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<!DOCTYPE html> <html lang="en"> <head> <title>skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix</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="Defines a skew polynomial ring via a skewing matrix" href="_skew__Polynomial__Ring.html">skewPolynomialRing</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="_skew__Polynomial__Ring_lp__Ring_cm__Ring__Element_cm__List_rp.html">next</a> | <a href="_size_lp__N__C__Ring__Element_rp.html">previous</a> | <a href="_skew__Polynomial__Ring_lp__Ring_cm__Ring__Element_cm__List_rp.html">forward</a> | <a href="_size_lp__N__C__Ring__Element_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">B = skewPolynomialRing(R,M,L)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">R</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, </span></li> <li><span><span class="tt">M</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, </span></li> <li><span><span class="tt">L</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">B</span>, <span>an instance of the type <a title="Type of a noncommutative ring" href="___N__C__Ring.html">NCRing</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This method constructs a skew polynomial ring with coefficients in the ring R and generators from the list L. A valid input matrix is a square matrix over R with at least #L rows such that M_{ij} = M_{ji}^{(-1)} and M_{ii}=1. The relations of the resulting ring have the form g_i*g_j - M_{ij}*g_j*g_i. If R is a Bergman coefficient ring, an NCGroebnerBasis is computed for B.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[q]/ideal{q^4+q^3+q^2+q+1} o1 = R o1 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}} o2 = | 1 q q | | -q3-q2-q-1 1 1 | | -q3-q2-q-1 1 1 | 3 3 o2 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : B = skewPolynomialRing(R,M,{x,y,z}) o3 = B o3 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : x*y == q^4*y*x o4 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : N = matrix{{1,1,1,1},{1,1,1,1},{1,1,1,1},{1,1,1,1}} o5 = | 1 1 1 1 | | 1 1 1 1 | | 1 1 1 1 | | 1 1 1 1 | 4 4 o5 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : C = skewPolynomialRing(QQ,promote(N,QQ), {a,b,c,d}) --Calling Bergman for NCGB calculation. Complete! o6 = C o6 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : isCommutative C o7 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : isCommutative B o8 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : Bop = oppositeRing B o9 = Bop o9 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : y*x == q^4*x*y o10 = true</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <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> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">skewPolynomialRing</span>:</h2> <ul> <li><kbd>skewPolynomialRing(Ring,Matrix,List)</kbd></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> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Defines a skew polynomial ring via a skewing matrix" href="_skew__Polynomial__Ring.html">skewPolynomialRing</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:3387:0</span>.</p> </div> </div> </div> </body> </html>