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___N__C__Ring__Element.html
<!DOCTYPE html> <html lang="en"> <head> <title>NCRingElement -- Type of an element in a noncommutative ring</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 title="Type of an element in a noncommutative ring" href="___N__C__Ring__Element.html">NCRingElement</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="___N__C__Ring__Element_sp_pc_sp__N__C__Groebner__Basis.html">next</a> | <a href="___N__C__Ring.html">previous</a> | <a href="___N__C__Ring__Element_sp_pc_sp__N__C__Groebner__Basis.html">forward</a> | <a href="___N__C__Ring.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NCRingElement -- Type of an element in a noncommutative ring</h1> <div> <h2>Description</h2> <div> <p>This is the type of an element in a noncommutative graded ring. One can deal with these elements in much the same way as in the commutative case. See <a title="the class of all ring elements handled by the engine" href="../../Macaulay2Doc/html/___Ring__Element.html">RingElement</a> for details.</p> </div> </div> <div> <div class="waystouse"> <h2>Methods that use an object of class NCRingElement:</h2> <ul> <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><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>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 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><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><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="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="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="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>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><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="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><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="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><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="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>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>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>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>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><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="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>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="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><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> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Type of an element in a noncommutative ring" href="___N__C__Ring__Element.html">NCRingElement</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> < <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:1115:0</span>.</p> </div> </div> </div> </body> </html>
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