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Edit File: _NCRingMap.html

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    <title>NCRingMap -- Type of a map to or from a noncommutative ring.</title>
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o1 = A

o1 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : B = QQ{a,b,c}

o2 = B

o2 : NCPolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i3 : f = ncMap(B,A,{a^3,b^2,a+b,a-b})

o3 = NCRingMap B &lt;--- A

o3 : NCRingMap</code></pre>
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              <pre><code class="language-macaulay2">i4 : target f

o4 = B

o4 : NCPolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i5 : source f

o5 = A

o5 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i6 : matrix f

o6 = | a^3 b^2 b+a -b+a |

o6 : NCMatrix</code></pre>
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          <p>Note that NCRingMaps need not be well-defined or homogeneous. Apply a function to an element or a matrix using the usual function notation. NCRingMaps are linear and multiplicative by definition.</p>
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              <pre><code class="language-macaulay2">i7 : f(w*x+2*y)

3 2
o7 = 2b+a b +2a

o7 : B</code></pre>
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              <pre><code class="language-macaulay2">i8 : isWellDefined f

o8 = false</code></pre>
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              <pre><code class="language-macaulay2">i9 : isHomogeneous f

o9 = false</code></pre>
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          <p>The user has the option to define an NCRingMap to be a derivation. Of course, such a map must have the same source and target.</p>
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              <pre><code class="language-macaulay2">i10 : g = ncMap(B,B,{a*b,b^2,c*a*c},Derivation=>true)

o10 = NCRingMap B &lt;--- B

o10 : NCRingMap</code></pre>
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              <pre><code class="language-macaulay2">i11 : g(a*b)==g(a)*b+a*g(b)

o11 = true</code></pre>
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              <pre><code class="language-macaulay2">i12 : g(promote(1,B))

o12 = 0

o12 : B</code></pre>
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              <pre><code class="language-macaulay2">i13 : g(c*a+2*b)

2
o13 = caca+cab+2b

o13 : B</code></pre>
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        <div class="waystouse">
          <h2>Methods that use an object of class NCRingMap:</h2>
          <ul>
            <li><span><a title="Extends an NCRingMap to the ambient ring of the source." href="_ambient_lp__N__C__Ring__Map_rp.html">ambient(NCRingMap)</a> -- Extends an NCRingMap to the ambient ring of the source.</span></li>
            <li><span><kbd>gddKernel(ZZ,NCRingMap)</kbd> -- see <span><a title="Computes a homogeneous generating set of the kernel of a ring map." href="_gdd__Kernel.html">gddKernel</a> -- Computes a homogeneous generating set of the kernel of a ring map.</span></span></li>
            <li><span><a title="Determines if an NCRingMap preserves the natural grading" href="_is__Homogeneous_lp__N__C__Ring__Map_rp.html">isHomogeneous(NCRingMap)</a> -- Determines if an NCRingMap preserves the natural grading</span></li>
            <li><span><a title="Determines if an NCRingMap is well-defined." href="_is__Well__Defined_lp__N__C__Ring__Map_rp.html">isWellDefined(NCRingMap)</a> -- Determines if an NCRingMap is well-defined.</span></li>
            <li><span><kbd>kernelComponent(ZZ,NCRingMap)</kbd> -- see <span><a title="Computes a basis of the kernel of a ring map in a specified degree." href="_kernel__Component.html">kernelComponent</a> -- Computes a basis of the kernel of a ring map in a specified degree.</span></span></li>
            <li><span><a title="Applies an NCRingMap to each element of a list" href="___List_sp_sl_sp__N__C__Ring__Map.html">List / NCRingMap</a> -- Applies an NCRingMap to each element of a list</span></li>
            <li><span><a title="An NCMatrix associated to an NCRingMap." href="_matrix_lp__N__C__Ring__Map_rp.html">matrix(NCRingMap)</a> -- An NCMatrix associated to an NCRingMap.</span></li>
            <li><span><a title="Basic operations with NCRingMaps" href="___N__C__Ring__Map_sp_pl_sp__N__C__Ring__Map.html">NCRingMap + NCRingMap</a> -- Basic operations with NCRingMaps</span></li>
            <li><span><kbd>NCRingMap ^ ZZ</kbd> -- see <span><a title="Basic operations with NCRingMaps" href="___N__C__Ring__Map_sp_pl_sp__N__C__Ring__Map.html">NCRingMap + NCRingMap</a> -- Basic operations with NCRingMaps</span></span></li>
            <li><span><kbd>QQ * NCRingMap</kbd> -- see <span><a title="Basic operations with NCRingMaps" href="___N__C__Ring__Map_sp_pl_sp__N__C__Ring__Map.html">NCRingMap + NCRingMap</a> -- Basic operations with NCRingMaps</span></span></li>
            <li><span><kbd>RingElement * NCRingMap</kbd> -- see <span><a title="Basic operations with NCRingMaps" href="___N__C__Ring__Map_sp_pl_sp__N__C__Ring__Map.html">NCRingMap + NCRingMap</a> -- Basic operations with NCRingMaps</span></span></li>
            <li><span><kbd>ZZ * NCRingMap</kbd> -- see <span><a title="Basic operations with NCRingMaps" href="___N__C__Ring__Map_sp_pl_sp__N__C__Ring__Map.html">NCRingMap + NCRingMap</a> -- Basic operations with NCRingMaps</span></span></li>
            <li><span><a title="Compose two NCRingMaps" href="___N__C__Ring__Map_sp_at_at_sp__N__C__Ring__Map.html">NCRingMap @@ NCRingMap</a> -- Compose two NCRingMaps</span></li>
            <li><span><a title="Matrix of one homogeneous component of an NCRingMap" href="___N__C__Ring__Map_sp_us_sp__Z__Z.html">NCRingMap _ ZZ</a> -- Matrix of one homogeneous component of an NCRingMap</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><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></li>
            <li><span><kbd>NCRingMap NCMatrix</kbd> -- see <span><a title="Apply an NCRingMap to an element or matrix" href="___N__C__Ring__Map_sp__N__C__Ring__Element.html">NCRingMap NCRingElement</a> -- Apply an NCRingMap to an element or matrix</span></span></li>
            <li><span><a title="Apply an NCRingMap to an element or matrix" href="___N__C__Ring__Map_sp__N__C__Ring__Element.html">NCRingMap NCRingElement</a> -- Apply an NCRingMap to an element or matrix</span></li>
            <li><span><kbd>NCRingMap RingElement</kbd> -- see <span><a title="Apply an NCRingMap to an element or matrix" href="___N__C__Ring__Map_sp__N__C__Ring__Element.html">NCRingMap NCRingElement</a> -- Apply an NCRingMap to an element or matrix</span></span></li>
            <li><span><a title="Finds elements normalized by a ring map" href="_normal__Elements_lp__N__C__Ring__Map_cm__Z__Z_rp.html">normalElements(NCRingMap,ZZ)</a> -- Finds elements normalized by a ring map</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><a title="Source of a map" href="_source_lp__N__C__Ring__Map_rp.html">source(NCRingMap)</a> -- Source of a map</span></li>
            <li><span><a title="Target of a map" href="_target_lp__N__C__Ring__Map_rp.html">target(NCRingMap)</a> -- Target of a map</span></li>
          </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Type of a map to or from a noncommutative ring." href="___N__C__Ring__Map.html">NCRingMap</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.m2:2948:0</span>.</p>
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