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# GradedRingForHomalg

## Endow Commutative Rings with an Abelian Grading

2022.11-01

23 November 2022

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57072 Siegen
Germany

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Markus Kirschmer
Email: markus.kischmer@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Markus.Kirschmer/
Address:
Markus Kirschmer
Lehrstuhl D fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

Sebastian Jambor
Email: sebastian.jambor@rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~sebastian/
Address:
Sebastian Jambor
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@hs-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Daniel Robertz
Email: daniel@momo.math.rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~daniel/
Address:
Daniel Robertz
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

### Contents

1 Introduction
1.1 What is the Role of the GradedRingForHomalg Package in the homalg Project?
1.2 Functionality
2 Installation of the GradedRingForHomalg Package
3 Graded Rings
3.1 Graded Rings: Category and Representations
3.1-1 IsHomalgGradedRingRep
3.1-2 IsHomalgGradedRingElementRep
3.2 Graded Rings: Constructors
3.2-1 HomalgGradedRingElement
3.3 Graded Rings: Attributes and Properties
3.3-1 DegreeGroup
3.3-2 CommonNonTrivialWeightOfIndeterminates
3.3-3 WeightsOfIndeterminates
3.3-4 IsHomogeneousRingElement
3.4 Graded Rings: Operations and Functions
3.4-1 UnderlyingNonGradedRing
3.4-2 UnderlyingNonGradedRing
3.4-3 Name
3.4-4 HomogeneousPartOfRingElement
4 Homogeneous Matrices
4.1 Homogeneous Matrices: Category and Representations
4.1-1 IsHomalgMatrixOverGradedRingRep
4.2 Homogeneous Matrices: Constructors
4.2-1 MatrixOverGradedRing
4.3 Homogeneous Matrices: Attributes
4.3-1 DegreesOfEntries
4.3-2 NonTrivialDegreePerRow
4.3-3 NonTrivialDegreePerColumn
4.3-4 HomogeneousPartOfMatrix
4.3-5 IsMatrixOverGradedRingWithHomogeneousEntries
4.4 Homogeneous Matrices: Operations and Functions
4.4-1 UnderlyingNonGradedRing
4.4-2 SetMatElm
4.4-3 AddToMatElm
4.4-4 MatElmAsString
4.4-5 MatElm
A The Matrix Tool Operations
A.1 The Tool Operations without a Fallback Method
A.2 The Tool Operations with a Fallback Method
A.2-1 MonomialMatrix
A.2-2 RandomMatrixBetweenGradedFreeLeftModules
A.2-3 RandomMatrixBetweenGradedFreeRightModules
A.2-4 Diff
B Overview of the GradedRingForHomalg Package Source Code
B.1 The generic Methods
B.1-1 BasisOfRowModule
B.1-2 DecideZeroRows
B.1-3 SyzygiesGeneratorsOfRows
B.2 Tools
References
Index
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