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<!DOCTYPE html> <html lang="en"> <head> <title>NCAlgebra</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a href="index.html">NCAlgebra</a> :: <a href="index.html">NCAlgebra</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_-_sp__N__C__Matrix.html">next</a> | previous | <a href="___Basic_spoperations_spon_spnoncommutative_spalgebras.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NCAlgebra</h1> <div> <h2>Description</h2> <div> <p>This package is used to define and manipulate noncommutative algebras. Many of the commands contain calls to the existing noncommutative algebra package Bergman.</p> <p>Detailed instructions for installing Bergman, as well as the NCAlgebra system, can be found in the file installNCAlgebra.txt file contained in the NCAlgebra package directory. It may also be found at <a href="https://github.com/Macaulay2/M2/blob/stable/M2/Macaulay2/packages/NCAlgebra/installNCAlgebra.txt">https://github.com/Macaulay2/M2/blob/stable/M2/Macaulay2/packages/NCAlgebra/installNCAlgebra.txt</a>.</p> <p></p> </div> </div> <div> <h3>Menu</h3> <ul> <li><span><a href="___Basic_spoperations_spon_spnoncommutative_spalgebras.html">Basic operations on noncommutative algebras</a></span></li> <li><span><a href="___General_spsetup_spinformation.html">General setup information</a></span></li> <li><span><a href="___Using_spthe_sp__Bergman_spinterface.html">Using the Bergman interface</a></span></li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.math.wfu.edu/Faculty/Moore.html">Frank Moore</a><span> <<a href="mailto:moorewf%40wfu.edu">moorewf@wfu.edu</a>></span></li> <li><a href="http://www.stmarys-ca.edu/math-and-computer-science/faculty%3Ffacid%3D142546">Andrew Conner</a><span> <<a href="mailto:abc12%40stmarys-ca.edu">abc12@stmarys-ca.edu</a>></span></li> <li><a href="http://people.hamilton.edu/cgibbons/index.html">Courtney Gibbons</a><span> <<a href="mailto:crgibbon%40hamilton.edu">crgibbon@hamilton.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.0</b> of NCAlgebra, released <b>July 2, 2016</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{NCAlgebraSource, title = {{NCAlgebra: data types for noncommutative algebras. Version~1.0}}, author = {Frank Moore and Andrew Conner and Courtney Gibbons}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="Type of a Groebner basis for an NCIdeal in an NCRing." href="___N__C__Groebner__Basis.html">NCGroebnerBasis</a> -- Type of a Groebner basis for an NCIdeal in an NCRing.</span></li> <li><span><a title="Type of a two-sided ideal in a noncommutative ring" href="___N__C__Ideal.html">NCIdeal</a> -- Type of a two-sided ideal in a noncommutative ring</span></li> <li><span><a title="Type of a left ideal in a noncommutative ring" href="___N__C__Left__Ideal.html">NCLeftIdeal</a> -- Type of a left ideal in a noncommutative ring</span></li> <li><span><a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</a> -- Type of a matrix over a noncommutative ring</span></li> <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> <li><span><a title="Type of a right ideal in a noncommutative ring" href="___N__C__Right__Ideal.html">NCRightIdeal</a> -- Type of a right ideal in a noncommutative ring</span></li> <li><span><a title="Type of a noncommutative ring" href="___N__C__Ring.html">NCRing</a> -- Type of a noncommutative ring</span></li> <li><span><a title="Type of an element in a noncommutative ring" href="___N__C__Ring__Element.html">NCRingElement</a> -- Type of an element in a noncommutative ring</span></li> <li><span><a title="Type of a map to or from a noncommutative ring." href="___N__C__Ring__Map.html">NCRingMap</a> -- Type of a map to or from a noncommutative ring.</span></li> <li><span><kbd>NCChainComplex</kbd> -- see <span><a title="Compute the resolution of coker M as a map of free right modules" href="_resolution_lp__N__C__Matrix_rp.html">resolution(NCMatrix)</a> -- Compute the resolution of coker M as a map of free right modules</span></span></li> </ul> </li> <li>Functions and commands <ul> <li><span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></li> <li><span><a title="Finds central elements in a given degree" href="_central__Elements.html">centralElements</a> -- Finds central elements in a given degree</span></li> <li><span><a title="Computes coordinates relative to a given basis" href="_coordinates.html">coordinates</a> -- Computes coordinates relative to a given basis</span></li> <li><span><a title="Methods for creating endomorphism rings of modules over a commutative ring" href="_endomorphism__Ring.html">endomorphismRing</a> -- Methods for creating endomorphism rings of modules over a commutative ring</span></li> <li><span><a title="Create the enveloping algebra" href="_enveloping__Algebra.html">envelopingAlgebra</a> -- Create the enveloping algebra</span></li> <li><span><a title="Defines a four-dimensional Sklyanin with given parameters" href="_four__Dim__Sklyanin.html">fourDimSklyanin</a> -- Defines a four-dimensional Sklyanin with given parameters</span></li> <li><span><a title="Define the free product of two algebras" href="_free__Product.html">freeProduct</a> -- Define the free product of two algebras</span></li> <li><span><a title="Read in a NCGroebnerBasis from a Bergman output file." href="_gb__From__Output__File.html">gbFromOutputFile</a> -- Read in a NCGroebnerBasis from a Bergman output file.</span></li> <li><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></li> <li><span><a title="Calls Bergman to compute the Hilbert series of an NCQuotientRing" href="_hilbert__Bergman.html">hilbertBergman</a> -- Calls Bergman to compute the Hilbert series of an NCQuotientRing</span></li> <li><span><a title="Computes the dual of a pure homogeneous ideal" href="_homog__Dual.html">homogDual</a> -- Computes the dual of a pure homogeneous ideal</span></li> <li><span><a title="Determines if an element is central" href="_is__Central.html">isCentral</a> -- Determines if an element is central</span></li> <li><span><kbd>isExterior</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><a title="Determines if a given (homogeneous) element is regular in a given degree" href="_is__Left__Regular.html">isLeftRegular</a> -- Determines if a given (homogeneous) element is regular in a given degree</span></li> <li><span><kbd>isRightRegular</kbd> -- see <span><a title="Determines if a given (homogeneous) element is regular in a given degree" href="_is__Left__Regular.html">isLeftRegular</a> -- Determines if a given (homogeneous) element is regular in a given degree</span></span></li> <li><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></li> <li><span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></li> <li><span><kbd>rightMultiplicationMap</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><a title="Minimizes a list of NCRingElements" href="_minimize__Relations.html">minimizeRelations</a> -- Minimizes a list of NCRingElements</span></li> <li><span><a title="Compute a noncommutative Groebner basis." href="_nc__Groebner__Basis.html">ncGroebnerBasis</a> -- Compute a noncommutative Groebner basis.</span></li> <li><span><a title="Define a two-sided ideal in a noncommutative ring" href="_nc__Ideal.html">ncIdeal</a> -- Define a two-sided ideal in a noncommutative ring</span></li> <li><span><a title="Define a left ideal in a noncommutative ring" href="_nc__Left__Ideal.html">ncLeftIdeal</a> -- Define a left ideal in a noncommutative ring</span></li> <li><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></li> <li><span><a title="Create an NCMatrix" href="_nc__Matrix.html">ncMatrix</a> -- Create an NCMatrix</span></li> <li><span><a title="Define a right ideal in a noncommutative ring" href="_nc__Right__Ideal.html">ncRightIdeal</a> -- Define a right ideal in a noncommutative ring</span></li> <li><span><a title="Computes the automorphism determined by a normal homogeneous element" href="_normal__Automorphism.html">normalAutomorphism</a> -- Computes the automorphism determined by a normal homogeneous element</span></li> <li><span><a title="Finds normal elements" href="_normal__Elements.html">normalElements</a> -- Finds normal elements</span></li> <li><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></li> <li><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></li> <li><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></li> <li><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></li> <li><span><a title="Define the (q-)commuting tensor product" href="_q__Tensor__Product.html">qTensorProduct</a> -- Define the (q-)commuting tensor product</span></li> <li><span><a title="Creates the subideal generated by quadratic elements of a given ideal" href="_quadratic__Closure.html">quadraticClosure</a> -- Creates the subideal generated by quadratic elements of a given ideal</span></li> <li><span><a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel</a> -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</span></li> <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> <li><span><a title="Set a nonstandard grading for a NCRing." href="_set__Weights.html">setWeights</a> -- Set a nonstandard grading for a NCRing.</span></li> <li><span><a title="Defines a skew polynomial ring via a skewing matrix" href="_skew__Polynomial__Ring.html">skewPolynomialRing</a> -- Defines a skew polynomial ring via a skewing matrix</span></li> <li><span><a title="Converts ring elements into vectors over the coefficient ring" href="_sparse__Coeffs.html">sparseCoeffs</a> -- Converts ring elements into vectors over the coefficient ring</span></li> <li><span><a title="Defines a three-dimensional Sklyanin with given parameters" href="_three__Dim__Sklyanin.html">threeDimSklyanin</a> -- Defines a three-dimensional Sklyanin with given parameters</span></li> <li><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></li> <li><span><a title="Converts a Ring to an NCRing" href="_to__N__C__Ring.html">toNCRing</a> -- Converts a Ring to an NCRing</span></li> <li><span><a title="Calls Bergman to compute a two sided noncommutative Groebner Basis." href="_two__Sided__N__C__Groebner__Basis__Bergman.html">twoSidedNCGroebnerBasisBergman</a> -- Calls Bergman to compute a two sided noncommutative Groebner Basis.</span></li> </ul> </li> <li>Methods <ul> <li><span><a title="Negates NCMatrices" href="_-_sp__N__C__Matrix.html">- NCMatrix</a> -- Negates NCMatrices</span></li> <li><span><span class="tt">ambient(NCPolynomialRing)</span> (missing documentation)<!--tag: (ambient,NCPolynomialRing)--> </span></li> <li><span><a title="Ambient ring of an NCQuotientRing" href="_ambient_lp__N__C__Quotient__Ring_rp.html">ambient(NCQuotientRing)</a> -- Ambient ring of an NCQuotientRing</span></li> <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>assignDegrees(NCMatrix)</kbd> -- see <span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></span></li> <li><span><kbd>assignDegrees(NCMatrix,List,List)</kbd> -- see <span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></span></li> <li><span><a title="Returns the base name of a generator of an NCRing" href="_base__Name_lp__N__C__Ring__Element_rp.html">baseName(NCRingElement)</a> -- Returns the base name of a generator of an NCRing</span></li> <li><span><a title="Returns a basis of an NCIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Ideal_rp.html">basis(ZZ,NCIdeal)</a> -- Returns a basis of an NCIdeal in a particular degree.</span></li> <li><span><a title="Returns a basis of an NCLeftIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Left__Ideal_rp.html">basis(ZZ,NCLeftIdeal)</a> -- Returns a basis of an NCLeftIdeal in a particular degree.</span></li> <li><span><a title="Returns a basis of an NCRightIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Right__Ideal_rp.html">basis(ZZ,NCRightIdeal)</a> -- Returns a basis of an NCRightIdeal in a particular degree.</span></li> <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>coordinates(List)</kbd> -- see <span><a title="Computes coordinates relative to a given basis" href="_coordinates.html">coordinates</a> -- Computes coordinates relative to a given basis</span></span></li> <li><span><kbd>coordinates(NCRingElement)</kbd> -- see <span><a title="Computes coordinates relative to a given basis" href="_coordinates.html">coordinates</a> -- Computes coordinates relative to a given basis</span></span></li> <li><span><a title="Returns the degree of an NCRingElement" href="_degree_lp__N__C__Ring__Element_rp.html">degree(NCRingElement)</a> -- Returns the degree of an NCRingElement</span></li> <li><span><kbd>endomorphismRing(Module,Symbol)</kbd> -- see <span><a title="Methods for creating endomorphism rings of modules over a commutative ring" href="_endomorphism__Ring.html">endomorphismRing</a> -- Methods for creating endomorphism rings of modules over a commutative ring</span></span></li> <li><span><a title="Returns the entries of the NCMatrix" href="_entries_lp__N__C__Matrix_rp.html">entries(NCMatrix)</a> -- Returns the entries of the NCMatrix</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>fourDimSklyanin(Ring,List)</kbd> -- see <span><a title="Defines a four-dimensional Sklyanin with given parameters" href="_four__Dim__Sklyanin.html">fourDimSklyanin</a> -- Defines a four-dimensional Sklyanin with given parameters</span></span></li> <li><span><kbd>fourDimSklyanin(Ring,List,List)</kbd> -- see <span><a title="Defines a four-dimensional Sklyanin with given parameters" href="_four__Dim__Sklyanin.html">fourDimSklyanin</a> -- Defines a four-dimensional Sklyanin with given parameters</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><kbd>gbFromOutputFile(NCPolynomialRing,String)</kbd> -- see <span><a title="Read in a NCGroebnerBasis from a Bergman output file." href="_gb__From__Output__File.html">gbFromOutputFile</a> -- Read in a NCGroebnerBasis from a Bergman output file.</span></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="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><a title="Returns the generators of an NCIdeal" href="_generators_lp__N__C__Ideal_rp.html">generators(NCIdeal)</a> -- Returns the generators of an NCIdeal</span></li> <li><span><a title="Returns the generators of an NCLeftIdeal" href="_generators_lp__N__C__Left__Ideal_rp.html">generators(NCLeftIdeal)</a> -- Returns the generators of an NCLeftIdeal</span></li> <li><span><a title="Returns the generators of an NCRightIdeal" href="_generators_lp__N__C__Right__Ideal_rp.html">generators(NCRightIdeal)</a> -- Returns the generators of an NCRightIdeal</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><kbd>hilbertBergman(NCQuotientRing)</kbd> -- see <span><a title="Calls Bergman to compute the Hilbert series of an NCQuotientRing" href="_hilbert__Bergman.html">hilbertBergman</a> -- Calls Bergman to compute the Hilbert series of an NCQuotientRing</span></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="Compute a graded component of Hom(M,N)" href="___Hom_lp__Z__Z_cm__N__C__Matrix_cm__N__C__Matrix_rp.html">Hom(ZZ,NCMatrix,NCMatrix)</a> -- Compute a graded component of Hom(M,N)</span></li> <li><span><kbd>homogDual(NCIdeal)</kbd> -- see <span><a title="Computes the dual of a pure homogeneous ideal" href="_homog__Dual.html">homogDual</a> -- Computes the dual of a pure homogeneous ideal</span></span></li> <li><span><kbd>homogDual(NCQuotientRing)</kbd> -- see <span><a title="Computes the dual of a pure homogeneous ideal" href="_homog__Dual.html">homogDual</a> -- Computes the dual of a pure homogeneous ideal</span></span></li> <li><span><kbd>homogDual(Ring)</kbd> -- see <span><a title="Computes the dual of a pure homogeneous ideal" href="_homog__Dual.html">homogDual</a> -- Computes the dual of a pure homogeneous ideal</span></span></li> <li><span><a title="The defining ideal of an NCPolynomialRing" href="_ideal_lp__N__C__Polynomial__Ring_rp.html">ideal(NCPolynomialRing)</a> -- The defining ideal of an NCPolynomialRing</span></li> <li><span><a title="Defining ideal of an NCQuotientRing in its ambient ring" href="_ideal_lp__N__C__Quotient__Ring_rp.html">ideal(NCQuotientRing)</a> -- Defining ideal of an NCQuotientRing in its ambient ring</span></li> <li><span><kbd>isCentral(NCRingElement)</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><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="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>isExterior(Ring)</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><a title="Returns whether the NCRingElement is constant" href="_is__Constant_lp__N__C__Ring__Element_rp.html">isConstant(NCRingElement)</a> -- Returns whether the NCRingElement is constant</span></li> <li><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></li> <li><span><kbd>isHomogeneous(NCLeftIdeal)</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>isHomogeneous(NCMatrix)</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>isHomogeneous(NCRightIdeal)</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>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>isHomogeneous(NCRingElement)</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><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><kbd>isLeftRegular(NCRingElement,ZZ)</kbd> -- see <span><a title="Determines if a given (homogeneous) element is regular in a given degree" href="_is__Left__Regular.html">isLeftRegular</a> -- Determines if a given (homogeneous) element is regular in a given degree</span></span></li> <li><span><kbd>isRightRegular(NCRingElement,ZZ)</kbd> -- see <span><a title="Determines if a given (homogeneous) element is regular in a given degree" href="_is__Left__Regular.html">isLeftRegular</a> -- Determines if a given (homogeneous) element is regular in a given degree</span></span></li> <li><span><a title="Determines if a given NCRingElement is normal" href="_is__Normal_lp__N__C__Ring__Element_rp.html">isNormal(NCRingElement)</a> -- Determines if a given NCRingElement is normal</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="Returns the lead monomial of an NCRingElement" href="_lead__Coefficient_lp__N__C__Ring__Element_rp.html">leadCoefficient(NCRingElement)</a> -- Returns the lead monomial of an NCRingElement</span></li> <li><span><a title="Returns the lead monomial of an NCRingElement" href="_lead__Monomial_lp__N__C__Ring__Element_rp.html">leadMonomial(NCRingElement)</a> -- Returns the lead monomial of an NCRingElement</span></li> <li><span><a title="Returns the lead term of an NCRingElement" href="_lead__Term_lp__N__C__Ring__Element_rp.html">leadTerm(NCRingElement)</a> -- Returns the lead term of an NCRingElement</span></li> <li><span><kbd>leftMultiplicationMap(NCRingElement,List,List)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><kbd>leftMultiplicationMap(NCRingElement,ZZ)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><kbd>leftMultiplicationMap(NCRingElement,ZZ,ZZ)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><kbd>rightMultiplicationMap(NCRingElement,List,List)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><kbd>rightMultiplicationMap(NCRingElement,ZZ)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><kbd>rightMultiplicationMap(NCRingElement,ZZ,ZZ)</kbd> -- see <span><a title="Computes a matrix for left or right multiplication by a homogeneous element" href="_left__Multiplication__Map.html">leftMultiplicationMap</a> -- Computes a matrix for left or right multiplication by a homogeneous element</span></span></li> <li><span><a title="Lifts an NCMatrix" href="_lift_lp__N__C__Matrix_rp.html">lift(NCMatrix)</a> -- Lifts an NCMatrix</span></li> <li><span><a title="Scales a list by an NCRingElement on the right" href="___List_sp_st_sp__N__C__Ring__Element.html">List * NCRingElement</a> -- Scales a list by an NCRingElement on the right</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="Product of NCMatrices" href="___Matrix_sp_st_sp__N__C__Matrix.html">Matrix * NCMatrix</a> -- Product of NCMatrices</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><kbd>minimizeRelations(List)</kbd> -- see <span><a title="Minimizes a list of NCRingElements" href="_minimize__Relations.html">minimizeRelations</a> -- Minimizes a list of NCRingElements</span></span></li> <li><span><a title="Returns the monomials appearing in the NCRingElement" href="_monomials_lp__N__C__Ring__Element_rp.html">monomials(NCRingElement)</a> -- Returns the monomials appearing in the NCRingElement</span></li> <li><span><kbd>ncGroebnerBasis(List)</kbd> -- see <span><a title="Compute a noncommutative Groebner basis." href="_nc__Groebner__Basis.html">ncGroebnerBasis</a> -- Compute a noncommutative Groebner basis.</span></span></li> <li><span><kbd>ncGroebnerBasis(NCIdeal)</kbd> -- see <span><a title="Compute a noncommutative Groebner basis." href="_nc__Groebner__Basis.html">ncGroebnerBasis</a> -- Compute a noncommutative Groebner basis.</span></span></li> <li><span><kbd>ncIdeal(List)</kbd> -- see <span><a title="Define a two-sided ideal in a noncommutative ring" href="_nc__Ideal.html">ncIdeal</a> -- Define a two-sided ideal in a noncommutative ring</span></span></li> <li><span><kbd>ncIdeal(NCRingElement)</kbd> -- see <span><a title="Define a two-sided ideal in a noncommutative ring" href="_nc__Ideal.html">ncIdeal</a> -- Define a two-sided ideal in a noncommutative ring</span></span></li> <li><span><a title="Sum of NCIdeals" href="___N__C__Ideal_sp_pl_sp__N__C__Ideal.html">NCIdeal + NCIdeal</a> -- Sum of NCIdeals</span></li> <li><span><kbd>ncLeftIdeal(List)</kbd> -- see <span><a title="Define a left ideal in a noncommutative ring" href="_nc__Left__Ideal.html">ncLeftIdeal</a> -- Define a left ideal in a noncommutative ring</span></span></li> <li><span><kbd>ncLeftIdeal(NCRingElement)</kbd> -- see <span><a title="Define a left ideal in a noncommutative ring" href="_nc__Left__Ideal.html">ncLeftIdeal</a> -- Define a left ideal in a noncommutative ring</span></span></li> <li><span><a title="Sum of NCLeftIdeals" href="___N__C__Left__Ideal_sp_pl_sp__N__C__Left__Ideal.html">NCLeftIdeal + NCLeftIdeal</a> -- Sum of NCLeftIdeals</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(List)</kbd> -- see <span><a title="Create an NCMatrix" href="_nc__Matrix.html">ncMatrix</a> -- Create an NCMatrix</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="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="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Matrix.html">NCMatrix * Matrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Matrix.html">NCMatrix * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Ring__Element.html">NCMatrix * NCRingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Q__Q.html">NCMatrix * QQ</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Ring__Element.html">NCMatrix * RingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Z__Z.html">NCMatrix * ZZ</a> -- Product of NCMatrices</span></li> <li><span><span class="tt">NCMatrix ** NCMatrix</span> (missing documentation)<!--tag: (**,NCMatrix,NCMatrix)--> </span></li> <li><span><a title="Add NCMatrices" href="___N__C__Matrix_sp_pl_sp__N__C__Matrix.html">NCMatrix + NCMatrix</a> -- Add NCMatrices</span></li> <li><span><span class="tt">NCMatrix ++ NCMatrix</span> (missing documentation)<!--tag: (++,NCMatrix,NCMatrix)--> </span></li> <li><span><a title="Subtract NCMatrices" href="___N__C__Matrix_sp-_sp__N__C__Matrix.html">NCMatrix - NCMatrix</a> -- Subtract NCMatrices</span></li> <li><span><a title="Factor one map through another" href="___N__C__Matrix_sp_sl_sl_sp__N__C__Matrix.html">NCMatrix // NCMatrix</a> -- Factor one map through another</span></li> <li><span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></li> <li><span><kbd>NCMatrix == ZZ</kbd> -- see <span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></span></li> <li><span><kbd>ZZ == NCMatrix</kbd> -- see <span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></span></li> <li><span><a title="Select some rows of an NCMatrix" href="___N__C__Matrix_sp%5E_sp__List.html">NCMatrix ^ List</a> -- Select some rows of an NCMatrix</span></li> <li><span><a title="Exponentiate an NCMatrix" href="___N__C__Matrix_sp%5E_sp__Z__Z.html">NCMatrix ^ ZZ</a> -- Exponentiate an NCMatrix</span></li> <li><span><a title="Select some columns of an NCMatrix" href="___N__C__Matrix_sp_us_sp__List.html">NCMatrix _ List</a> -- Select some columns of an NCMatrix</span></li> <li><span><a title="Induced map in homogeneous degree n" href="___N__C__Matrix_sp_us_sp__Z__Z.html">NCMatrix _ ZZ</a> -- Induced map in homogeneous degree n</span></li> <li><span><a title="Join NCMatrices horizontally" href="___N__C__Matrix_sp_vb_sp__N__C__Matrix.html">NCMatrix | NCMatrix</a> -- Join NCMatrices horizontally</span></li> <li><span><a title="Join NCMatrices vertically" href="___N__C__Matrix_sp_vb_vb_sp__N__C__Matrix.html">NCMatrix || NCMatrix</a> -- Join NCMatrices vertically</span></li> <li><span><a title="Graded shift of an NCMatrix." href="___N__C__Matrix_sp__Array.html">NCMatrix Array</a> -- Graded shift of an NCMatrix.</span></li> <li><span><a title="Construct a NCQuotientRing" href="___N__C__Polynomial__Ring_sp_sl_sp__N__C__Ideal.html">NCPolynomialRing / NCIdeal</a> -- Construct a NCQuotientRing</span></li> <li><span><kbd>ncRightIdeal(List)</kbd> -- see <span><a title="Define a right ideal in a noncommutative ring" href="_nc__Right__Ideal.html">ncRightIdeal</a> -- Define a right ideal in a noncommutative ring</span></span></li> <li><span><kbd>ncRightIdeal(NCRingElement)</kbd> -- see <span><a title="Define a right ideal in a noncommutative ring" href="_nc__Right__Ideal.html">ncRightIdeal</a> -- Define a right ideal in a noncommutative ring</span></span></li> <li><span><a title="Sum of NCRightIdeals" href="___N__C__Right__Ideal_sp_pl_sp__N__C__Right__Ideal.html">NCRightIdeal + NCRightIdeal</a> -- Sum of NCRightIdeals</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><a title="Scales a list by an NCRingElement on the left" href="___N__C__Ring__Element_sp_st_sp__List.html">NCRingElement * List</a> -- Scales a list by an NCRingElement on the left</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Ring__Element_sp_st_sp__N__C__Matrix.html">NCRingElement * NCMatrix</a> -- Product of NCMatrices</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><kbd>normalAutomorphism(NCRingElement)</kbd> -- see <span><a title="Computes the automorphism determined by a normal homogeneous element" href="_normal__Automorphism.html">normalAutomorphism</a> -- Computes the automorphism determined by a normal homogeneous element</span></span></li> <li><span><kbd>normalElements(NCQuotientRing,ZZ,Symbol,Symbol)</kbd> -- see <span><a title="Finds normal elements" href="_normal__Elements.html">normalElements</a> -- Finds normal elements</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>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><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><a title="Product of NCMatrices" href="___Q__Q_sp_st_sp__N__C__Matrix.html">QQ * NCMatrix</a> -- Product of NCMatrices</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>quadraticClosure(NCIdeal)</kbd> -- see <span><a title="Creates the subideal generated by quadratic elements of a given ideal" href="_quadratic__Closure.html">quadraticClosure</a> -- Creates the subideal generated by quadratic elements of a given ideal</span></span></li> <li><span><kbd>quadraticClosure(NCQuotientRing)</kbd> -- see <span><a title="Creates the subideal generated by quadratic elements of a given ideal" href="_quadratic__Closure.html">quadraticClosure</a> -- Creates the subideal generated by quadratic elements of a given ideal</span></span></li> <li><span><kbd>betti(NCChainComplex)</kbd> -- see <span><a title="Compute the resolution of coker M as a map of free right modules" href="_resolution_lp__N__C__Matrix_rp.html">resolution(NCMatrix)</a> -- Compute the resolution of coker M as a map of free right modules</span></span></li> <li><span><a title="Compute the resolution of coker M as a map of free right modules" href="_resolution_lp__N__C__Matrix_rp.html">resolution(NCMatrix)</a> -- Compute the resolution of coker M as a map of free right modules</span></li> <li><span><kbd>rightKernel(NCMatrix,ZZ)</kbd> -- see <span><a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel</a> -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</span></span></li> <li><span><kbd>rightKernelBergman(NCMatrix)</kbd> -- see <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></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> <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> <li><span><a title="Product of NCMatrices" href="___Ring__Element_sp_st_sp__N__C__Matrix.html">RingElement * NCMatrix</a> -- Product of NCMatrices</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><a title="Returns the number of terms of an NCRingElement" href="_size_lp__N__C__Ring__Element_rp.html">size(NCRingElement)</a> -- Returns the number of terms of an NCRingElement</span></li> <li><span><kbd>skewPolynomialRing(Ring,Matrix,List)</kbd> -- see <span><a title="Defines a skew polynomial ring via a skewing matrix" href="_skew__Polynomial__Ring.html">skewPolynomialRing</a> -- Defines a skew polynomial ring via a skewing matrix</span></span></li> <li><span><kbd>skewPolynomialRing(Ring,QQ,List)</kbd> -- see <span><a title="Defines a skew polynomial ring via a scaling factor" href="_skew__Polynomial__Ring_lp__Ring_cm__Ring__Element_cm__List_rp.html">skewPolynomialRing(Ring,RingElement,List)</a> -- Defines a skew polynomial ring via a scaling factor</span></span></li> <li><span><a title="Defines a skew polynomial ring via a scaling factor" href="_skew__Polynomial__Ring_lp__Ring_cm__Ring__Element_cm__List_rp.html">skewPolynomialRing(Ring,RingElement,List)</a> -- Defines a skew polynomial ring via a scaling factor</span></li> <li><span><kbd>skewPolynomialRing(Ring,ZZ,List)</kbd> -- see <span><a title="Defines a skew polynomial ring via a scaling factor" href="_skew__Polynomial__Ring_lp__Ring_cm__Ring__Element_cm__List_rp.html">skewPolynomialRing(Ring,RingElement,List)</a> -- Defines a skew polynomial ring via a scaling factor</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><kbd>sparseCoeffs(List)</kbd> -- see <span><a title="Converts ring elements into vectors over the coefficient ring" href="_sparse__Coeffs.html">sparseCoeffs</a> -- Converts ring elements into vectors over the coefficient ring</span></span></li> <li><span><kbd>sparseCoeffs(NCRingElement)</kbd> -- see <span><a title="Converts ring elements into vectors over the coefficient ring" href="_sparse__Coeffs.html">sparseCoeffs</a> -- Converts ring elements into vectors over the coefficient ring</span></span></li> <li><span><a title="Returns the variables appearing in the NCRingElement" href="_support_lp__N__C__Ring__Element_rp.html">support(NCRingElement)</a> -- Returns the variables appearing in the NCRingElement</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> <li><span><a title="Returns the terms of an NCRingElement" href="_terms_lp__N__C__Ring__Element_rp.html">terms(NCRingElement)</a> -- Returns the terms of an NCRingElement</span></li> <li><span><kbd>threeDimSklyanin(Ring,List)</kbd> -- see <span><a title="Defines a three-dimensional Sklyanin with given parameters" href="_three__Dim__Sklyanin.html">threeDimSklyanin</a> -- Defines a three-dimensional Sklyanin with given parameters</span></span></li> <li><span><kbd>threeDimSklyanin(Ring,List,List)</kbd> -- see <span><a title="Defines a three-dimensional Sklyanin with given parameters" href="_three__Dim__Sklyanin.html">threeDimSklyanin</a> -- Defines a three-dimensional Sklyanin with given parameters</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><kbd>toNCRing(Ring)</kbd> -- see <span><a title="Converts a Ring to an NCRing" href="_to__N__C__Ring.html">toNCRing</a> -- Converts a Ring to an NCRing</span></span></li> <li><span><a title="Converts an NCRingElement to a string" href="_to__String_lp__N__C__Ring__Element_rp.html">toString(NCRingElement)</a> -- Converts an NCRingElement to a string</span></li> <li><span><a title="Transposes an NCMatrix" href="_transpose_lp__N__C__Matrix_rp.html">transpose(NCMatrix)</a> -- Transposes an NCMatrix</span></li> <li><span><kbd>twoSidedNCGroebnerBasisBergman(List)</kbd> -- see <span><a title="Calls Bergman to compute a two sided noncommutative Groebner Basis." href="_two__Sided__N__C__Groebner__Basis__Bergman.html">twoSidedNCGroebnerBasisBergman</a> -- Calls Bergman to compute a two sided noncommutative Groebner Basis.</span></span></li> <li><span><kbd>twoSidedNCGroebnerBasisBergman(NCIdeal)</kbd> -- see <span><a title="Calls Bergman to compute a two sided noncommutative Groebner Basis." href="_two__Sided__N__C__Groebner__Basis__Bergman.html">twoSidedNCGroebnerBasisBergman</a> -- Calls Bergman to compute a two sided noncommutative Groebner Basis.</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> <li><span><a title="Product of NCMatrices" href="___Z__Z_sp_st_sp__N__C__Matrix.html">ZZ * NCMatrix</a> -- Product of NCMatrices</span></li> </ul> </li> <li>Symbols <ul> <li><span><kbd>endomorphismRingGens</kbd> -- see <span><a title="Methods for creating endomorphism rings of modules over a commutative ring" href="_endomorphism__Ring.html">endomorphismRing</a> -- Methods for creating endomorphism rings of modules over a commutative ring</span></span></li> <li><span><kbd>rightKernelDegreeLimit</kbd> -- see <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></span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a href="index.html">NCAlgebra</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">NCAlgebra.m2</span>, with auxiliary files in <span class="tt">NCAlgebra/</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:49:0</span>.</p> </div> </div> </div> </body> </html>
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