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<h1>GradedRingForHomalg</h1>
<h2>Endow Commutative Rings with an Abelian Grading</h2>
<p>
2024.07-01</p>
<p>
9 July 2024
</p>
</div>
<p><b>
Mohamed Barakat
</b>
<br />Email: <span class="URL"><a href="mailto:mohamed.barakat@uni-siegen.de">mohamed.barakat@uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://mohamed-barakat.github.io">https://mohamed-barakat.github.io</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57072 Siegen<br /> Germany<br />
</p><p><b>
Sebastian Gutsche
</b>
<br />Email: <span class="URL"><a href="mailto:gutsche@mathematik.uni-siegen.de">gutsche@mathematik.uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://sebasguts.github.io">https://sebasguts.github.io</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57072 Siegen<br /> Germany<br />
</p><p><b>
Markus Kirschmer
</b>
<br />Email: <span class="URL"><a href="mailto:markus.kischmer@math.rwth-aachen.de">markus.kischmer@math.rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://www.math.rwth-aachen.de/~Markus.Kirschmer/">http://www.math.rwth-aachen.de/~Markus.Kirschmer/</a></span>
<br />Address: <br />Markus Kirschmer<br /> Lehrstuhl D fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<br /> Germany<br />
</p><p><b>
Sebastian Jambor
</b>
<br />Email: <span class="URL"><a href="mailto:sebastian.jambor@rwth-aachen.de">sebastian.jambor@rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://wwwb.math.rwth-aachen.de/~sebastian/">http://wwwb.math.rwth-aachen.de/~sebastian/</a></span>
<br />Address: <br />Sebastian Jambor<br /> Lehrstuhl B fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<br /> Germany<br />
</p><p><b>
Markus Lange-Hegermann
</b>
<br />Email: <span class="URL"><a href="mailto:markus.lange-hegermann@hs-owl.de">markus.lange-hegermann@hs-owl.de</a></span>
<br />Homepage: <span class="URL"><a href="https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/">https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/</a></span>
<br />Address: <br />Markus Lange-Hegermann<br /> Hochschule Ostwestfalen-Lippe<br /> Liebigstraße 87<br /> 32657 Lemgo<br /> Germany<br />
</p><p><b>
Daniel Robertz
</b>
<br />Email: <span class="URL"><a href="mailto:daniel@momo.math.rwth-aachen.de">daniel@momo.math.rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://wwwb.math.rwth-aachen.de/~daniel/">http://wwwb.math.rwth-aachen.de/~daniel/</a></span>
<br />Address: <br />Daniel Robertz<br /> Lehrstuhl B fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<br /> Germany<br />
</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7F7255BA7A43157E">1.1 <span class="Heading">What is the Role of the <strong class="pkg">GradedRingForHomalg</strong> Package in
the <strong class="pkg">homalg</strong> Project?</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X87F1120883F5B4D0">1.2 <span class="Heading">Functionality</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X83CA25F97F1EECDB">2 <span class="Heading">Installation of the <strong class="pkg">GradedRingForHomalg</strong> Package</span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X810D68A7794474F9">3 <span class="Heading">Graded Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A16AC227F0B7E6B">3.1 <span class="Heading">Graded Rings: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8025623B822D44AF">3.1-1 IsHomalgGradedRingRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X84BC7F817BF737F7">3.1-2 IsHomalgGradedRingElementRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8538A7C985FE0E46">3.2 <span class="Heading">Graded Rings: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X823140CE8286BA90">3.2-1 HomalgGradedRingElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7CA805077B75EAAD">3.3 <span class="Heading">Graded Rings: Attributes and Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X79649FC47A744BD2">3.3-1 DegreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X826C131D83796CC5">3.3-2 CommonNonTrivialWeightOfIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7E43637F8431FB69">3.3-3 WeightsOfIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X807A4268816DD69D">3.3-4 IsHomogeneousRingElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B86A3DD7D2CE35F">3.4 <span class="Heading">Graded Rings: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7DA1D14F87DE72D8">3.4-1 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8708576B84997122">3.4-2 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X83971F1879017E3D">3.4-3 Name</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X840864B38184FF3E">3.4-4 HomogeneousPartOfRingElement</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X81274D247D293332">4 <span class="Heading">Homogeneous Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7912A7EE7DD9C130">4.1 <span class="Heading">Homogeneous Matrices: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X860DABC0806C4064">4.1-1 IsHomalgMatrixOverGradedRingRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7E3D72117F84D517">4.2 <span class="Heading">Homogeneous Matrices: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AC02E03868CB664">4.2-1 MatrixOverGradedRing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C1C5AA2813F4F73">4.3 <span class="Heading">Homogeneous Matrices: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8608086E82701204">4.3-1 DegreesOfEntries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86C3DD8479FDDB36">4.3-2 NonTrivialDegreePerRow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X801EBD4D7DAB0672">4.3-3 NonTrivialDegreePerColumn</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B109D738475FED4">4.3-4 HomogeneousPartOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B28E6B97D1A5E09">4.3-5 IsMatrixOverGradedRingWithHomogeneousEntries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81897DD8835ACE8C">4.4 <span class="Heading">Homogeneous Matrices: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7DC25A5E83DC3E85">4.4-1 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D0D1E1E784C3BC7">4.4-2 SetMatElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B56EC5E8545C1B6">4.4-3 AddToMatElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X865E51967E6D0AD3">4.4-4 MatElmAsString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8028986083BEC896">4.4-5 MatElm</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapA_mj.html#X7B2993CB7B012115">A <span class="Heading">The Matrix Tool Operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7988F0AF7D87FD23">A.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7912E42C81296637">A.2 <span class="Heading">The Tool Operations with a Fallback Method</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA_mj.html#X84C78C6B7C0890BF">A.2-1 MonomialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA_mj.html#X7F9F9C978703E871">A.2-2 RandomMatrixBetweenGradedFreeLeftModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA_mj.html#X82E813D982331C0B">A.2-3 RandomMatrixBetweenGradedFreeRightModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA_mj.html#X7EA1DC07822DE322">A.2-4 Diff</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapB_mj.html#X78C55DF7875560DD">B <span class="Heading">Overview of the <strong class="pkg">GradedRingForHomalg</strong> Package Source Code</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapB_mj.html#X87807E467C364A00">B.1 <span class="Heading">The generic Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB_mj.html#X7D9FBBDE794DE447">B.1-1 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB_mj.html#X84C649FA7C820D49">B.1-2 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB_mj.html#X87D2B8748076B5DD">B.1-3 SyzygiesGeneratorsOfRows</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapB_mj.html#X8508AEF8845565A1">B.2 <span class="Heading">Tools</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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