  
  
                              [1X[5XGradedRingForHomalg[105X[101X
  
  
                [1XEndow Commutative Rings with an Abelian Grading[101X
  
  
                               Version 2020.01.02
  
  
                                 September 2015
  
  
                                Mohamed Barakat
  
                               Sebastian Gutsche
  
                             Markus Lange-Hegermann
  
  
  
            [33X[0;10YThis  manual  is  best  viewed  as an [12XHTML[112X document. The
            latest version is available [12Xonline[112X at:[133X
            [33X[0;10Y[7Xhttp://homalg.math.rwth-aachen.de/~markus/GradedRingForHomalg/chap0.html[107X[133X
            [33X[0;10YAn   [12Xoffline[112X   version   should   be   included  in  the
            documentation subfolder of the package.[133X
  
  
  
  Mohamed Barakat
      Email:    [7Xmailto:mohamed.barakat@uni-siegen.de[107X
      Homepage: [7Xhttps://mohamed-barakat.github.io[107X
      Address:  [33X[0;14YDepartment of Mathematics,[133X
                [33X[0;14YUniversity of Siegen,[133X
                [33X[0;14Y57072 Siegen,[133X
                [33X[0;14YGermany[133X
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:sebastian.gutsche@rwth-aachen.de[107X
      Homepage: [7Xhttp://wwwb.math.rwth-aachen.de/~gutsche/[107X
      Address:  [33X[0;14YLehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany[133X
  
  
  Markus Lange-Hegermann
      Email:    [7Xmailto:markus.lange.hegermann@rwth-aachen.de[107X
      Homepage: [7Xhttp://wwwb.math.rwth-aachen.de/~markus[107X
      Address:  [33X[0;14YLehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©   2008-2015   by   Mohamed   Barakat,   Sebastian   Gutsche,   and  Markus
  Lange-Hegermann[133X
  
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2 or (at your option) any later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  would  like  to  thank the Aachen forest for being such a nice place for
  jogging.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (GradedRingForHomalg)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YWhat is the Role of the [5XGradedRingForHomalg[105X Package in the [5Xhomalg[105X
    Project?[133X
    1.2 [33X[0;0YFunctionality[133X
    1.3 [33X[0;0YThe Math Behind This Package[133X
  2 [33X[0;0YInstallation of the [5XGradedRingForHomalg[105X Package[133X
  3 [33X[0;0YQuick Start[133X
  4 [33X[0;0YGraded Rings[133X
    4.1 [33X[0;0YGraded Rings: Category and Representations[133X
      4.1-1 IsHomalgGradedRingRep
      4.1-2 IsHomalgGradedRingElementRep
    4.2 [33X[0;0YGraded Rings: Constructors[133X
      4.2-1 HomalgGradedRingElement
    4.3 [33X[0;0YGraded Rings: Attributes and Properties[133X
      4.3-1 DegreeGroup
      4.3-2 CommonNonTrivialWeightOfIndeterminates
      4.3-3 WeightsOfIndeterminates
      4.3-4 IsHomogeneousRingElement
    4.4 [33X[0;0YGraded Rings: Operations and Functions[133X
      4.4-1 UnderlyingNonGradedRing
      4.4-2 UnderlyingNonGradedRing
      4.4-3 Name
      4.4-4 HomogeneousPartOfRingElement
  5 [33X[0;0YHomogeneous Matrices[133X
    5.1 [33X[0;0YHomogeneous Matrices: Category and Representations[133X
      5.1-1 IsHomalgMatrixOverGradedRingRep
    5.2 [33X[0;0YHomogeneous Matrices: Constructors[133X
      5.2-1 MatrixOverGradedRing
    5.3 [33X[0;0YHomogeneous Matrices: Attributes[133X
      5.3-1 DegreesOfEntries
      5.3-2 NonTrivialDegreePerRow
      5.3-3 NonTrivialDegreePerColumn
      5.3-4 HomogeneousPartOfMatrix
      5.3-5 IsMatrixOverGradedRingWithHomogeneousEntries
    5.4 [33X[0;0YHomogeneous Matrices: Operations and Functions[133X
      5.4-1 UnderlyingNonGradedRing
      5.4-2 SetMatElm
      5.4-3 AddToMatElm
      5.4-4 MatElmAsString
      5.4-5 MatElm
  6 [33X[0;0YExamples[133X
  A [33X[0;0YThe Matrix Tool Operations[133X
    A.1 [33X[0;0YThe Tool Operations [13Xwithout[113X a Fallback Method[133X
    A.2 [33X[0;0YThe Tool Operations with a Fallback Method[133X
      A.2-1 MonomialMatrix
      A.2-2 RandomMatrixBetweenGradedFreeLeftModules
      A.2-3 RandomMatrixBetweenGradedFreeRightModules
      A.2-4 Diff
  B [33X[0;0YOverview of the [5XGradedRingForHomalg[105X Package Source Code[133X
    B.1 [33X[0;0YThe generic Methods[133X
      B.1-1 BasisOfRowModule
      B.1-2 DecideZeroRows
      B.1-3 SyzygiesGeneratorsOfRows
    B.2 [33X[0;0YTools[133X
  
  
  [32X
