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<title>GAP (GradedRingForHomalg) - Chapter 3: Graded Rings</title>
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< class=ContSS">br/<span =""&;nbsp;<span><a href=chap3.html#X84BC7F817BF737F7">3.1-2 IsHomalgGradedRingElementRep</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E43637F8431FB69">3.3-3 WeightsOfIndeterminates</a></span>
<spanclass="ontSS>< /><span class="nocss>nbsp; </span><a href="chap3.html#X807A4268816DD69D">3.3-4 IsHomogeneousRingElement</a></span>
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<div class="ContSect"><< class=ContSS"< />span class=nocss"&;&bsp;/span>ahref=chap3.#X8025623B822D44AF">3.1-1 IsHomalgGradedRingRep</a></span>
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<span class="ContSS"><br /><span class="nocss"><div class""<span ""< "nocss"&;/<a ="chap3.#X8538A7C985FE0E46">3.2 <span class="Heading">Graded Rings: Constructors</span></a>
<pan ">br/< class"nocss"&;nbsp;<span< hrefhref=chap3.html#X8708576B84997122">3.4-2 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="span class"ContSS">br/<span ""&&;<span>< href=chap3.html#X823140CE8286BA90">3.2-1 HomalgGradedRingElement</a></span>
< class=ContSS">br/><class"">nbsp;&bsp;</< "chap3html#X840864B38184FF3E">3.4-4 HomogeneousPartOfRingElement</a></span>
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<h3>3 <span class="Heading">Graded Rings</span></h3>

<p>The package <strong spanclass=ContSS"<br >spanclass="nocss>nbsp;nbsp;/span>< href=chap3.#X807A4268816DD69D">3.3-4 IsHomogeneousRingElement</a></span>

pThegradedrings prominently  beused with  known from  strong=pkg>/strong>rings.The methods  doingthe  arepresented  theappendix <a =chapB.java.lang.StringIndexOutOfBoundsException: Range [365, 231) out of bounds for length 365

<p>Since the objects inplemented here are representations from objects /span>

<p>Operations within <strong class="pkg">MatricesForHomalg< classclass=>nbsp&<span>a href"hap3html#X7DA1D14F87DE72D8">3.4-1 UnderlyingNonGradedRing</a></span>


<ul>
<li><p><code class="code">SyzygiesGeneratorsOfRows</code></p>

</li>
<li><p><code class="code"><span class="ContSS"><br /><span class="nocss"> /><a href="chap3.html#X840864B38184FF3E">3.4-4 HomogeneousPartOfRingElement</a></span>div

</li>
<li><p>

<li>
<li><p><code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
<li><p><code class="code">BasisOfRowModule</code></p>

</li>
<li><p><code class="code">

</li>
<li><p><code class

</li>
<li><p><code class=<><>< class=code"SyzygiesGeneratorsOfRows</code>p>

</li>
<li><p><code class="code">DecideZeroRows</java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0

</li>
<li><p><code class="code">DecideZeroColumns</code></p>

<li>
<li><p><code class="code">LeftDivide</codeli<><code =code"ReducedSyzygiesGeneratorsOfColumnscode<p>

</li>
<li><p><code class="

</li>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 computationsin< class=pkg></strong,which always forced to  performed with atail reduction by<strongclass=pkg"homalg/strong> Inparticular,the  elements  the Grö baseshave   .</>

<p><a id="X7A16AC227F0B7E6B" name="X7A16AC227F0B7E6B"></a></p>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>3.1-1 IsHomalgGradedRingRep</h5>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
pReturns: or false<>

pTherepresentation of strong class=pkg><strong> graded rings.</p>

<p>(It is a subrepresentation of the <strong class="pkg">GAP</strong> representation <br /> <code class="code">IsHomalgRingOrFinitelyPresentedModuleRep</code>.)</p>


<div class="example"><preLeftDivide<code</>
<li>p<code class""></code<p>
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        IsHomalgGradedRingOrGradedModuleRep
        [ p>< =X7A16AC227F0B7E6B"name="7A16AC227F0B7E6B">/a>/p>
</pre></div>

<p><a id="X84BC7F817BF737F7" name="X84BC7F817BF737F7"></a></p>

<h5>3.1-2 IsHomalgGradedRingElementRep</h5>

divclass=func> class"func"width"%>tr<td class=tdleft"<code =func>#8227;IsHomalgGradedRingElementRep<code( < class=">r/> <td<td =t">&; </d<tr>/>/div>
<p>Returns: true or false</p>

<p>The java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0

<p>(It is a representation of the <strong class="pkg">GAP</strong> categoryp>Returns:true />


<pThejava.lang.StringIndexOutOfBoundsException: Range [25, 21) out of bounds for length 78
 "sHomalgGradedRingElementRep
        IsHomalgGradedRingElement,
        [ ] );
</pre></div>

<p><a id="X8538A7C985FE0E46" name="X8538A7C985FE0E46"></a></p>

<h4>3.2 <span class="Heading">Graded Rings: Constructors</span></h4>

<p><a id="X823140CE8286BA90" name="X823140CE8286BA90"></a></p>

<h5>3.2-1 HomalgGradedRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; HomalgGradedRingElement</code>( <var class="Arg">numer</var>, <var class="Arg">denom</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; HomalgGradedRingElement</code>( <var class="Arg">numer</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a graded ring element</p>

<p>Creates the graded ring element <span class="SimpleMath"><var class="Arg">numer</var>/<var class="Arg">denom</var></span> or in the second case <span class="SimpleMath"><var class="Arg">numer</var>/1</span> for the graded ring <var class="Arg">R</var>. Both <var class="Arg">numer</var> and <var class="Arg">denom</var> may either be a string describing a valid global ring element or from the global ring or computation ring.</p>

<p><a id="X7CA805077B75EAAD" name="X7CA805077B75EAAD"></a></p>

<h4>3.3 <span class="Heading">Graded Rings: Attributes and Properties</span></h4>

<p><a id="X79649FC47A744BD2" name="X79649FC47A744BD2"></a></p>

<h5>3.3-1 DegreeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; DegreeGroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a left ℤ-module</p>

<p>The degree Abelian group of the commutative graded ring <var class="Arg">S</var>.</p>

<p><a id="X826C131D83796CC5" name="X826C131D83796CC5"></a></p>

<h5>3.3-2 CommonNonTrivialWeightOfIndeterminates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; CommonNonTrivialWeightOfIndeterminates</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a degree</p>

<p>The common nontrivial weight of the indeterminates of the graded ring <var class="Arg">S</var> if it exists. Otherwise an error is issued. WARNING: Since the DegreeGroup and WeightsOfIndeterminates are in some cases bound together, you MUST not set the DegreeGroup by hand and let the algorithm create the weights. Set both by hand, set only weights or use the method WeightsOfIndeterminates to set both. Never set the DegreeGroup without the WeightsOfIndeterminates, because it simply wont work!</p>

<p><a id="X7E43637F8431FB69" name="X7E43637F8431FB69"></a></p>

<h5>3.3-3 WeightsOfIndeterminates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; WeightsOfIndeterminates</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list or listlist of integers</p>

<p>The list of degrees of the indeterminates of the graded ring <var class="Arg">S</var>.</p>

<p><a id="X807A4268816DD69D" name="X807A4268816DD69D"></a></p>

<h5>3.3-4 IsHomogeneousRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; IsHomogeneousRingElement</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>returns whether the graded ring element <var class="Arg">r</var> is homogeneous or not.</p>

<p><a id="X7B86A3DD7D2CE35F" name="X7B86A3DD7D2CE35F"></a></p>

<h4>3.4 <span class="Heading">Graded Rings: Operations and Functions</span></h4>

<p><a id="X7DA1D14F87DE72D8" name="X7DA1D14F87DE72D8"></a></p>

<h5>3.4-1 UnderlyingNonGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; UnderlyingNonGradedRing</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> ring</p>

<p>Internally there is a ring, in which computations take place.</p>

<p><a id="X8708576B84997122" name="X8708576B84997122"></a></p>

<h5>3.4-2 UnderlyingNonGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; UnderlyingNonGradedRing</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> ring</p>

<p>Internally there is a ring, in which computations take place.</p>

<p><a id="X83971F1879017E3D" name="X83971F1879017E3D"></a></p>

<h5>3.4-3 Name</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; Name</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a string</p>

<p>The name of the graded ring element <var class="Arg">r</var>.</p>

<p><a id="X840864B38184FF3E" name="X840864B38184FF3E"></a></p>

<h5>3.4-4 HomogeneousPartOfRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; HomogeneousPartOfRingElement</code>( <var class="Arg">r</var>, <var class="Arg">degree</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a graded ring element</p>

<p>returns the summand of <var class="Arg">r</var> whose monomials have the given degree <var class="Arg">degree</var> and if <var class="Arg">r</var> has no such monomials then it returns the zero element of the ring.</p>


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