0  |

o8 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i9 : M4 = rightKernel(M3,1)

o9 = | -w 0  0  0  0  0  0  0 |
     | -y 0  w  0  0  0  0  0 |
     | -x 0  0  -w 0  0  0  0 |
     | z  -w 0  0  0  0  0  0 |
     | 0  z  -x -y 0  0  -w 0 |
     | 0  -y z  -x -w 0  0  0 |
     | 0  x  y  z  0  -w 0  0 |
     | 0  0  0  0  x  y  z  w |

o9 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i10 : rightKernelBergman(M3)
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!

o10 = | 0 0  0  0  0  0  0  w  |
      | 0 0  0  0  0  -w 0  y  |
      | 0 0  0  0  w  0  0  x  |
      | 0 0  0  0  0  0  w  -z |
      | 0 w  0  0  y  x  -z 0  |
      | 0 0  0  w  x  -z y  0  |
      | 0 0  w  0  -z -y -x 0  |
      | w -z -y -x 0  0  0  0  |

o10 : NCMatrix</code></pre>
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        <h2>See also</h2>
        <ul>
          <li><span><a title="Methods for computing kernels of matrices over noncommutative rings using Bergman" href="_right__Kernel__Bergman.html">rightKernelBergman</a> -- Methods for computing kernels of matrices over noncommutative rings using Bergman</span></li>
        </ul>
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        <div class="waystouse">
          <h2>Ways to use <span class="tt">rightKernel</span>:</h2>
          <ul>
            <li><kbd>rightKernel(NCMatrix,ZZ)</kbd></li>
          </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
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        <div class="waystouse">
          <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:2718:0</span>.</p>
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