GAP (MatricesForHomalg) - Chapter 3: Rings

  3.3-1 IsZero
  3.3-2 ContainsAField
  3.3-3 IsRationalsForHomalg
  3.3-4 IsFieldForHomalg
  3.3-5 IsDivisionRingForHomalg
  3.3-6 IsIntegersForHomalg
  3.3-7 IsResidueClassRingOfTheIntegers
  3.3-8 IsBezoutRing
  3.3-9 IsIntegrallyClosedDomain
  3.3-10 IsUniqueFactorizationDomain
  3.3-11 IsKaplanskyHermite
  3.3-12 IsDedekindDomain
  3.3-13 IsDiscreteValuationRing
  3.3-14 IsFreePolynomialRing
  3.3-15 IsWeylRing
  3.3-16 IsLocalizedWeylRing
  3.3-17 IsGlobalDimensionFinite
  3.3-18 IsLeftGlobalDimensionFinite
  3.3-19 IsRightGlobalDimensionFinite
  3.3-20 HasInvariantBasisProperty
  3.3-21 HasLeftInvariantBasisProperty
  3.3-22 HasRightInvariantBasisProperty
  3.3-23 IsLocal
  3.3-24 IsSemiLocalRing
  3.3-25 IsIntegralDomain
  3.3-26 IsHereditary
  3.3-27 IsLeftHereditary
  3.3-28 IsRightHereditary
  3.3-29 IsHermite
  3.3-30 IsLeftHermite
  3.3-31 IsRightHermite
  3.3-32 IsNoetherian
  3.3-33 IsLeftNoetherian
  3.3-34 IsRightNoetherian
  3.3-35 IsCohenMacaulay
  3.3-36 IsGorenstein
  3.3-37 IsKoszul
  3.3-38 IsArtinian
  3.3-39 IsLeftArtinian
  3.3-40 IsRightArtinian
  3.3-41 IsOreDomain
  3.3-42 IsLeftOreDomain
  3.3-43 IsRightOreDomain
  3.3-44 IsPrincipalIdealRing
  3.3-45 IsLeftPrincipalIdealRing
  3.3-46 IsRightPrincipalIdealRing
  3.3-47 IsRegular
  3.3-48 IsFiniteFreePresentationRing
  3.3-49 IsLeftFiniteFreePresentationRing
  3.3-50 IsRightFiniteFreePresentationRing
  3.3-51 IsSimpleRing
  3.3-52 IsSemiSimpleRing
  3.3-53 IsSuperCommutative
  3.3-54 BasisAlgorithmRespectsPrincipalIdeals
  3.3-55 AreUnitsCentral
  3.3-56 IsMinusOne
  3.3-57 IsMonic
  3.3-58 IsMonicUptoUnit
  3.3-59 IsLeftRegular
  3.3-60 IsRightRegular
  3.3-61 IsRegular

3.4 Rings: Attributes

  3.4-1 Inverse
  3.4-2 homalgTable
  3.4-3 RingElementConstructor
  3.4-4 TypeOfHomalgMatrix
  3.4-5 ConstructorForHomalgMatrices
  3.4-6 Zero
  3.4-7 One
  3.4-8 MinusOne
  3.4-9 ProductOfIndeterminates
  3.4-10 RationalParameters
  3.4-11 IndeterminatesOfPolynomialRing
  3.4-12 RelativeIndeterminatesOfPolynomialRing
  3.4-13 IndeterminateCoordinatesOfRingOfDerivations
  3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations
  3.4-15 IndeterminateDerivationsOfRingOfDerivations
  3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations
  3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing
  3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing
  3.4-19 IndeterminatesOfExteriorRing
  3.4-20 CoefficientsRing
  3.4-21 KrullDimension
  3.4-22 LeftGlobalDimension
  3.4-23 RightGlobalDimension
  3.4-24 GlobalDimension
  3.4-25 GeneralLinearRank
  3.4-26 ElementaryRank
  3.4-27 StableRank
  3.4-28 AssociatedGradedRing

3.5 Rings: Operations and Functions

3 Rings

3.1 Rings: Category and Representations

3.1-1 IsHomalgRing

‣ IsHomalgRing( R ) ( category )

(It is a subcategory of the GAP categories IsStructureObject and IsHomalgRingOrModule.)

DeclareCategory( "IsHomalgRing",
        IsStructureObject and
        IsRingWithOne and
        IsHomalgRingOrModule );

3.1-2 IsPreHomalgRing

‣ IsPreHomalgRing( R ) ( category )

(It is a subcategory of the GAP category IsHomalgRing.)

These are rings with an incomplete homalgTable. They provide flexibility for developers to support a wider class of rings, as was necessary for the development of the LocalizeRingForHomalg package. They are not suited for direct usage.

DeclareCategory( "IsPreHomalgRing",
        IsHomalgRing );

3.1-3 IsHomalgRingElement

‣ IsHomalgRingElement( r ) ( category )

The GAP category of elements of homalg rings which are not GAP4 built-in.

DeclareCategory( "IsHomalgRingElement",
        IsExtAElement and
        IsExtLElement and
        IsExtRElement and
        IsAdditiveElementWithInverse and
        IsMultiplicativeElementWithInverse and
        IsAssociativeElement and
        IsAdditivelyCommutativeElement and
        ## all the above guarantees IsHomalgRingElement => IsRingElement (in GAP4)
        IsAttributeStoringRep );

3.1-4 IsHomalgInternalRingRep

‣ IsHomalgInternalRingRep( R ) ( representation )

3.2 Rings: Constructors

This section describes how to construct rings for use with MatricesForHomalg, which exploit the GAP4-built-in abilities to perform the necessary ring operations. By this we also mean necessary matrix operations over such rings. For the purposes of MatricesForHomalg only the ring of integers is properly supported in GAP4. The GAP4 extension packages Gauss and GaussForHomalg extend these built-in abilities to operations with sparse matrices over the ring \(ℤ / p^n\) for \(p\) prime and \(n\) positive.

If a ring \(R\) is supported in MatricesForHomalg any of its residue class rings \(R/I\) is supported as well, provided the ideal \(I\) of relations admits a finite set of generators as a left resp. right ideal (--> \/ (3.2-3)). This is immediate for commutative noetherian rings.

3.2-1 HomalgRingOfIntegers

‣ HomalgRingOfIntegers( ) ( function )

‣ HomalgRingOfIntegers( c ) ( function )

The one-argument form accepts an integer c and returns the ring \(ℤ / c \) for homalg:

The operation SetRingProperties is automatically invoked to set the ring properties.

If for some reason you don't want to use the GaussForHomalg package (maybe because you didn't install it), then use

3.2-2 HomalgFieldOfRationals

‣ HomalgFieldOfRationals( ) ( function )

The package GaussForHomalg is loaded and the field of rationals \(ℚ\) is returned. If GaussForHomalg fails to load an error is issued.

The operation SetRingProperties is automatically invoked to set the ring properties.

3.2-3 \/

‣ \/( R, ring_rel ) ( operation )

This is the homalg constructor for residue class rings R \(/ I\), where R is a homalg ring and \(I=\)ring_rel is the ideal of relations generated by ring_rel. ring_rel might be:

For noncommutative rings: In the first case the set of ring relations should generate the ideal of relations \(I\) as left resp. right ideal, and their involutions should generate \(I\) as right resp. left ideal. If ring_rel is not a set of relations, a left set of relations is constructed.

The operation SetRingProperties is automatically invoked to set the ring properties.

gap> ZZ := HomalgRingOfIntegers( );
Z
gap> Display( ZZ );
<An internal ring>
gap> Z256 := ZZ / 2^8;
Z/( 256 )
gap> Display( Z256 );
<A residue class ring>
gap> Z2 := Z256 / 6;
Z/( 256, 6 )
gap> BasisOfRows( MatrixOfRelations( Z2 ) );
<An unevaluated non-zero 1 x 1 matrix over an internal ring>
gap> Z2;
Z/( 2 )
gap> Display( Z2 );
<A residue class ring>

3.3 Rings: Properties

The following properties are declared for homalg rings. Note that (apart from so-called true and immediate methods (--> C.1)) there are no methods installed for ring properties. This means that if the value of the ring property Prop is not set for a homalg ring R, then

will cause an error. One can use the usual GAP4 mechanism to check if the value of the property is set or not

If you discover that a specific property Prop is missing for a certain homalg ring R you can it add using the usual GAP4 mechanism

Be very cautious with setting "missing" properties to homalg objects: If the value you set is mathematically wrong homalg will probably draw wrong conclusions and might return wrong results.

3.3-1 IsZero

‣ IsZero( R ) ( property )

Check if the ring R is a zero, i.e., if One\((\)R\()=\)Zero\((\)R\()\).

3.3-2 ContainsAField

‣ ContainsAField( R ) ( property )

3.3-3 IsRationalsForHomalg

‣ IsRationalsForHomalg( R ) ( property )

3.3-4 IsFieldForHomalg

‣ IsFieldForHomalg( R ) ( property )

3.3-5 IsDivisionRingForHomalg

‣ IsDivisionRingForHomalg( R ) ( property )

3.3-6 IsIntegersForHomalg

‣ IsIntegersForHomalg( R ) ( property )

3.3-7 IsResidueClassRingOfTheIntegers

‣ IsResidueClassRingOfTheIntegers( R ) ( property )

3.3-8 IsBezoutRing

‣ IsBezoutRing( R ) ( property )

3.3-9 IsIntegrallyClosedDomain

‣ IsIntegrallyClosedDomain( R ) ( property )

3.3-10 IsUniqueFactorizationDomain

‣ IsUniqueFactorizationDomain( R ) ( property )

3.3-11 IsKaplanskyHermite

‣ IsKaplanskyHermite( R ) ( property )

3.3-12 IsDedekindDomain

‣ IsDedekindDomain( R ) ( property )

3.3-13 IsDiscreteValuationRing

‣ IsDiscreteValuationRing( R ) ( property )

3.3-14 IsFreePolynomialRing

‣ IsFreePolynomialRing( R ) ( property )

3.3-15 IsWeylRing

‣ IsWeylRing( R ) ( property )

3.3-16 IsLocalizedWeylRing

‣ IsLocalizedWeylRing( R ) ( property )

3.3-17 IsGlobalDimensionFinite

‣ IsGlobalDimensionFinite( R ) ( property )

3.3-18 IsLeftGlobalDimensionFinite

‣ IsLeftGlobalDimensionFinite( R ) ( property )

3.3-19 IsRightGlobalDimensionFinite

‣ IsRightGlobalDimensionFinite( R ) ( property )

3.3-20 HasInvariantBasisProperty

‣ HasInvariantBasisProperty( R ) ( property )

3.3-21 HasLeftInvariantBasisProperty

‣ HasLeftInvariantBasisProperty( R ) ( property )

3.3-22 HasRightInvariantBasisProperty

‣ HasRightInvariantBasisProperty( R ) ( property )

3.3-23 IsLocal

‣ IsLocal( R ) ( property )

3.3-24 IsSemiLocalRing

‣ IsSemiLocalRing( R ) ( property )

3.3-25 IsIntegralDomain

‣ IsIntegralDomain( R ) ( property )

3.3-26 IsHereditary

‣ IsHereditary( R ) ( property )

3.3-27 IsLeftHereditary

‣ IsLeftHereditary( R ) ( property )

3.3-28 IsRightHereditary

‣ IsRightHereditary( R ) ( property )

3.3-29 IsHermite

‣ IsHermite( R ) ( property )

3.3-30 IsLeftHermite

‣ IsLeftHermite( R ) ( property )

3.3-31 IsRightHermite

‣ IsRightHermite( R ) ( property )

3.3-32 IsNoetherian

‣ IsNoetherian( R ) ( property )

3.3-33 IsLeftNoetherian

‣ IsLeftNoetherian( R ) ( property )

3.3-34 IsRightNoetherian

‣ IsRightNoetherian( R ) ( property )

3.3-35 IsCohenMacaulay

‣ IsCohenMacaulay( R ) ( property )

3.3-36 IsGorenstein

‣ IsGorenstein( R ) ( property )

3.3-37 IsKoszul

‣ IsKoszul( R ) ( property )

3.3-38 IsArtinian

‣ IsArtinian( R ) ( property )

3.3-39 IsLeftArtinian

‣ IsLeftArtinian( R ) ( property )

3.3-40 IsRightArtinian

‣ IsRightArtinian( R ) ( property )

3.3-41 IsOreDomain

‣ IsOreDomain( R ) ( property )

3.3-42 IsLeftOreDomain

‣ IsLeftOreDomain( R ) ( property )

3.3-43 IsRightOreDomain

‣ IsRightOreDomain( R ) ( property )

3.3-44 IsPrincipalIdealRing

‣ IsPrincipalIdealRing( R ) ( property )

3.3-45 IsLeftPrincipalIdealRing

‣ IsLeftPrincipalIdealRing( R ) ( property )

3.3-46 IsRightPrincipalIdealRing

‣ IsRightPrincipalIdealRing( R ) ( property )

3.3-47 IsRegular

‣ IsRegular( R ) ( property )

3.3-48 IsFiniteFreePresentationRing

‣ IsFiniteFreePresentationRing( R ) ( property )

3.3-49 IsLeftFiniteFreePresentationRing

‣ IsLeftFiniteFreePresentationRing( R ) ( property )

3.3-50 IsRightFiniteFreePresentationRing

‣ IsRightFiniteFreePresentationRing( R ) ( property )

3.3-51 IsSimpleRing

‣ IsSimpleRing( R ) ( property )

3.3-52 IsSemiSimpleRing

‣ IsSemiSimpleRing( R ) ( property )

3.3-53 IsSuperCommutative

‣ IsSuperCommutative( R ) ( property )

3.3-54 BasisAlgorithmRespectsPrincipalIdeals

‣ BasisAlgorithmRespectsPrincipalIdeals( R ) ( property )

3.3-55 AreUnitsCentral

‣ AreUnitsCentral( R ) ( property )

3.3-56 IsMinusOne

‣ IsMinusOne( r ) ( property )

3.3-57 IsMonic

‣ IsMonic( r ) ( property )

3.3-58 IsMonicUptoUnit

‣ IsMonicUptoUnit( r ) ( property )

3.3-59 IsLeftRegular

‣ IsLeftRegular( r ) ( property )

3.3-60 IsRightRegular

‣ IsRightRegular( r ) ( property )

3.3-61 IsRegular

‣ IsRegular( r ) ( property )

Check if the homalg ring element r is regular, i.e. left and right regular.

3.4 Rings: Attributes

3.4-1 Inverse

‣ Inverse( r ) ( attribute )

gap> ZZ := HomalgRingOfIntegers( );;
gap> R := ZZ / 2^8;
Z/( 256 )
gap> r := (1/3*One(R)+1/5)+3/7;
|[ 157 ]|
gap> 1 / r;	## = r^-1;
|[ 181 ]|
gap> s := (1/3*One(R)+2/5)+3/7;
|[ 106 ]|
gap> 1 / s;
fail

3.4-2 homalgTable

‣ homalgTable( R ) ( attribute )

The homalg table of R is a ring dictionary, i.e. the translator between homalg and the (specific implementation of the) ring.

3.4-3 RingElementConstructor

‣ RingElementConstructor( R ) ( attribute )

3.4-4 TypeOfHomalgMatrix

‣ TypeOfHomalgMatrix( R ) ( attribute )

3.4-5 ConstructorForHomalgMatrices

‣ ConstructorForHomalgMatrices( R ) ( attribute )

3.4-6 Zero

‣ Zero( R ) ( attribute )

3.4-7 One

‣ One( R ) ( attribute )

3.4-8 MinusOne

‣ MinusOne( R ) ( attribute )

3.4-9 ProductOfIndeterminates

‣ ProductOfIndeterminates( R ) ( attribute )

3.4-10 RationalParameters

‣ RationalParameters( R ) ( attribute )

3.4-11 IndeterminatesOfPolynomialRing

‣ IndeterminatesOfPolynomialRing( R ) ( attribute )

3.4-12 RelativeIndeterminatesOfPolynomialRing

‣ RelativeIndeterminatesOfPolynomialRing( R ) ( attribute )

3.4-13 IndeterminateCoordinatesOfRingOfDerivations

‣ IndeterminateCoordinatesOfRingOfDerivations( R ) ( attribute )

3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations

‣ RelativeIndeterminateCoordinatesOfRingOfDerivations( R ) ( attribute )

3.4-15 IndeterminateDerivationsOfRingOfDerivations

‣ IndeterminateDerivationsOfRingOfDerivations( R ) ( attribute )

3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations

‣ RelativeIndeterminateDerivationsOfRingOfDerivations( R ) ( attribute )

3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing

‣ IndeterminateAntiCommutingVariablesOfExteriorRing( R ) ( attribute )

3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing

‣ RelativeIndeterminateAntiCommutingVariablesOfExteriorRing( R ) ( attribute )

The list of anti-commuting relative indeterminates of the homalg exterior ring R.

3.4-19 IndeterminatesOfExteriorRing

‣ IndeterminatesOfExteriorRing( R ) ( attribute )

The list of all indeterminates (commuting and anti-commuting) of the homalg exterior ring R.

3.4-20 CoefficientsRing

‣ CoefficientsRing( R ) ( attribute )

3.4-21 KrullDimension

‣ KrullDimension( R ) ( attribute )

3.4-22 LeftGlobalDimension

‣ LeftGlobalDimension( R ) ( attribute )

3.4-23 RightGlobalDimension

‣ RightGlobalDimension( R ) ( attribute )

3.4-24 GlobalDimension

‣ GlobalDimension( R ) ( attribute )

The global dimension of the homalg ring R. The global dimension is defined, only if the left and right global dimensions coincide.

3.4-25 GeneralLinearRank

‣ GeneralLinearRank( R ) ( attribute )

3.4-26 ElementaryRank

‣ ElementaryRank( R ) ( attribute )

3.4-27 StableRank

‣ StableRank( R ) ( attribute )

3.4-28 AssociatedGradedRing

‣ AssociatedGradedRing( R ) ( attribute )