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<!DOCTYPE html> <html lang="en"> <head> <title>RingElement -- the class of all ring elements handled by the engine</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="index.html">Documentation </a> <br><a href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Macaulay2 documentation" href="index.html">Macaulay2Doc</a> » <a href="___The_sp__Macaulay2_splanguage.html">The Macaulay2 language</a> » <a title="a detailed overview of lists and sequences in Macaulay2" href="_lists_spand_spsequences.html">lists and sequences</a> » <a title="the class of all basic lists" href="___Basic__List.html">BasicList</a> » <a title="the class of all ring elements handled by the engine" href="___Ring__Element.html">RingElement</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="___Option.html">next</a> | <a href="___U__R__L.html">previous</a> | <a href="_is__Unit.html">forward</a> | <a href="___U__R__L.html">backward</a> | <a href="___Basic__List.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>RingElement -- the class of all ring elements handled by the engine</h1> <div> <!--empty documentation body--> </div> <div> <h2>See also</h2> <ul> <li><span><a title="the class of rings handled by the engine" href="___Engine__Ring.html">EngineRing</a> -- the class of rings handled by the engine</span></li> </ul> </div> <div> <h3>Menu</h3> <ul> <li><span><a title="whether a ring element is a unit" href="_is__Unit.html">isUnit</a> -- whether a ring element is a unit</span></li> <li><span><a title="get the greatest term" href="_lead__Term_lp__Ring__Element_rp.html">leadTerm(RingElement)</a> -- get the greatest term</span></li> <li><span><a title="get the lead polynomials using part of the monomial order" href="_lead__Term_lp__Z__Z_cm__Ring__Element_rp.html">leadTerm(ZZ,RingElement)</a> -- get the lead polynomials using part of the monomial order</span></li> <li><span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></li> <li><span><a href="_degree_lp__Ring__Element_rp.html">degree(RingElement)</a></span></li> <li><span><a title="degree with respect to a variable" href="_degree_lp__Ring__Element_cm__Ring__Element_rp.html">degree(RingElement,RingElement)</a> -- degree with respect to a variable</span></li> <li><span><a title="factor a ring element" href="_factor_lp__Ring__Element_rp.html">factor(RingElement)</a> -- factor a ring element</span></li> <li><span><a title="indices of variables occurring in a polynomial" href="_indices_lp__Ring__Element_rp.html">indices(RingElement)</a> -- indices of variables occurring in a polynomial</span></li> <li><span><a title="power" href="___Ring__Element_sp%5E_sp__Z__Z.html">RingElement ^ ZZ</a> -- power</span></li> <li><span><a title="fraction" href="___Ring__Element_sp_sl_sp__Ring__Element.html">RingElement / RingElement</a> -- fraction</span></li> <li><span><a title="a sequence of consecutive generators of a polynomial ring" href="___Ring__Element_sp.._sp__Ring__Element.html">RingElement .. RingElement</a> -- a sequence of consecutive generators of a polynomial ring</span></li> <li><span><a title="a sequence of consecutive generators of a polynomial ring" href="___Ring__Element_sp.._lt_sp__Ring__Element.html">RingElement ..< RingElement</a> -- a sequence of consecutive generators of a polynomial ring</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Functions and methods returning a ring element:</h2> <ul> <li><span><kbd>Number % GroebnerBasis</kbd> -- see <span><a title="a binary operator, usually used for remainder and reduction" href="__pc.html">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li> <li><span><kbd>ZZ % MonomialIdeal</kbd> -- see <span><a title="a binary operator, usually used for remainder and reduction" href="__pc.html">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li> <li><span><kbd>RingElement * RingElement</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>- RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement // RingElement</kbd> -- see <span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></span></li> <li><span><kbd>ZZ // MonomialIdeal</kbd> -- see <span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></span></li> <li><span><a title="coefficient of a monomial" href="_coefficient.html">coefficient</a> -- coefficient of a monomial</span></li> <li><span><kbd>contract(Number,RingElement)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>contract(RingElement,Number)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>contract(RingElement,RingElement)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>determinant(Matrix)</kbd> -- see <span><a title="determinant of a matrix" href="_determinant.html">determinant</a> -- determinant of a matrix</span></span></li> <li><span><a title="differentiation" href="_diff_lp__Ring__Element_cm__Ring__Element_rp.html">diff(RingElement,RingElement)</a> -- differentiation</span></li> <li><span><kbd>gcd(RingElement,RingElement)</kbd> -- see <span><a title="greatest common divisor" href="_gcd.html">gcd</a> -- greatest common divisor</span></span></li> <li><span><kbd>generator(Ideal)</kbd> -- see <span><a title="provide a single generator" href="_generator.html">generator</a> -- provide a single generator</span></span></li> <li><span><kbd>generator(Module)</kbd> -- see <span><a title="provide a single generator" href="_generator.html">generator</a> -- provide a single generator</span></span></li> <li><span><kbd>Ideal _ ZZ</kbd> -- see <span><a href="_generators_spof_springs_cm_spideals_cm_spand_spmodules.html">generators of rings, ideals, and modules</a></span></span></li> <li><span><kbd>Ring _ List</kbd> -- see <span><a title="make a monomial from a list of exponents" href="_get_spa_spmonomial_spby_spexponent_spvector.html">get a monomial by exponent vector</a> -- make a monomial from a list of exponents</span></span></li> <li><span><kbd>Ring _ ZZ</kbd> -- see <span><a title="get a ring variable by index" href="_get_spa_spring_spvariable_spby_spindex.html">get a ring variable by index</a> -- get a ring variable by index</span></span></li> <li><span><kbd>String _ Ring</kbd> -- see <span><a title="get a ring variable by name" href="_get_spa_spring_spvariable_spby_spname.html">get a ring variable by name</a> -- get a ring variable by name</span></span></li> <li><span><kbd>homogenize(RingElement,RingElement,List)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><a title="get a ring variable by name" href="___Indexed__Variable_sp_us_sp__Ring.html">IndexedVariable _ Ring</a> -- get a ring variable by name</span></li> <li><span><kbd>leadCoefficient(RingElement)</kbd> -- see <span><a title="the coefficient of the leading term" href="_lead__Coefficient.html">leadCoefficient</a> -- the coefficient of the leading term</span></span></li> <li><span><kbd>leadMonomial(RingElement)</kbd> -- see <span><a title="the leading monomial of a ring element" href="_lead__Monomial.html">leadMonomial</a> -- the leading monomial of a ring element</span></span></li> <li><span><a title="get the greatest term" href="_lead__Term_lp__Ring__Element_rp.html">leadTerm(RingElement)</a> -- get the greatest term</span></li> <li><span><a title="get the lead polynomials using part of the monomial order" href="_lead__Term_lp__Z__Z_cm__Ring__Element_rp.html">leadTerm(ZZ,RingElement)</a> -- get the lead polynomials using part of the monomial order</span></li> <li><span><kbd>RingElement % GroebnerBasis</kbd> -- see <span><a title="compute the normal form of a ring element or matrix using a Gröbner basis" href="___Matrix_sp_pc_sp__Groebner__Basis.html">Matrix % GroebnerBasis</a> -- compute the normal form of a ring element or matrix using a Gröbner basis</span></span></li> <li><span><a title="get entry of matrix" href="___Matrix_sp_us_sp__Sequence.html">Matrix _ Sequence</a> -- get entry of matrix</span></li> <li><span><kbd>RingElement % MonomialIdeal</kbd> -- see <span><a title="normal form of ring elements and matrices" href="_methods_spfor_spnormal_spforms_spand_spremainder.html">methods for normal forms and remainder</a> -- normal form of ring elements and matrices</span></span></li> <li><span><kbd>RingElement % RingElement</kbd> -- see <span><a title="normal form of ring elements and matrices" href="_methods_spfor_spnormal_spforms_spand_spremainder.html">methods for normal forms and remainder</a> -- normal form of ring elements and matrices</span></span></li> <li><span><kbd>part(InfiniteNumber,InfiniteNumber,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(InfiniteNumber,InfiniteNumber,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(InfiniteNumber,ZZ,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(InfiniteNumber,ZZ,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(List,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(Nothing,Nothing,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(Nothing,Nothing,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(Nothing,ZZ,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(Nothing,ZZ,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,InfiniteNumber,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,InfiniteNumber,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,Nothing,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,Nothing,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,ZZ,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><kbd>part(ZZ,ZZ,VisibleList,RingElement)</kbd> -- see <span><a title="select terms of a polynomial by degree(s) or weight(s)" href="_part.html">part</a> -- select terms of a polynomial by degree(s) or weight(s)</span></span></li> <li><span><a title="assemble degrees of a ring, module, or ideal into a polynomial" href="_poincare.html">poincare</a> -- assemble degrees of a ring, module, or ideal into a polynomial</span></li> <li><span><a title="assemble degrees into polynomial" href="_poincare__N.html">poincareN</a> -- assemble degrees into polynomial</span></li> <li><span><kbd>RingElement _ Ring</kbd> -- see <span><a title="promote to another ring" href="_promote.html">promote</a> -- promote to another ring</span></span></li> <li><span><span class="tt">promote(IndexedVariable,type of RingElement)</span> (missing documentation)<!--tag: (promote,IndexedVariable,RingElement)--> </span></li> <li><span><kbd>pseudoRemainder(RingElement,RingElement)</kbd> -- see <span><a title="compute the pseudo-remainder" href="_pseudo__Remainder.html">pseudoRemainder</a> -- compute the pseudo-remainder</span></span></li> <li><span><kbd>RingElement // MonomialIdeal</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><a title="get a random homogeneous element from a graded ideal" href="_random_lp__Z__Z_cm__Ideal_rp.html">random(ZZ,Ideal)</a> -- get a random homogeneous element from a graded ideal</span></li> <li><span><kbd>random(List,Ring)</kbd> -- see <span><a title="get a random homogeneous element from a graded ring" href="_random_lp__Z__Z_cm__Ring_rp.html">random(ZZ,Ring)</a> -- get a random homogeneous element from a graded ring</span></span></li> <li><span><a title="get a random homogeneous element from a graded ring" href="_random_lp__Z__Z_cm__Ring_rp.html">random(ZZ,Ring)</a> -- get a random homogeneous element from a graded ring</span></li> <li><span><a title="fraction" href="___Ring__Element_sp_sl_sp__Ring__Element.html">RingElement / RingElement</a> -- fraction</span></li> <li><span><a title="power" href="___Ring__Element_sp%5E_sp__Z__Z.html">RingElement ^ ZZ</a> -- power</span></li> <li><span><a title="apply a ring map" href="___Ring__Map_sp__Ring__Element.html">RingMap RingElement</a> -- apply a ring map</span></li> <li><span><kbd>someTerms(RingElement,ZZ,ZZ)</kbd> -- see <span><a title="select some terms of a polynomial" href="_some__Terms.html">someTerms</a> -- select some terms of a polynomial</span></span></li> <li><span><kbd>substitute(Number,Ring)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>substitute(Number,RingFamily)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>substitute(RingElement,List)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>substitute(RingElement,Matrix)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>substitute(RingElement,Ring)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>substitute(RingElement,RingFamily)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><a title="get a ring variable by name" href="___Symbol_sp_us_sp__Ring.html">Symbol _ Ring</a> -- get a ring variable by name</span></li> <li><span><a title="trace of a matrix" href="_trace_lp__Matrix_rp.html">trace(Matrix)</a> -- trace of a matrix</span></li> </ul> <h2>Methods that use a ring element:</h2> <ul> <li><span><kbd>Number % RingElement</kbd> -- see <span><a title="a binary operator, usually used for remainder and reduction" href="__pc.html">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li> <li><span><kbd>RingElement % Number</kbd> -- see <span><a title="a binary operator, usually used for remainder and reduction" href="__pc.html">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li> <li><span><kbd>Matrix * RingElement</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>Ring * RingElement</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * Ideal</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * Matrix</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * Module</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * MonomialIdeal</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * MutableMatrix</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>RingElement * Vector</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>Vector * RingElement</kbd> -- see <span><a title="a binary operator, usually used for multiplication" href="__st.html">*</a> -- a binary operator, usually used for multiplication</span></span></li> <li><span><kbd>+ RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Constant + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Ideal + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>InexactNumber + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Matrix + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Number + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + Constant</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + Ideal</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + InexactNumber</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + Matrix</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + Number</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>RingElement + Vector</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Vector + RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for addition" href="__pl.html">+</a> -- a unary or binary operator, usually used for addition</span></span></li> <li><span><kbd>Constant - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>InexactNumber - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>Matrix - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>Number - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - Constant</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - InexactNumber</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - Matrix</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - Number</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>RingElement - Vector</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>Vector - RingElement</kbd> -- see <span><a title="a unary or binary operator, usually used for negation or subtraction" href="_-.html">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li> <li><span><kbd>List // RingElement</kbd> -- see <span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></span></li> <li><span><kbd>Number // RingElement</kbd> -- see <span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></span></li> <li><span><kbd>RingElement // Number</kbd> -- see <span><a title="a binary operator, usually used for quotient" href="__sl_sl.html">//</a> -- a binary operator, usually used for quotient</span></span></li> <li><span><kbd>annihilator(RingElement)</kbd> -- see <span><a title="the annihilator ideal" href="../../Saturation/html/_annihilator.html">annihilator</a> -- the annihilator ideal</span></span></li> <li><span><kbd>antipode(RingElement)</kbd> -- see <span><a title="antipode for skew commuting polynomial rings" href="_antipode.html">antipode</a> -- antipode for skew commuting polynomial rings</span></span></li> <li><span><kbd>baseName(RingElement)</kbd> -- see <span><a title="the base name of a generator" href="_base__Name.html">baseName</a> -- the base name of a generator</span></span></li> <li><span><kbd>binomial(RingElement,ZZ)</kbd> -- see <span><a title="binomial coefficient" href="_binomial.html">binomial</a> -- binomial coefficient</span></span></li> <li><span><kbd>clean(RR,RingElement)</kbd> -- see <span><a title="set to zero elements that are approximately zero" href="_clean.html">clean</a> -- set to zero elements that are approximately zero</span></span></li> <li><span><kbd>coefficients(RingElement)</kbd> -- see <span><a title="get the monomials and coefficients of a polynomial or matrix" href="_coefficients.html">coefficients</a> -- get the monomials and coefficients of a polynomial or matrix</span></span></li> <li><span><kbd>cokernel(RingElement)</kbd> -- see <span><a title="cokernel of a map of modules" href="_cokernel_lp__Matrix_rp.html">cokernel(Matrix)</a> -- cokernel of a map of modules</span></span></li> <li><span><kbd>columnAdd(MutableMatrix,ZZ,RingElement,ZZ)</kbd> -- see <span><a title="add a multiple of one column to another" href="_column__Add.html">columnAdd</a> -- add a multiple of one column to another</span></span></li> <li><span><kbd>columnMult(MutableMatrix,ZZ,RingElement)</kbd> -- see <span><a title="multiply a column by a ring element" href="_column__Mult.html">columnMult</a> -- multiply a column by a ring element</span></span></li> <li><span><kbd>content(RingElement)</kbd> -- see <span><a title="the content of a polynomial" href="_content.html">content</a> -- the content of a polynomial</span></span></li> <li><span><kbd>content(RingElement,RingElement)</kbd> -- see <span><a title="the content of a polynomial" href="_content.html">content</a> -- the content of a polynomial</span></span></li> <li><span><kbd>contract(Matrix,RingElement)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>contract(RingElement,Matrix)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>contract(RingElement,Vector)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><kbd>contract(Vector,RingElement)</kbd> -- see <span><a title="contract a matrix by a matrix" href="_contract_lp__Matrix_cm__Matrix_rp.html">contract(Matrix,Matrix)</a> -- contract a matrix by a matrix</span></span></li> <li><span><a href="_degree_lp__Ring__Element_rp.html">degree(RingElement)</a></span></li> <li><span><a title="degree with respect to a variable" href="_degree_lp__Ring__Element_cm__Ring__Element_rp.html">degree(RingElement,RingElement)</a> -- degree with respect to a variable</span></li> <li><span><kbd>diff(RingElement,Vector)</kbd> -- see <span><a title="differentiate a matrix by a matrix" href="_diff_lp__Matrix_cm__Matrix_rp.html">diff(Matrix,Matrix)</a> -- differentiate a matrix by a matrix</span></span></li> <li><span><kbd>diff(Vector,RingElement)</kbd> -- see <span><a title="differentiate a matrix by a matrix" href="_diff_lp__Matrix_cm__Matrix_rp.html">diff(Matrix,Matrix)</a> -- differentiate a matrix by a matrix</span></span></li> <li><span><a title="differentiation" href="_diff_lp__Matrix_cm__Ring__Element_rp.html">diff(Matrix,RingElement)</a> -- differentiation</span></li> <li><span><a title="differentiate each entry of a matrix" href="_diff_lp__Ring__Element_cm__Matrix_rp.html">diff(RingElement,Matrix)</a> -- differentiate each entry of a matrix</span></li> <li><span><kbd>divideByVariable(Matrix,RingElement)</kbd> -- see <span><a title="divide all columns by a (power of a) variable" href="_divide__By__Variable.html">divideByVariable</a> -- divide all columns by a (power of a) variable</span></span></li> <li><span><kbd>divideByVariable(Matrix,RingElement,ZZ)</kbd> -- see <span><a title="divide all columns by a (power of a) variable" href="_divide__By__Variable.html">divideByVariable</a> -- divide all columns by a (power of a) variable</span></span></li> <li><span><a title="the Alexander dual" href="_dual_lp__Monomial__Ideal_cm__Ring__Element_rp.html">dual(MonomialIdeal,RingElement)</a> -- the Alexander dual</span></li> <li><span><a href="_exp_lp__Ring__Element_rp.html">exp(RingElement)</a></span></li> <li><span><kbd>exponents(RingElement)</kbd> -- see <span><a title="the exponents of a polynomial" href="_exponents.html">exponents</a> -- the exponents of a polynomial</span></span></li> <li><span><a title="factor a ring element" href="_factor_lp__Ring__Element_rp.html">factor(RingElement)</a> -- factor a ring element</span></li> <li><span><kbd>fraction(RingElement,RingElement)</kbd> -- see <span><a href="_fraction.html">fraction</a></span></span></li> <li><span><kbd>gcd(RingElement)</kbd> -- see <span><a title="greatest common divisor" href="_gcd.html">gcd</a> -- greatest common divisor</span></span></li> <li><span><kbd>gcd(RingElement,ZZ)</kbd> -- see <span><a title="greatest common divisor" href="_gcd.html">gcd</a> -- greatest common divisor</span></span></li> <li><span><kbd>gcd(ZZ,RingElement)</kbd> -- see <span><a title="greatest common divisor" href="_gcd.html">gcd</a> -- greatest common divisor</span></span></li> <li><span><kbd>gcdCoefficients(RingElement,RingElement)</kbd> -- see <span><a title="greatest common divisor with coefficients" href="_gcd__Coefficients.html">gcdCoefficients</a> -- greatest common divisor with coefficients</span></span></li> <li><span><kbd>gcdCoefficients(RingElement,ZZ)</kbd> -- see <span><a title="greatest common divisor with coefficients" href="_gcd__Coefficients.html">gcdCoefficients</a> -- greatest common divisor with coefficients</span></span></li> <li><span><kbd>gcdCoefficients(ZZ,RingElement)</kbd> -- see <span><a title="greatest common divisor with coefficients" href="_gcd__Coefficients.html">gcdCoefficients</a> -- greatest common divisor with coefficients</span></span></li> <li><span><kbd>genericMatrix(Ring,RingElement,ZZ,ZZ)</kbd> -- see <span><a title="make a generic matrix of variables" href="_generic__Matrix.html">genericMatrix</a> -- make a generic matrix of variables</span></span></li> <li><span><kbd>genericSkewMatrix(Ring,RingElement,ZZ)</kbd> -- see <span><a title="make a generic skew symmetric matrix of variables" href="_generic__Skew__Matrix.html">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></span></li> <li><span><kbd>genericSymmetricMatrix(Ring,RingElement,ZZ)</kbd> -- see <span><a title="make a generic symmetric matrix" href="_generic__Symmetric__Matrix.html">genericSymmetricMatrix</a> -- make a generic symmetric matrix</span></span></li> <li><span><kbd>homogenize(Ideal,RingElement)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Matrix,RingElement)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Matrix,RingElement,List)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Module,RingElement)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Module,RingElement,List)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(RingElement,RingElement)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Vector,RingElement)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><kbd>homogenize(Vector,RingElement,List)</kbd> -- see <span><a title="homogenize with respect to a variable" href="_homogenize.html">homogenize</a> -- homogenize with respect to a variable</span></span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">Ideal : RingElement</a></span></li> <li><span><a title="make an ideal" href="_ideal_lp__Ring__Element_rp.html">ideal(RingElement)</a> -- make an ideal</span></li> <li><span><kbd>image(RingElement)</kbd> -- see <span><a title="image of a map" href="_image_lp__Matrix_rp.html">image(Matrix)</a> -- image of a map</span></span></li> <li><span><kbd>index(RingElement)</kbd> -- see <span><a title="numeric index of a ring variable" href="_index.html">index</a> -- numeric index of a ring variable</span></span></li> <li><span><a title="indices of variables occurring in a polynomial" href="_indices_lp__Ring__Element_rp.html">indices(RingElement)</a> -- indices of variables occurring in a polynomial</span></li> <li><span><span class="tt">installHilbertFunction(Ideal,RingElement)</span> (missing documentation)<!--tag: (installHilbertFunction,Ideal,RingElement)--> </span></li> <li><span><span class="tt">installHilbertFunction(Matrix,RingElement)</span> (missing documentation)<!--tag: (installHilbertFunction,Matrix,RingElement)--> </span></li> <li><span><span class="tt">installHilbertFunction(Module,RingElement)</span> (missing documentation)<!--tag: (installHilbertFunction,Module,RingElement)--> </span></li> <li><span><kbd>isConstant(RingElement)</kbd> -- see <span><a title="whether a ring element is constant" href="_is__Constant.html">isConstant</a> -- whether a ring element is constant</span></span></li> <li><span><kbd>isHomogeneous(RingElement)</kbd> -- see <span><a title="whether something is homogeneous (graded)" href="_is__Homogeneous.html">isHomogeneous</a> -- whether something is homogeneous (graded)</span></span></li> <li><span><a title="test membership in an ideal" href="_is__Member_lp__Ring__Element_cm__Ideal_rp.html">isMember(RingElement,Ideal)</a> -- test membership in an ideal</span></li> <li><span><span class="tt">isNormal(RingElement)</span> (missing documentation)<!--tag: (isNormal,RingElement)--> </span></li> <li><span><kbd>isPrime(RingElement)</kbd> -- see <span><a title="whether a integer or polynomial is prime" href="_is__Prime.html">isPrime</a> -- whether a integer or polynomial is prime</span></span></li> <li><span><kbd>isUnit(RingElement)</kbd> -- see <span><a title="whether a ring element is a unit" href="_is__Unit.html">isUnit</a> -- whether a ring element is a unit</span></span></li> <li><span><kbd>jacobian(RingElement)</kbd> -- see <span><a title="the matrix of partial derivatives of polynomials in a matrix" href="_jacobian_lp__Matrix_rp.html">jacobian(Matrix)</a> -- the matrix of partial derivatives of polynomials in a matrix</span></span></li> <li><span><kbd>kernel(RingElement)</kbd> -- see <span><a title="kernel of a matrix" href="_kernel_lp__Matrix_rp.html">kernel(Matrix)</a> -- kernel of a matrix</span></span></li> <li><span><kbd>lcm(RingElement)</kbd> -- see <span><a title="least common multiple" href="_lcm.html">lcm</a> -- least common multiple</span></span></li> <li><span><kbd>lcm(RingElement,RingElement)</kbd> -- see <span><a title="least common multiple" href="_lcm.html">lcm</a> -- least common multiple</span></span></li> <li><span><kbd>lcm(RingElement,ZZ)</kbd> -- see <span><a title="least common multiple" href="_lcm.html">lcm</a> -- least common multiple</span></span></li> <li><span><kbd>lcm(ZZ,RingElement)</kbd> -- see <span><a title="least common multiple" href="_lcm.html">lcm</a> -- least common multiple</span></span></li> <li><span><kbd>lift(Ideal,type of RingElement)</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>lift(Matrix,type of RingElement)</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>lift(Module,type of RingElement)</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>lift(MutableMatrix,type of RingElement)</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>lift(Vector,type of RingElement)</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>listForm(RingElement)</kbd> -- see <span><a title="convert to list form" href="_list__Form.html">listForm</a> -- convert to list form</span></span></li> <li><span><a title="construct the map induced by multiplication by a ring element on the generators" href="_map_lp__Module_cm__Module_cm__Ring__Element_rp.html">map(Module,Module,RingElement)</a> -- construct the map induced by multiplication by a ring element on the generators</span></li> <li><span><a href="___Matrix_sp_st_st_sp__Ring__Element.html">Matrix ** RingElement</a></span></li> <li><span><kbd>Matrix ++ RingElement</kbd> -- see <span><a title="direct sum of maps" href="___Matrix_sp_pl_pl_sp__Matrix.html">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li> <li><span><kbd>RingElement ++ Matrix</kbd> -- see <span><a title="direct sum of maps" href="___Matrix_sp_pl_pl_sp__Matrix.html">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li> <li><span><kbd>RingElement ++ RingElement</kbd> -- see <span><a title="direct sum of maps" href="___Matrix_sp_pl_pl_sp__Matrix.html">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li> <li><span><kbd>List / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Matrix / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Vector / RingElement</kbd> -- see <span><a title="scalar division" href="___Matrix_sp_sl_sp__Number.html">Matrix / Number</a> -- scalar division</span></span></li> <li><span><kbd>Matrix | RingElement</kbd> -- see <span><a title="join matrices horizontally" href="___Matrix_sp_vb_sp__Matrix.html">Matrix | Matrix</a> -- join matrices horizontally</span></span></li> <li><span><kbd>RingElement | Matrix</kbd> -- see <span><a title="join matrices horizontally" href="___Matrix_sp_vb_sp__Matrix.html">Matrix | Matrix</a> -- join matrices horizontally</span></span></li> <li><span><kbd>RingElement | RingElement</kbd> -- see <span><a title="join matrices horizontally" href="___Matrix_sp_vb_sp__Matrix.html">Matrix | Matrix</a> -- join matrices horizontally</span></span></li> <li><span><kbd>Matrix || RingElement</kbd> -- see <span><a title="join matrices vertically" href="___Matrix_sp_vb_vb_sp__Matrix.html">Matrix || Matrix</a> -- join matrices vertically</span></span></li> <li><span><kbd>RingElement || Matrix</kbd> -- see <span><a title="join matrices vertically" href="___Matrix_sp_vb_vb_sp__Matrix.html">Matrix || Matrix</a> -- join matrices vertically</span></span></li> <li><span><kbd>RingElement || RingElement</kbd> -- see <span><a title="join matrices vertically" href="___Matrix_sp_vb_vb_sp__Matrix.html">Matrix || Matrix</a> -- join matrices vertically</span></span></li> <li><span><a title="make a matrix from a ring element" href="_matrix_lp__Ring__Element_rp.html">matrix(RingElement)</a> -- make a matrix from a ring element</span></li> <li><span><kbd>Matrix % RingElement</kbd> -- see <span><a title="normal form of ring elements and matrices" href="_methods_spfor_spnormal_spforms_spand_spremainder.html">methods for normal forms and remainder</a> -- normal form of ring elements and matrices</span></span></li> <li><span><kbd>RingElement % Ideal</kbd> -- see <span><a title="normal form of ring elements and matrices" href="_methods_spfor_spnormal_spforms_spand_spremainder.html">methods for normal forms and remainder</a> -- normal form of ring elements and matrices</span></span></li> <li><span><kbd>RingElement % Matrix</kbd> -- see <span><a title="normal form of ring elements and matrices" href="_methods_spfor_spnormal_spforms_spand_spremainder.html">methods for normal forms and remainder</a> -- normal form of ring elements and matrices</span></span></li> <li><span><kbd>Module / RingElement</kbd> -- see <span><a title="quotient module" href="___Module_sp_sl_sp__Module.html">Module / Module</a> -- quotient module</span></span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">Module : RingElement</a></span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">MonomialIdeal : RingElement</a></span></li> <li><span><kbd>monomialIdeal(RingElement)</kbd> -- see <span><a title="monomial ideal of lead monomials" href="_monomial__Ideal_lp__Matrix_rp.html">monomialIdeal(Matrix)</a> -- monomial ideal of lead monomials</span></span></li> <li><span><kbd>monomials(RingElement)</kbd> -- see <span><a title="matrix of monomials in a ring element or matrix" href="_monomials.html">monomials</a> -- matrix of monomials in a ring element or matrix</span></span></li> <li><span><kbd>norm(InfiniteNumber,RingElement)</kbd> -- see <span><a href="_norm.html">norm</a></span></span></li> <li><span><kbd>norm(RingElement)</kbd> -- see <span><a href="_norm.html">norm</a></span></span></li> <li><span><kbd>norm(RR,RingElement)</kbd> -- see <span><a href="_norm.html">norm</a></span></span></li> <li><span><a title="The normal cone of a subscheme" href="../../ReesAlgebra/html/_associated__Graded__Ring.html">normalCone(Ideal,RingElement)</a></span></li> <li><span><kbd>parts(RingElement)</kbd> -- see <span><a title="display terms of a polynomial degree by degree" href="_parts.html">parts</a> -- display terms of a polynomial degree by degree</span></span></li> <li><span><kbd>precision(RingElement)</kbd> -- see <span><a href="_precision.html">precision</a></span></span></li> <li><span><span class="tt">promote(Constant,type of RingElement)</span> (missing documentation)<!--tag: (promote,Constant,RingElement)--> </span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">quotient(Ideal,RingElement)</a></span></li> <li><span><kbd>Matrix // RingElement</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>RingElement // GroebnerBasis</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><kbd>RingElement // Matrix</kbd> -- see <span><a title="factor a map through another with the same target" href="_quotient_lp__Matrix_cm__Matrix_rp.html">quotient(Matrix,Matrix)</a> -- factor a map through another with the same target</span></span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">quotient(Module,RingElement)</a></span></li> <li><span><a title="ideal or submodule quotient" href="../../Saturation/html/_quotient_lp__Module_cm__Module_rp.html">quotient(MonomialIdeal,RingElement)</a></span></li> <li><span><kbd>quotientRemainder(InexactNumber,RingElement)</kbd> -- see <span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li> <li><span><kbd>quotientRemainder(Number,RingElement)</kbd> -- see <span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li> <li><span><kbd>quotientRemainder(RingElement,InexactNumber)</kbd> -- see <span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li> <li><span><kbd>quotientRemainder(RingElement,Number)</kbd> -- see <span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li> <li><span><a title="quotient and remainder" href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></li> <li><span><kbd>ring(RingElement)</kbd> -- see <span><a title="get the associated ring of an object" href="_ring.html">ring</a> -- get the associated ring of an object</span></span></li> <li><span><kbd>Ring / RingElement</kbd> -- see <span><a title="make a quotient ring" href="___Ring_sp_sl_sp__Ideal.html">Ring / Ideal</a> -- make a quotient ring</span></span></li> <li><span><a title="a sequence of consecutive generators of a polynomial ring" href="___Ring__Element_sp.._sp__Ring__Element.html">RingElement .. RingElement</a> -- a sequence of consecutive generators of a polynomial ring</span></li> <li><span><a title="a sequence of consecutive generators of a polynomial ring" href="___Ring__Element_sp.._lt_sp__Ring__Element.html">RingElement ..< RingElement</a> -- a sequence of consecutive generators of a polynomial ring</span></li> <li><span><kbd>RingElement Number</kbd> -- see <span><a title="evaluation of polynomials" href="___Ring__Element_sp__Sequence.html">RingElement Sequence</a> -- evaluation of polynomials</span></span></li> <li><span><kbd>RingElement RingElement</kbd> -- see <span><a title="evaluation of polynomials" href="___Ring__Element_sp__Sequence.html">RingElement Sequence</a> -- evaluation of polynomials</span></span></li> <li><span><a title="evaluation of polynomials" href="___Ring__Element_sp__Sequence.html">RingElement Sequence</a> -- evaluation of polynomials</span></li> <li><span><a title="compute the roots of a polynomial" href="_roots_lp__Ring__Element_rp.html">roots(RingElement)</a> -- compute the roots of a polynomial</span></li> <li><span><kbd>rowAdd(MutableMatrix,ZZ,RingElement,ZZ)</kbd> -- see <span><a title="add a multiple of one row to another" href="_row__Add.html">rowAdd</a> -- add a multiple of one row to another</span></span></li> <li><span><kbd>rowMult(MutableMatrix,ZZ,RingElement)</kbd> -- see <span><a title="multiply a row by a ring element" href="_row__Mult.html">rowMult</a> -- multiply a row by a ring element</span></span></li> <li><span><kbd>saturate(Ideal,RingElement)</kbd> -- see <span><a title="saturation of ideal or submodule" href="../../Saturation/html/_saturate.html">saturate</a> -- saturation of ideal or submodule</span></span></li> <li><span><kbd>saturate(Module,RingElement)</kbd> -- see <span><a title="saturation of ideal or submodule" href="../../Saturation/html/_saturate.html">saturate</a> -- saturation of ideal or submodule</span></span></li> <li><span><kbd>saturate(MonomialIdeal,RingElement)</kbd> -- see <span><a title="saturation of ideal or submodule" href="../../Saturation/html/_saturate.html">saturate</a> -- saturation of ideal or submodule</span></span></li> <li><span><kbd>saturate(Vector,RingElement)</kbd> -- see <span><a title="saturation of ideal or submodule" href="../../Saturation/html/_saturate.html">saturate</a> -- saturation of ideal or submodule</span></span></li> <li><span><kbd>size(RingElement)</kbd> -- see <span><a title="the size of an object" href="_size.html">size</a> -- the size of an object</span></span></li> <li><span><kbd>standardForm(RingElement)</kbd> -- see <span><a title="convert to standard form" href="_standard__Form.html">standardForm</a> -- convert to standard form</span></span></li> <li><span><kbd>substitute(RingElement,Option)</kbd> -- see <span><a title="substituting values for variables" href="_substitute.html">substitute</a> -- substituting values for variables</span></span></li> <li><span><kbd>support(RingElement)</kbd> -- see <span><a title="list of variables occurring in a polynomial or matrix" href="_support.html">support</a> -- list of variables occurring in a polynomial or matrix</span></span></li> <li><span><kbd>terms(Ring,RingElement)</kbd> -- see <span><a title="provide a list of terms of a polynomial" href="_terms.html">terms</a> -- provide a list of terms of a polynomial</span></span></li> <li><span><kbd>terms(RingElement)</kbd> -- see <span><a title="provide a list of terms of a polynomial" href="_terms.html">terms</a> -- provide a list of terms of a polynomial</span></span></li> <li><span><kbd>topCoefficients(RingElement)</kbd> -- see <span><a title="first variable and its coefficient of a polynomial or matrix" href="_top__Coefficients.html">topCoefficients</a> -- first variable and its coefficient of a polynomial or matrix</span></span></li> <li><span><kbd>vector(Module,RingElement)</kbd> -- see <span><a title="make a vector" href="_vector.html">vector</a> -- make a vector</span></span></li> <li><span><kbd>vector(Ring,RingElement)</kbd> -- see <span><a title="make a vector" href="_vector.html">vector</a> -- make a vector</span></span></li> <li><span><kbd>vector(RingElement)</kbd> -- see <span><a title="make a vector" href="_vector.html">vector</a> -- make a vector</span></span></li> <li><span><kbd>vector(RingFamily,RingElement)</kbd> -- see <span><a title="make a vector" href="_vector.html">vector</a> -- make a vector</span></span></li> <li><span><kbd>weightRange(List,RingElement)</kbd> -- see <span><a title="the pair of lowest and highest weights of the monomials" href="_weight__Range.html">weightRange</a> -- the pair of lowest and highest weights of the monomials</span></span></li> <li><span><kbd>weightRange(RingElement)</kbd> -- see <span><a title="the pair of lowest and highest weights of the monomials" href="_weight__Range.html">weightRange</a> -- the pair of lowest and highest weights of the monomials</span></span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="the class of all ring elements handled by the engine" href="___Ring__Element.html">RingElement</a> is <span>a <a title="the class of all mutable types" href="___Type.html">type</a></span>, with ancestor classes <a title="the class of all basic lists" href="___Basic__List.html">BasicList</a> < <a title="the class of all things" href="___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Macaulay2Doc/doc_rings.m2:177:0</span>.</p> </div> </div> </div> </body> </html>
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