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    <title>NCRing -- Type of a noncommutative ring</title>
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o1 = A

o1 : NCPolynomialRing</code></pre>
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          <p>Then input the defining relations, and put them in an ideal:</p>
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              <pre><code class="language-macaulay2">i2 : f = y*z + z*y - x^2

2
o2 = zy+yz-x

o2 : A</code></pre>
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              <pre><code class="language-macaulay2">i3 : g = x*z + z*x - y^2

2
o3 = zx-y +xz

o3 : A</code></pre>
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              <pre><code class="language-macaulay2">i4 : h = z^2 - x*y - y*x

2
o4 = z -yx-xy

o4 : A</code></pre>
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              <pre><code class="language-macaulay2">i5 : I=ncIdeal{f,g,h}

2      2      2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o5 : NCIdeal</code></pre>
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          <p>Next, define the quotient ring (as well as try a few functions on this new ring).  Note that when the quotient ring is defined, a call is made to Bergman to compute the Groebner basis of I (out to a certain degree, should the Groebner basis be infinite).</p>
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              <pre><code class="language-macaulay2">i6 : B=A/I
--Calling Bergman for NCGB calculation.
Complete!

o6 = B

o6 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i7 : generators B

o7 = {x, y, z}

o7 : List</code></pre>
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              <pre><code class="language-macaulay2">i8 : numgens B

o8 = 3</code></pre>
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              <pre><code class="language-macaulay2">i9 : isCommutative B

o9 = false</code></pre>
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              <pre><code class="language-macaulay2">i10 : coefficientRing B

o10 = QQ

o10 : Ring</code></pre>
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          <p>As we can see, x is an element of B.</p>
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              <pre><code class="language-macaulay2">i11 : x

o11 = x

o11 : B</code></pre>
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          <p>If we define a new ring containing x, x is now part of that new ring</p>
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              <pre><code class="language-macaulay2">i12 : C = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
--Calling Bergman for NCGB calculation.
Complete!

o12 = C

o12 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i13 : x

o13 = x

o13 : C</code></pre>
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          <p>We can 'go back' to B using the command <a title="Brings the variables of a particular NCRing in scope" href="_use_lp__N__C__Ring_rp.html">use(NCRing)</a>.</p>
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              <pre><code class="language-macaulay2">i14 : use B

o14 = B

o14 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i15 : x

o15 = x

o15 : B</code></pre>
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              <pre><code class="language-macaulay2">i16 : use C

o16 = C

o16 : NCQuotientRing</code></pre>
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          <p>We can also create an Ore extension.  First define a <a title="Type of a map to or from a noncommutative ring." href="___N__C__Ring__Map.html">NCRingMap</a> with <a title="Make a map to or from an NCRing" href="_nc__Map.html">ncMap</a>.</p>
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              <pre><code class="language-macaulay2">i17 : sigma = ncMap(C,C,{y,z,w,x})

o17 = NCRingMap C &lt;--- C

o17 : NCRingMap</code></pre>
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          <p>Then call the command <a title="Creates an Ore extension of a noncommutative ring" href="_ore__Extension.html">oreExtension</a>.</p>
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              <pre><code class="language-macaulay2">i18 : D = oreExtension(C,sigma,a)
--Calling Bergman for NCGB calculation.
Complete!

o18 = D

o18 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i19 : generators D

o19 = {x, y, z, w, a}

o19 : List</code></pre>
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              <pre><code class="language-macaulay2">i20 : numgens D

o20 = 5</code></pre>
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        <h2>See also</h2>
        <ul>
          <li><span><a href="___Basic_spoperations_spon_spnoncommutative_spalgebras.html">Basic operations on noncommutative algebras</a></span></li>
        </ul>
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        <div class="waystouse">
          <h2>Types of <span class="tt">NCRing</span>:</h2>
          <ul>
            <li><span><a title="Type of a noncommutative polynomial ring" href="___N__C__Polynomial__Ring.html">NCPolynomialRing</a> -- Type of a noncommutative polynomial ring</span></li>
            <li><span><a title="Type of a noncommutative ring" href="___N__C__Quotient__Ring.html">NCQuotientRing</a> -- Type of a noncommutative ring</span></li>
          </ul>
          <h2>Functions and methods returning an object of class NCRing:</h2>
          <ul>
            <li><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></li>
            <li><span><a title="Returns the ring of an NCLeftIdeal" href="_ring_lp__N__C__Left__Ideal_rp.html">ring(NCLeftIdeal)</a> -- Returns the ring of an NCLeftIdeal</span></li>
            <li><span><a title="Gives the ring of the NCMatrix" href="_ring_lp__N__C__Matrix_rp.html">ring(NCMatrix)</a> -- Gives the ring of the NCMatrix</span></li>
            <li><span><a title="Returns the ring of an NCRightIdeal" href="_ring_lp__N__C__Right__Ideal_rp.html">ring(NCRightIdeal)</a> -- Returns the ring of an NCRightIdeal</span></li>
            <li><span><a title="Returns the NCRing of an NCRingElement" href="_ring_lp__N__C__Ring__Element_rp.html">ring(NCRingElement)</a> -- Returns the NCRing of an NCRingElement</span></li>
          </ul>
          <h2>Methods that use an object of class NCRing:</h2>
          <ul>
            <li><span><a title="Returns a basis of an NCRing in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Ring_rp.html">basis(ZZ,NCRing)</a> -- Returns a basis of an NCRing in a particular degree.</span></li>
            <li><span><kbd>centralElements(NCRing,ZZ)</kbd> -- see <span><a title="Finds central elements in a given degree" href="_central__Elements.html">centralElements</a> -- Finds central elements in a given degree</span></span></li>
            <li><span><a title="Returns the base ring of an NCRing" href="_coefficient__Ring_lp__N__C__Ring_rp.html">coefficientRing(NCRing)</a> -- Returns the base ring of an NCRing</span></li>
            <li><span><kbd>envelopingAlgebra(NCRing,Symbol)</kbd> -- see <span><a title="Create the enveloping algebra" href="_enveloping__Algebra.html">envelopingAlgebra</a> -- Create the enveloping algebra</span></span></li>
            <li><span><kbd>freeProduct(NCRing,NCRing)</kbd> -- see <span><a title="Define the free product of two algebras" href="_free__Product.html">freeProduct</a> -- Define the free product of two algebras</span></span></li>
            <li><span><a title="The list of algebra generators of an NCRing" href="_generators_lp__N__C__Ring_rp.html">generators(NCRing)</a> -- The list of algebra generators of an NCRing</span></li>
            <li><span><a title="Computes the Hilbert series of an NCRing" href="_hilbert__Series_lp__N__C__Ring_rp.html">hilbertSeries(NCRing)</a> -- Computes the Hilbert series of an NCRing</span></li>
            <li><span><a title="Returns whether an NCRing is commutative" href="_is__Commutative_lp__N__C__Ring_rp.html">isCommutative(NCRing)</a> -- Returns whether an NCRing is commutative</span></li>
            <li><span><kbd>isExterior(NCRing)</kbd> -- see <span><a title="Returns whether an NCRing is commutative" href="_is__Commutative_lp__N__C__Ring_rp.html">isCommutative(NCRing)</a> -- Returns whether an NCRing is commutative</span></span></li>
            <li><span><kbd>isHomogeneous(NCRing)</kbd> -- see <span><a title="Determines whether the input defines a homogeneous object" href="_is__Homogeneous_lp__N__C__Ideal_rp.html">isHomogeneous(NCIdeal)</a> -- Determines whether the input defines a homogeneous object</span></span></li>
            <li><span><kbd>ncMap(NCRing,NCRing,List)</kbd> -- see <span><a title="Make a map to or from an NCRing" href="_nc__Map.html">ncMap</a> -- Make a map to or from an NCRing</span></span></li>
            <li><span><kbd>ncMap(NCRing,Ring,List)</kbd> -- see <span><a title="Make a map to or from an NCRing" href="_nc__Map.html">ncMap</a> -- Make a map to or from an NCRing</span></span></li>
            <li><span><kbd>ncMap(Ring,NCRing,List)</kbd> -- see <span><a title="Make a map to or from an NCRing" href="_nc__Map.html">ncMap</a> -- Make a map to or from an NCRing</span></span></li>
            <li><span><kbd>ncMatrix(NCRing,List,List)</kbd> -- see <span><a title="Create an NCMatrix" href="_nc__Matrix.html">ncMatrix</a> -- Create an NCMatrix</span></span></li>
            <li><span><a title="The number of algebra generators of an NCRing" href="_numgens_lp__N__C__Ring_rp.html">numgens(NCRing)</a> -- The number of algebra generators of an NCRing</span></li>
            <li><span><kbd>oppositeRing(NCRing)</kbd> -- see <span><a title="Creates the opposite ring of a noncommutative ring" href="_opposite__Ring.html">oppositeRing</a> -- Creates the opposite ring of a noncommutative ring</span></span></li>
            <li><span><kbd>oreExtension(NCRing,NCRingMap,NCRingElement)</kbd> -- see <span><a title="Creates an Ore extension of a noncommutative ring" href="_ore__Extension.html">oreExtension</a> -- Creates an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement)</kbd> -- see <span><a title="Creates an Ore extension of a noncommutative ring" href="_ore__Extension.html">oreExtension</a> -- Creates an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreExtension(NCRing,NCRingMap,NCRingMap,Symbol)</kbd> -- see <span><a title="Creates an Ore extension of a noncommutative ring" href="_ore__Extension.html">oreExtension</a> -- Creates an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreExtension(NCRing,NCRingMap,Symbol)</kbd> -- see <span><a title="Creates an Ore extension of a noncommutative ring" href="_ore__Extension.html">oreExtension</a> -- Creates an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreIdeal(NCRing,NCRingMap,NCRingElement)</kbd> -- see <span><a title="Creates the defining ideal of an Ore extension of a noncommutative ring" href="_ore__Ideal.html">oreIdeal</a> -- Creates the defining ideal of an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement)</kbd> -- see <span><a title="Creates the defining ideal of an Ore extension of a noncommutative ring" href="_ore__Ideal.html">oreIdeal</a> -- Creates the defining ideal of an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreIdeal(NCRing,NCRingMap,NCRingMap,Symbol)</kbd> -- see <span><a title="Creates the defining ideal of an Ore extension of a noncommutative ring" href="_ore__Ideal.html">oreIdeal</a> -- Creates the defining ideal of an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>oreIdeal(NCRing,NCRingMap,Symbol)</kbd> -- see <span><a title="Creates the defining ideal of an Ore extension of a noncommutative ring" href="_ore__Ideal.html">oreIdeal</a> -- Creates the defining ideal of an Ore extension of a noncommutative ring</span></span></li>
            <li><span><kbd>NCRing ** NCRing</kbd> -- see <span><a title="Define the (q-)commuting tensor product" href="_q__Tensor__Product.html">qTensorProduct</a> -- Define the (q-)commuting tensor product</span></span></li>
            <li><span><kbd>qTensorProduct(NCRing,NCRing,QQ)</kbd> -- see <span><a title="Define the (q-)commuting tensor product" href="_q__Tensor__Product.html">qTensorProduct</a> -- Define the (q-)commuting tensor product</span></span></li>
            <li><span><kbd>qTensorProduct(NCRing,NCRing,RingElement)</kbd> -- see <span><a title="Define the (q-)commuting tensor product" href="_q__Tensor__Product.html">qTensorProduct</a> -- Define the (q-)commuting tensor product</span></span></li>
            <li><span><kbd>qTensorProduct(NCRing,NCRing,ZZ)</kbd> -- see <span><a title="Define the (q-)commuting tensor product" href="_q__Tensor__Product.html">qTensorProduct</a> -- Define the (q-)commuting tensor product</span></span></li>
            <li><span><kbd>setWeights(NCRing,List)</kbd> -- see <span><a title="Set a nonstandard grading for a NCRing." href="_set__Weights.html">setWeights</a> -- Set a nonstandard grading for a NCRing.</span></span></li>
            <li><span><kbd>toM2Ring(NCRing)</kbd> -- see <span><a title="Compute the abelianization of an NCRing and returns a Ring." href="_to__M2__Ring.html">toM2Ring</a> -- Compute the abelianization of an NCRing and returns a Ring.</span></span></li>
            <li><span><a title="Brings the variables of a particular NCRing in scope" href="_use_lp__N__C__Ring_rp.html">use(NCRing)</a> -- Brings the variables of a particular NCRing in scope</span></li>
          </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Type of a noncommutative ring" href="___N__C__Ring.html">NCRing</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 rings" href="../../Macaulay2Doc/html/___Ring.html">Ring</a> &lt; <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">Type</a> &lt; <a title="the class of all mutable hash tables" href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html">MutableHashTable</a> &lt; <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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        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:111:0</span>.</p>
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Edit File: ___N__C__Ring.html

Edit File: _NC__Ring.html