chap3.txt

  
  3 Rings
  
  
  3.1 Rings: Category and Representations
  
  3.1-1 IsHomalgRing
  
  IsHomalgRing( R )  Category
  Returns:  true or false
  
  The GAP category of homalg rings.
  
  (It   is   a   subcategory  of  the  GAP  categories  IsStructureObject  and
  IsHomalgRingOrModule.)
  
    Code  
    DeclareCategory( "IsHomalgRing",
            IsStructureObject and
            IsRingWithOne and
            IsHomalgRingOrModule );
  
  
  3.1-2 IsPreHomalgRing
  
  IsPreHomalgRing( R )  Category
  Returns:  true or false
  
  The GAP category of pre homalg rings.
  
  (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.
  
    Code  
    DeclareCategory( "IsPreHomalgRing",
            IsHomalgRing );
  
  
  3.1-3 IsHomalgRingElement
  
  IsHomalgRingElement( r )  Category
  Returns:  true or false
  
  The GAP category of elements of homalg rings which are not GAP4 built-in.
  
    Code  
    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
  Returns:  true or false
  
  The internal representation of homalg rings.
  
  (It is a representation of the GAP category IsHomalgRing.)
  
  
  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
  Returns:  a homalg ring
  
  HomalgRingOfIntegers( c )  function
  Returns:  a homalg ring
  
  The no-argument form returns the ring of integers ℤ for homalg.
  
  The  one-argument  form  accepts an integer c and returns the ring ℤ / c for
  homalg:
  
      c= 0 defaults to ℤ
  
      if c is a prime power then the package GaussForHomalg is loaded (if it
        fails to load an error is issued)
  
      otherwise,  the  residue  class ring constructor / (--> \/ (3.2-3)) is
        invoked
  
  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
  
  HomalgRingOfIntegers( ) / c;
  
  but note that the computations will then be considerably slower.
  
  3.2-2 HomalgFieldOfRationals
  
  HomalgFieldOfRationals(  )  function
  Returns:  a homalg ring
  
  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
  Returns:  a homalg ring
  
  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:
  
      a set of ring relations of a left resp. right ideal
  
      a list of ring elements of R
  
      a ring element of R
  
  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.
  
    Example  
    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
  
  Prop( R );
  
  will  cause  an  error. One can use the usual GAP4 mechanism to check if the
  value of the property is set or not
  
  HasProp( R );
  
  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
  
  SetProp( R, true );
  
  or
  
  SetProp( R, false );
  
  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
  Returns:  true or false
  
  Check if the ring R is a zero, i.e., if One(R)=Zero(R).
  
  3.3-2 ContainsAField
  
  ContainsAField( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-3 IsRationalsForHomalg
  
  IsRationalsForHomalg( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-4 IsFieldForHomalg
  
  IsFieldForHomalg( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-5 IsDivisionRingForHomalg
  
  IsDivisionRingForHomalg( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-6 IsIntegersForHomalg
  
  IsIntegersForHomalg( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-7 IsResidueClassRingOfTheIntegers
  
  IsResidueClassRingOfTheIntegers( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-8 IsBezoutRing
  
  IsBezoutRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-9 IsIntegrallyClosedDomain
  
  IsIntegrallyClosedDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-10 IsUniqueFactorizationDomain
  
  IsUniqueFactorizationDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-11 IsKaplanskyHermite
  
  IsKaplanskyHermite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-12 IsDedekindDomain
  
  IsDedekindDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-13 IsDiscreteValuationRing
  
  IsDiscreteValuationRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-14 IsFreePolynomialRing
  
  IsFreePolynomialRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-15 IsWeylRing
  
  IsWeylRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-16 IsLocalizedWeylRing
  
  IsLocalizedWeylRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-17 IsGlobalDimensionFinite
  
  IsGlobalDimensionFinite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-18 IsLeftGlobalDimensionFinite
  
  IsLeftGlobalDimensionFinite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-19 IsRightGlobalDimensionFinite
  
  IsRightGlobalDimensionFinite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-20 HasInvariantBasisProperty
  
  HasInvariantBasisProperty( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-21 HasLeftInvariantBasisProperty
  
  HasLeftInvariantBasisProperty( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-22 HasRightInvariantBasisProperty
  
  HasRightInvariantBasisProperty( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-23 IsLocal
  
  IsLocal( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-24 IsSemiLocalRing
  
  IsSemiLocalRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-25 IsIntegralDomain
  
  IsIntegralDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-26 IsHereditary
  
  IsHereditary( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-27 IsLeftHereditary
  
  IsLeftHereditary( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-28 IsRightHereditary
  
  IsRightHereditary( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-29 IsHermite
  
  IsHermite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-30 IsLeftHermite
  
  IsLeftHermite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-31 IsRightHermite
  
  IsRightHermite( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-32 IsNoetherian
  
  IsNoetherian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-33 IsLeftNoetherian
  
  IsLeftNoetherian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-34 IsRightNoetherian
  
  IsRightNoetherian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-35 IsCohenMacaulay
  
  IsCohenMacaulay( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-36 IsGorenstein
  
  IsGorenstein( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-37 IsKoszul
  
  IsKoszul( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-38 IsArtinian
  
  IsArtinian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-39 IsLeftArtinian
  
  IsLeftArtinian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-40 IsRightArtinian
  
  IsRightArtinian( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-41 IsOreDomain
  
  IsOreDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-42 IsLeftOreDomain
  
  IsLeftOreDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-43 IsRightOreDomain
  
  IsRightOreDomain( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-44 IsPrincipalIdealRing
  
  IsPrincipalIdealRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-45 IsLeftPrincipalIdealRing
  
  IsLeftPrincipalIdealRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-46 IsRightPrincipalIdealRing
  
  IsRightPrincipalIdealRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-47 IsRegular
  
  IsRegular( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-48 IsFiniteFreePresentationRing
  
  IsFiniteFreePresentationRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-49 IsLeftFiniteFreePresentationRing
  
  IsLeftFiniteFreePresentationRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-50 IsRightFiniteFreePresentationRing
  
  IsRightFiniteFreePresentationRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-51 IsSimpleRing
  
  IsSimpleRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-52 IsSemiSimpleRing
  
  IsSemiSimpleRing( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-53 IsSuperCommutative
  
  IsSuperCommutative( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-54 BasisAlgorithmRespectsPrincipalIdeals
  
  BasisAlgorithmRespectsPrincipalIdeals( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-55 AreUnitsCentral
  
  AreUnitsCentral( R )  property
  Returns:  true or false
  
  R is a ring for homalg.
  
  3.3-56 IsMinusOne
  
  IsMinusOne( r )  property
  Returns:  true or false
  
  Check if the ring element r is the additive inverse of one.
  
  3.3-57 IsMonic
  
  IsMonic( r )  property
  Returns:  true or false
  
  Check if the homalg ring element r is monic.
  
  3.3-58 IsMonicUptoUnit
  
  IsMonicUptoUnit( r )  property
  Returns:  true or false
  
  Check if leading coefficient of the homalg ring element r is a unit.
  
  3.3-59 IsLeftRegular
  
  IsLeftRegular( r )  property
  Returns:  true or false
  
  Check if the homalg ring element r is left regular.
  
  3.3-60 IsRightRegular
  
  IsRightRegular( r )  property
  Returns:  true or false
  
  Check if the homalg ring element r is right regular.
  
  3.3-61 IsRegular
  
  IsRegular( r )  property
  Returns:  true or false
  
  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
  Returns:  a homalg ring element or fail
  
  The inverse of the homalg ring element r.
  
    Example  
    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
  Returns:  a homalg table
  
  The  homalg  table  of  R  is a ring dictionary, i.e. the translator between
  homalg and the (specific implementation of the) ring.
  
  Every homalg ring has a homalg table.
  
  3.4-3 RingElementConstructor
  
  RingElementConstructor( R )  attribute
  Returns:  a function
  
  The constructor of ring elements in the homalg ring R.
  
  3.4-4 TypeOfHomalgMatrix
  
  TypeOfHomalgMatrix( R )  attribute
  Returns:  a type
  
  The GAP4-type of homalg matrices over the homalg ring R.
  
  3.4-5 ConstructorForHomalgMatrices
  
  ConstructorForHomalgMatrices( R )  attribute
  Returns:  a type
  
  The constructor for homalg matrices over the homalg ring R.
  
  3.4-6 Zero
  
  Zero( R )  attribute
  Returns:  a homalg ring element
  
  The zero of the homalg ring R.
  
  3.4-7 One
  
  One( R )  attribute
  Returns:  a homalg ring element
  
  The one of the homalg ring R.
  
  3.4-8 MinusOne
  
  MinusOne( R )  attribute
  Returns:  a homalg ring element
  
  The minus one of the homalg ring R.
  
  3.4-9 ProductOfIndeterminates
  
  ProductOfIndeterminates( R )  attribute
  Returns:  a homalg ring element
  
  The product of indeterminates of the homalg ring R.
  
  3.4-10 RationalParameters
  
  RationalParameters( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of rational parameters of the homalg ring R.
  
  3.4-11 IndeterminatesOfPolynomialRing
  
  IndeterminatesOfPolynomialRing( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of indeterminates of the homalg polynomial ring R.
  
  3.4-12 RelativeIndeterminatesOfPolynomialRing
  
  RelativeIndeterminatesOfPolynomialRing( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of relative indeterminates of the homalg polynomial ring R.
  
  3.4-13 IndeterminateCoordinatesOfRingOfDerivations
  
  IndeterminateCoordinatesOfRingOfDerivations( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of indeterminate coordinates of the homalg Weyl ring R.
  
  3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations
  
  RelativeIndeterminateCoordinatesOfRingOfDerivations( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of relative indeterminate coordinates of the homalg Weyl ring R.
  
  3.4-15 IndeterminateDerivationsOfRingOfDerivations
  
  IndeterminateDerivationsOfRingOfDerivations( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of indeterminate derivations of the homalg Weyl ring R.
  
  3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations
  
  RelativeIndeterminateDerivationsOfRingOfDerivations( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of relative indeterminate derivations of the homalg Weyl ring R.
  
  3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing
  
  IndeterminateAntiCommutingVariablesOfExteriorRing( R )  attribute
  Returns:  a list of homalg ring elements
  
  The list of anti-commuting indeterminates of the homalg exterior ring R.
  
  3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing
  
  RelativeIndeterminateAntiCommutingVariablesOfExteriorRing( R )  attribute
  Returns:  a list of homalg ring elements
  
  The  list  of  anti-commuting relative indeterminates of the homalg exterior
  ring R.
  
  3.4-19 IndeterminatesOfExteriorRing
  
  IndeterminatesOfExteriorRing( R )  attribute
  Returns:  a list of homalg ring elements
  
  The  list of all indeterminates (commuting and anti-commuting) of the homalg
  exterior ring R.
  
  3.4-20 CoefficientsRing
  
  CoefficientsRing( R )  attribute
  Returns:  a homalg ring
  
  The ring of coefficients of the homalg ring R.
  
  3.4-21 KrullDimension
  
  KrullDimension( R )  attribute
  Returns:  a non-negative integer
  
  The Krull dimension of the commutative homalg ring R.
  
  3.4-22 LeftGlobalDimension
  
  LeftGlobalDimension( R )  attribute
  Returns:  a non-negative integer
  
  The left global dimension of the homalg ring R.
  
  3.4-23 RightGlobalDimension
  
  RightGlobalDimension( R )  attribute
  Returns:  a non-negative integer
  
  The right global dimension of the homalg ring R.
  
  3.4-24 GlobalDimension
  
  GlobalDimension( R )  attribute
  Returns:  a non-negative integer
  
  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
  Returns:  a non-negative integer
  
  The general linear rank of the homalg ring R ([MR01], 11.1.14).
  
  3.4-26 ElementaryRank
  
  ElementaryRank( R )  attribute
  Returns:  a non-negative integer
  
  The elementary rank of the homalg ring R ([MR01], 11.3.10).
  
  3.4-27 StableRank
  
  StableRank( R )  attribute
  Returns:  a non-negative integer
  
  The stable rank of the homalg ring R ([MR01], 11.3.4).
  
  3.4-28 AssociatedGradedRing
  
  AssociatedGradedRing( R )  attribute
  Returns:  a homalg ring
  
  The graded ring associated to the filtered ring R.
  
  
  3.5 Rings: Operations and Functions