CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutSign UpSign In

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place.

| Download

GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

Views: 418346
Tweet
Share
Share
1
<?xml version="1.0" encoding="UTF-8"?>

2


3
<!-- 

4


5
  Rings.xml            MatricesForHomalg package documentation            Mohamed Barakat

6


7
         Copyright (C) 2007-2010, Mohamed Barakat, RWTH-Aachen University

8


9
-->

10


11
<Chapter Label="Rings">

12
<Heading>Rings</Heading>

13


14
<Section Label="Rings:Category">

15
<Heading>Rings: Category and Representations</Heading>

16


17
<#Include Label="IsHomalgRing">

18


19
<#Include Label="IsPreHomalgRing">

20


21
<#Include Label="IsHomalgRingElement">

22


23
<#Include Label="IsHomalgInternalRingRep">

24


25
</Section>

26


27
<Section Label="Rings:Constructors">

28
<Heading>Rings: Constructors</Heading>

29


30
This section describes how to construct rings for use with &MatricesForHomalg;,

31
which exploit the &GAP4;-built-in abilities to perform the necessary

32
ring operations. By this we also mean necessary matrix operations over

33
such rings. For the purposes of &MatricesForHomalg; only the ring of integers is

34
properly supported in &GAP4;. The &GAP4; extension packages &Gauss;

35
and &GaussForHomalg; extend these built-in abilities to operations

36
with sparse matrices over the ring <M>&ZZ; / p^n</M> for <M>p</M>

37
prime and <M>n</M> positive.<P/>

38


39
If a ring <M>R</M> is supported in &MatricesForHomalg; any of its residue class

40
rings <M>R/I</M> is supported as well, provided the ideal <M>I</M> of

41
relations admits a finite set of generators as a left resp. right

42
ideal (&see; <Ref Oper="\/"

43
Label="constructor for residue class rings" Style="Number"/>).

44
This is immediate for commutative noetherian rings.

45


46
<#Include Label="HomalgRingOfIntegers">

47
<#Include Label="HomalgFieldOfRationals">

48
<#Include Label="ResidueClassRingConstructor">

49


50
</Section>

51


52
<Section Label="Rings:Properties">

53
<Heading>Rings: Properties</Heading>

54


55
The following properties are declared for &homalg; rings. Note that

56
(apart from so-called true and immediate methods (&see;

57
<Ref Sect="Rings:LIRNG"/>)) there are no methods installed for ring

58
properties. This means that if the value of the ring

59
property <C>Prop</C> is not set for a &homalg; ring <A>R</A>, then

60
<P/>

61


62
<C>Prop</C>( <A>R</A> ); <P/>

63


64
will cause an error. One can use the usual &GAP4; mechanism to check if

65
the value of the property is set or not <P/>

66


67
<C>HasProp</C>( <A>R</A> ); <P/>

68


69
If you discover that a specific property <C>Prop</C> is missing for a

70
certain &homalg; ring <A>R</A> you can it add using the usual &GAP4;

71
mechanism <P/>

72


73
<C>SetProp</C>( <A>R</A>, true ); <P/>

74


75
or <P/>

76


77
<C>SetProp</C>( <A>R</A>, false ); <P/>

78


79
Be very cautious with setting "missing" properties to &homalg;

80
objects: If the value you set is mathematically wrong &homalg; will

81
probably draw wrong conclusions and might return wrong results.

82


83
<#Include Label="IsZero:rings">

84
<#Include Label="ContainsAField">

85
<#Include Label="IsRationalsForHomalg">

86
<#Include Label="IsFieldForHomalg">

87
<#Include Label="IsDivisionRingForHomalg">

88
<#Include Label="IsIntegersForHomalg">

89
<#Include Label="IsResidueClassRingOfTheIntegers">

90
<#Include Label="IsBezoutRing">

91
<#Include Label="IsIntegrallyClosedDomain">

92
<#Include Label="IsUniqueFactorizationDomain">

93
<#Include Label="IsKaplanskyHermite">

94
<#Include Label="IsDedekindDomain">

95
<#Include Label="IsDiscreteValuationRing">

96
<#Include Label="IsFreePolynomialRing">

97
<#Include Label="IsWeylRing">

98
<#Include Label="IsLocalizedWeylRing">

99
<#Include Label="IsGlobalDimensionFinite">

100
<#Include Label="IsLeftGlobalDimensionFinite">

101
<#Include Label="IsRightGlobalDimensionFinite">

102
<#Include Label="HasInvariantBasisProperty">

103
<#Include Label="HasLeftInvariantBasisProperty">

104
<#Include Label="HasRightInvariantBasisProperty">

105
<#Include Label="IsLocal">

106
<#Include Label="IsSemiLocalRing">

107
<#Include Label="IsIntegralDomain">

108
<#Include Label="IsHereditary">

109
<#Include Label="IsLeftHereditary">

110
<#Include Label="IsRightHereditary">

111
<#Include Label="IsHermite">

112
<#Include Label="IsLeftHermite">

113
<#Include Label="IsRightHermite">

114
<#Include Label="IsNoetherian">

115
<#Include Label="IsLeftNoetherian">

116
<#Include Label="IsRightNoetherian">

117
<#Include Label="IsCohenMacaulay">

118
<#Include Label="IsGorenstein">

119
<#Include Label="IsKoszul">

120
<#Include Label="IsArtinian">

121
<#Include Label="IsLeftArtinian">

122
<#Include Label="IsRightArtinian">

123
<#Include Label="IsOreDomain">

124
<#Include Label="IsLeftOreDomain">

125
<#Include Label="IsRightOreDomain">

126
<#Include Label="IsPrincipalIdealRing">

127
<#Include Label="IsLeftPrincipalIdealRing">

128
<#Include Label="IsRightPrincipalIdealRing">

129
<#Include Label="IsRegular">

130
<#Include Label="IsFiniteFreePresentationRing">

131
<#Include Label="IsLeftFiniteFreePresentationRing">

132
<#Include Label="IsRightFiniteFreePresentationRing">

133
<#Include Label="IsSimpleRing">

134
<#Include Label="IsSemiSimpleRing">

135
<#Include Label="IsSuperCommutative">

136
<#Include Label="BasisAlgorithmRespectsPrincipalIdeals">

137
<#Include Label="AreUnitsCentral">

138
<#Include Label="IsMinusOne">

139
<#Include Label="IsMonic:ringelement">

140
<#Include Label="IsMonicUptoUnit:ringelement">

141
<#Include Label="IsLeftRegular:ringelement">

142
<#Include Label="IsRightRegular:ringelement">

143
<#Include Label="IsRegular:ringelement">

144


145
</Section>

146


147
<Section Label="Rings:Attributes">

148
<Heading>Rings: Attributes</Heading>

149


150
<#Include Label="Inverse:ring_element">

151


152
<#Include Label="homalgTable">

153
<#Include Label="RingElementConstructor">

154
<#Include Label="TypeOfHomalgMatrix">

155
<#Include Label="ConstructorForHomalgMatrices">

156
<#Include Label="Zero:ring">

157
<#Include Label="One:ring">

158
<#Include Label="MinusOne">

159
<#Include Label="ProductOfIndeterminates">

160
<#Include Label="RationalParameters">

161
<#Include Label="IndeterminatesOfPolynomialRing">

162
<#Include Label="RelativeIndeterminatesOfPolynomialRing">

163
<#Include Label="IndeterminateCoordinatesOfRingOfDerivations">

164
<#Include Label="RelativeIndeterminateCoordinatesOfRingOfDerivations">

165
<#Include Label="IndeterminateDerivationsOfRingOfDerivations">

166
<#Include Label="RelativeIndeterminateDerivationsOfRingOfDerivations">

167
<#Include Label="IndeterminateAntiCommutingVariablesOfExteriorRing">

168
<#Include Label="RelativeIndeterminateAntiCommutingVariablesOfExteriorRing">

169
<#Include Label="IndeterminatesOfExteriorRing">

170
<#Include Label="CoefficientsRing">

171
<#Include Label="KrullDimension">

172
<#Include Label="LeftGlobalDimension">

173
<#Include Label="RightGlobalDimension">

174
<#Include Label="GlobalDimension">

175
<#Include Label="GeneralLinearRank">

176
<#Include Label="ElementaryRank">

177
<#Include Label="StableRank">

178
<#Include Label="AssociatedGradedRing">

179


180
</Section>

181


182
<Section Label="Rings:Operations">

183
<Heading>Rings: Operations and Functions</Heading>

184


185
</Section>

186


187
<!-- ############################################################ -->

188


189
</Chapter>

190


191