|
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
##
#############################################################################
##
#F TwoSidedIdeal( <R>, <gens> )
#F TwoSidedIdeal( <R>, <gens>, "basis" )
##
InstallGlobalFunction( TwoSidedIdeal, function( arg )
local I;
if Length( arg ) <= 1
or not IsRing( arg[1] )
or not IsHomogeneousList( arg[2] ) then
Error( "first argument must be a ring,\n",
"second argument must be a list of generators" );
elif IsEmpty( arg[2] ) then
return TwoSidedIdealNC( arg[1], arg[2] );
elif IsIdenticalObj( FamilyObj( arg[1] ),
FamilyObj( arg[2] ) )
and ForAll( arg[2], v -> v in arg[1] ) then
I:= IdealByGenerators( arg[1], arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
fi;
Error( "usage: TwoSidedIdeal( <R>, <gens> [, \"basis\"] )" );
end );
#############################################################################
##
#F TwoSidedIdealNC( <R>, <gens>, "basis" )
#F TwoSidedIdealNC( <R>, <gens> )
##
InstallGlobalFunction( TwoSidedIdealNC, function( arg )
local I;
if IsEmpty( arg[2] ) then
# If <R> is a FLMLOR then also the ideal is a FLMLOR.
if IsFLMLOR( arg[1] ) then
I:= SubFLMLORNC( arg[1], arg[2] );
else
I:= Objectify( NewType( FamilyObj( arg[1] ),
IsRing
and IsTrivial
and IsAttributeStoringRep ),
rec() );
fi;
SetGeneratorsOfRing( I, AsList( arg[2] ) );
SetLeftActingRingOfIdeal( I, arg[1] );
SetRightActingRingOfIdeal( I, arg[1] );
else
I:= TwoSidedIdealByGenerators( arg[1], arg[2] );
fi;
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
end );
#############################################################################
##
#F LeftIdeal( <R>, <gens> )
#F LeftIdeal( <R>, <gens>, "basis" )
##
InstallGlobalFunction( LeftIdeal, function( arg )
local I;
if Length( arg ) <= 1
or not IsRing( arg[1] )
or not IsHomogeneousList( arg[2] ) then
Error( "first argument must be a ring,\n",
"second argument must be a list of generators" );
elif IsEmpty( arg[2] ) then
return TwoSidedIdealNC( arg[1], arg[2] );
elif IsIdenticalObj( FamilyObj( arg[1] ),
FamilyObj( arg[2] ) )
and ForAll( arg[2], v -> v in arg[1] ) then
I:= LeftIdealByGenerators( arg[1], arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
fi;
Error( "usage: LeftIdeal( <R>, <gens> [, \"basis\"] )" );
end );
#############################################################################
##
#F LeftIdealNC( <R>, <gens>, "basis" )
#F LeftIdealNC( <R>, <gens> )
##
InstallGlobalFunction( LeftIdealNC, function( arg )
local I;
if IsEmpty( arg[2] ) then
return TwoSidedIdealNC( arg[1], arg[2] );
fi;
I:= LeftIdealByGenerators( arg[1], arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
end );
#############################################################################
##
#F RightIdeal( <R>, <gens> )
#F RightIdeal( <R>, <gens>, "basis" )
##
InstallGlobalFunction( RightIdeal, function( arg )
local I;
if Length( arg ) <= 1
or not IsRing( arg[1] )
or not IsHomogeneousList( arg[2] ) then
Error( "first argument must be a ring,\n",
"second argument must be a list of generators" );
elif IsEmpty( arg[2] ) then
return TwoSidedIdealNC( arg[1], arg[2] );
elif IsIdenticalObj( FamilyObj( arg[1] ),
FamilyObj( arg[2] ) )
and ForAll( arg[2], v -> v in arg[1] ) then
I:= RightIdealByGenerators( arg[1], arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
fi;
Error( "usage: RightIdeal( <R>, <gens> [, \"basis\"] )" );
end );
#############################################################################
##
#F RightIdealNC( <R>, <gens>, "basis" )
#F RightIdealNC( <R>, <gens> )
##
InstallGlobalFunction( RightIdealNC, function( arg )
local I;
if IsEmpty( arg[2] ) then
return TwoSidedIdealNC( arg[1], arg[2] );
fi;
I:= RightIdealByGenerators( arg[1], arg[2] );
if Length( arg ) = 3 and arg[3] = "basis" then
UseBasis( I, arg[2] );
fi;
UseSubsetRelation( arg[1], I );
return I;
end );
#############################################################################
##
#M TwoSidedIdealByGenerators( <R>, <gens> ) . . . create an ideal in a ring
##
InstallMethod( TwoSidedIdealByGenerators,
"for ring and collection",
IsIdenticalObj,
[ IsRing, IsCollection ], 0,
function( R, gens )
local I;
I:= Objectify( NewType( FamilyObj( R ),
IsRing
and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfTwoSidedIdeal( I, gens );
SetLeftActingRingOfIdeal( I, R );
SetRightActingRingOfIdeal( I, R );
return I;
end );
#############################################################################
##
#M LeftIdealByGenerators( <R>, <gens> ) . . . create a left ideal in a ring
##
InstallMethod( LeftIdealByGenerators,
"for ring and collection",
IsIdenticalObj,
[ IsRing, IsCollection ], 0,
function( R, gens )
local I;
I:= Objectify( NewType( FamilyObj( R ),
IsRing
and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfLeftIdeal( I, gens );
SetLeftActingRingOfIdeal( I, R );
return I;
end );
#############################################################################
##
#M RightIdealByGenerators( <R>, <gens> ) . . create a right ideal in a ring
##
InstallMethod( RightIdealByGenerators,
"for ring and collection",
IsIdenticalObj,
[ IsRing, IsCollection ], 0,
function( R, gens )
local I;
I:= Objectify( NewType( FamilyObj( R ),
IsRing
and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfRightIdeal( I, gens );
SetRightActingRingOfIdeal( I, R );
return I;
end );
#############################################################################
##
#M LeftIdealByGenerators( <R>, <gens> ) . . . . . . for commutative rings
#M RightIdealByGenerators( <R>, <gens> ) . . . . . . for commutative rings
##
## If R is a commutative ring, then we create a two-sided ideal in a ring R
## instead of its left or right ideal
##
InstallMethod( LeftIdealByGenerators,
"to construct ideals of commutative rings",
true,
[ IsFLMLOR and IsCommutative, IsCollection ],
0,
function( R, gens )
return TwoSidedIdealByGenerators( R, gens );
end );
InstallMethod( RightIdealByGenerators,
"to construct ideals of commutative rings",
true,
[ IsFLMLOR and IsCommutative, IsCollection ],
0,
function( R, gens )
return TwoSidedIdealByGenerators( R, gens );
end );
#############################################################################
##
#M IsIdealInParent(<I>) . for left resp. right ideals in a commutative ring
##
InstallImmediateMethod( IsIdealInParent,
IsLeftIdealInParent and HasParent, 10,
function( I )
I:= Parent( I );
if ( HasIsCommutative( I ) and IsCommutative( I ) )
or ( HasIsAnticommutative( I ) and IsAnticommutative( I ) ) then
return true;
else
TryNextMethod();
fi;
end );
InstallImmediateMethod( IsIdealInParent,
IsRightIdealInParent and HasParent, 10,
function( I )
I:= Parent( I );
if ( HasIsCommutative( I ) and IsCommutative( I ) )
or ( HasIsAnticommutative( I ) and IsAnticommutative( I ) ) then
return true;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M PrintObj( <I> ) . . . . . . . . . . . . . . . . . . . . . . for an ideal
##
InstallMethod( PrintObj,
"for a left ideal with known generators",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasGeneratorsOfLeftIdeal ],
0,
function( I )
Print( "LeftIdeal( ", LeftActingRingOfIdeal( I ), ", ",
GeneratorsOfLeftIdeal( I ), " )" );
end );
InstallMethod( PrintObj,
"for a right ideal with known generators",
true,
[ IsRing and HasRightActingRingOfIdeal and HasGeneratorsOfRightIdeal ],
0,
function( I )
Print( "RightIdeal( ", RightActingRingOfIdeal( I ), ", ",
GeneratorsOfRightIdeal( I ), " )" );
end );
InstallMethod( PrintObj,
"for a two-sided ideal with known generators",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfTwoSidedIdeal ],
0,
function( I )
Print( "TwoSidedIdeal( ", RightActingRingOfIdeal( I ), ", ",
GeneratorsOfTwoSidedIdeal( I ), " )" );
end );
#############################################################################
##
#M ViewObj( <I> ) . . . . . . . . . . . . . . . . . . . . . . for an ideal
##
InstallMethod( ViewObj,
"for a left ideal with known generators",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasGeneratorsOfLeftIdeal ],
100, # stronger than methods for the different kinds of algebras
function( I )
Print( "\>\><left ideal in \>\>" );
View( LeftActingRingOfIdeal( I ) );
if HasDimension( I ) then
Print( "\<,\< \>\>(dimension ", Dimension( I ), "\<\<\<\<)>" );
else
Print( "\<,\< \>\>(",
Pluralize( Length( GeneratorsOfLeftIdeal( I ) ), "generator" ),
")\<\<\<\<>" );
fi;
end );
InstallMethod( ViewObj,
"for a right ideal with known generators",
true,
[ IsRing and HasRightActingRingOfIdeal and HasGeneratorsOfRightIdeal ],
100, # stronger than methods for the different kinds of algebras
function( I )
Print( "\>\><right ideal in \>\>" );
View( RightActingRingOfIdeal( I ) );
if HasDimension( I ) then
Print( "\<,\< \>\>(dimension ", Dimension( I ), "\<\<\<\<)>" );
else
Print( "\<,\< \>\>(",
Pluralize( Length( GeneratorsOfRightIdeal( I ) ), "generator" ),
")\<\<\<\<>" );
fi;
end );
InstallMethod( ViewObj,
"for a two-sided ideal with known generators",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfTwoSidedIdeal ],
100, # stronger than methods for the different kinds of algebras
function( I )
Print( "\>\><two-sided ideal in \>\>" );
View( RightActingRingOfIdeal( I ) );
if HasDimension( I ) then
Print( "\<,\< \>\>(dimension ", Dimension( I ), "\<\<\<\<)>" );
else
Print( "\<,\< \>\>(",
Pluralize( Length( GeneratorsOfTwoSidedIdeal( I ) ), "generator" ),
")\<\<\<\<>" );
fi;
end );
#############################################################################
##
#M Representative( <I> ) . . . . one element of a left/right/two sided ideal
##
InstallMethod( Representative,
"for left ideal with known generators",
[ IsRing and HasGeneratorsOfLeftIdeal ],
RepresentativeFromGenerators( GeneratorsOfLeftIdeal ) );
InstallMethod( Representative,
"for right ideal with known generators",
[ IsRing and HasGeneratorsOfRightIdeal ],
RepresentativeFromGenerators( GeneratorsOfRightIdeal ) );
InstallMethod( Representative,
"for two-sided ideal with known generators",
[ IsRing and HasGeneratorsOfTwoSidedIdeal ],
RepresentativeFromGenerators( GeneratorsOfTwoSidedIdeal ) );
#############################################################################
##
#M Zero( <I> ) . . . . . . . . . . . . . . . . . . . . . . . . for an ideal
##
InstallOtherMethod( Zero,
"for a left ideal",
true,
[ IsRing and HasLeftActingRingOfIdeal ], 0,
I -> Zero( LeftActingRingOfIdeal( I ) ) );
InstallOtherMethod( Zero,
"for a right ideal",
true,
[ IsRing and HasRightActingRingOfIdeal ], 0,
I -> Zero( RightActingRingOfIdeal( I ) ) );
#############################################################################
##
#M Enumerator( <I> ) . . . . . . . . . . . . . . . . . . . . . for an ideal
##
BindGlobal( "EnumeratorOfIdeal", function( I )
local left, # we must multiply with ring elements from the left
right, # we must multiply with ring elements from the right
elms, # elements of <I>, result
set, # set corresponding to <elms>
Igens, # ideal generators of <I>
R, # the acting ring
Rgens, # ring generators of `R'
elmsgens, # additive generators
elm, # one element of <elms>
gen, # one generator of <I>
new; # product or sum of <elm> and <gen>
# check that we can handle this ideal
if HasIsFinite( I ) and not IsFinite( I ) then
TryNextMethod();
fi;
# Check from what sides we must multiply with ring elements.
if HasGeneratorsOfLeftIdeal( I ) then
Igens := GeneratorsOfLeftIdeal( I );
R := LeftActingRingOfIdeal( I );
left := true;
right := false;
elif HasGeneratorsOfRightIdeal( I ) then
Igens := GeneratorsOfRightIdeal( I );
R := RightActingRingOfIdeal( I );
left := false;
right := true;
elif HasGeneratorsOfTwoSidedIdeal( I ) then
Igens := GeneratorsOfTwoSidedIdeal( I );
R := LeftActingRingOfIdeal( I );
left := true;
right := true;
else
TryNextMethod();
fi;
# the elements of the ideal are sums of elements of the form r*g*s where
# g is an ideal generator and r and s are ring elements. Therefore
# *first* compute the ring multiples of the generators and then form the
# additive closure.
elms := Set(ShallowCopy(Igens));
set := ShallowCopy( elms );
# Compute the closure under the action of the acting ring
# from the left and from the right.
# If this ring is associative then it is sufficient to multiply
# with generators, otherwise we act with all elements.
if HasIsAssociative( R ) and IsAssociative( R ) then
Rgens:= GeneratorsOfRing( R );
else
Rgens:= Enumerator( R );
fi;
for elm in elms do
for gen in Rgens do
if left then
new := gen * elm;
if not new in set then
Add( elms, new );
AddSet( set, new );
fi;
fi;
if right then
new := elm * gen;
if not new in set then
Add( elms, new );
AddSet( set, new );
fi;
fi;
od;
od;
elms := set;
elmsgens:=ShallowCopy(elms);
set := ShallowCopy( elms );
# Use an orbit like algorithm.
# Compute the additive closure of elms
for elm in elms do
for gen in elmsgens do
new := elm + gen;
if not new in set then
Add( elms, new );
AddSet( set, new );
fi;
od;
od;
return set;
end );
InstallMethod( Enumerator,
"generic method for a left ideal with known generators",
true,
[ IsRing and HasGeneratorsOfLeftIdeal ], 0,
EnumeratorOfIdeal );
InstallMethod( Enumerator,
"generic method for a right ideal with known generators",
true,
[ IsRing and HasGeneratorsOfRightIdeal ], 0,
EnumeratorOfIdeal );
InstallMethod( Enumerator,
"generic method for a two-sided ideal with known generators",
true,
[ IsRing and HasGeneratorsOfIdeal ], 0,
EnumeratorOfIdeal );
#############################################################################
##
#M GeneratorsOfRing( <I> ) . . . . . . . . . . . . . . . . . . for an ideal
##
BindGlobal( "GeneratorsOfRingForIdeal", function( I )
local left, # we must multiply with ring elements from the left
right, # we must multiply with ring elements from the right
Igens, # ideal generators of <I>
R, # the acting ring
Rgens, # ring generators of `R'
gens, # generators list, result
S, # subring generated by `gens'
s, r, # loop over lists
prod; # product of `s' and `r'
# check that we can handle this ideal
if HasIsFinite( I ) and not IsFinite( I ) then
TryNextMethod();
fi;
# Check from what sides we must multiply with elements
# of the acting ring.
if HasGeneratorsOfLeftIdeal( I ) then
Igens := GeneratorsOfLeftIdeal( I );
R := LeftActingRingOfIdeal( I );
left := true;
right := false;
elif HasGeneratorsOfRightIdeal( I ) then
Igens := GeneratorsOfRightIdeal( I );
R := RightActingRingOfIdeal( I );
left := false;
right := true;
elif HasGeneratorsOfTwoSidedIdeal( I ) then
Igens := GeneratorsOfTwoSidedIdeal( I );
R := LeftActingRingOfIdeal( I );
left := true;
right := true;
else
Error( "no ideal generators of <I> known" );
fi;
# Handle the case of trivial ideals.
if IsEmpty( Igens ) then
return [];
fi;
# Start with the ring generated by the ideal generators,
# and close it until it becomes stable.
S := SubringNC( R, Igens );
gens := ShallowCopy( Igens );
Rgens := GeneratorsOfRing( R );
for s in gens do
for r in Rgens do
if left then
prod:= r * s;
if not prod in S then
S:= ClosureRing( S, prod );
Add( gens, prod );
fi;
fi;
if right then
prod:= s * r;
if not prod in S then
S:= ClosureRing( S, prod );
Add( gens, prod );
fi;
fi;
od;
od;
return gens;
end );
InstallMethod( GeneratorsOfRing,
"generic method for a left ideal with known generators",
true,
[ IsRing and HasGeneratorsOfLeftIdeal ], 0,
GeneratorsOfRingForIdeal );
InstallMethod( GeneratorsOfRing,
"generic method for a right ideal with known generators",
true,
[ IsRing and HasGeneratorsOfRightIdeal ], 0,
GeneratorsOfRingForIdeal );
InstallMethod( GeneratorsOfRing,
"generic method for a two-sided ideal with known generators",
true,
[ IsRing and HasGeneratorsOfTwoSidedIdeal ], 0,
GeneratorsOfRingForIdeal );
#############################################################################
##
#M GeneratorsOfTwoSidedIdeal( <I> ) . . . for known left/right ideal gens.
#M GeneratorsOfLeftIdeal( <I> ) . . . . . . for known two-sided ideal gens.
#M GeneratorsOfRightIdeal( <I> ) . . . . . . for known two-sided ideal gens.
##
InstallMethod( GeneratorsOfTwoSidedIdeal,
"for a two-sided ideal with known `GeneratorsOfLeftIdeal'",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfLeftIdeal ], 0,
GeneratorsOfLeftIdeal );
InstallMethod( GeneratorsOfTwoSidedIdeal,
"for a two-sided ideal with known `GeneratorsOfRightIdeal'",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfRightIdeal ], 0,
GeneratorsOfRightIdeal );
InstallMethod( GeneratorsOfLeftIdeal,
"for an ideal with known `GeneratorsOfTwoSidedIdeal'",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfTwoSidedIdeal ], 0,
GeneratorsOfRing );
InstallMethod( GeneratorsOfRightIdeal,
"for an ideal with known `GeneratorsOfTwoSidedIdeal'",
true,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal
and HasGeneratorsOfTwoSidedIdeal ], 0,
GeneratorsOfRing );
#############################################################################
##
#M \+( <I1>, <I2> ) . . . . . . . . . . . . . . . . . . . sum of two ideals
##
InstallMethod( \+,
"method for two left ideals",
IsIdenticalObj,
[ IsRing and HasLeftActingRingOfIdeal,
IsRing and HasLeftActingRingOfIdeal ], 0,
function( I1, I2 )
if LeftActingRingOfIdeal( I1 ) <> LeftActingRingOfIdeal( I2 ) then
TryNextMethod();
else
return LeftIdealByGenerators( LeftActingRingOfIdeal( I1 ),
Concatenation( GeneratorsOfLeftIdeal( I1 ),
GeneratorsOfLeftIdeal( I2 ) ) );
fi;
end );
InstallMethod( \+,
"method for two right ideals",
IsIdenticalObj,
[ IsRing and HasRightActingRingOfIdeal,
IsRing and HasRightActingRingOfIdeal ], 0,
function( I1, I2 )
if RightActingRingOfIdeal( I1 ) <> RightActingRingOfIdeal( I2 ) then
TryNextMethod();
else
return RightIdealByGenerators( RightActingRingOfIdeal( I1 ),
Concatenation( GeneratorsOfRightIdeal( I1 ),
GeneratorsOfRightIdeal( I2 ) ) );
fi;
end );
InstallMethod( \+,
"method for two two-sided ideals",
IsIdenticalObj,
[ IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal,
IsRing and HasLeftActingRingOfIdeal and HasRightActingRingOfIdeal ], 0,
function( I1, I2 )
if RightActingRingOfIdeal( I1 ) <> RightActingRingOfIdeal( I2 ) then
TryNextMethod();
else
return TwoSidedIdealByGenerators( RightActingRingOfIdeal( I1 ),
Concatenation( GeneratorsOfTwoSidedIdeal( I1 ),
GeneratorsOfTwoSidedIdeal( I2 ) ) );
fi;
end );
#############################################################################
##
#M \*( <r>, <R> ) . . . . . . . . . . . . . . . . . construct a right ideal
#M \*( <R>, <r> ) . . . . . . . . . . . . . . . . . construct a left ideal
##
## If <r> is an element in <R> then the result is the right or left ideal in
## <R> spanned by <r>.
## If <r> is not contained in <R> then the product is in general not closed
## under multiplication, and the default is to return the strictly sorted
## (note that the result shall be regarded as equal to the result of a
## method that returns a domain object) list of elements.
## (If <R> is trivial then the result is also trivial.)
##
InstallMethod( \*,
"for ring element and ring (construct a right ideal)",
IsElmsColls,
[ IsRingElement, IsRing ], 0,
function( r, R )
local z;
if r in R then
return RightIdealByGenerators( R, [ r ] );
fi;
if IsTrivial( R ) then
z:= Zero( R );
if r * z = z then
return R;
else
return [ r * z ];
fi;
elif IsFinite( R ) then
return Set( Enumerator( R ), elm -> r * elm );
else
TryNextMethod();
fi;
end );
InstallMethod( \*,
"for ring and ring element (construct a left ideal)",
IsCollsElms,
[ IsRing, IsRingElement ], 0,
function( R, r )
local z;
if r in R then
return LeftIdealByGenerators( R, [ r ] );
fi;
if IsTrivial( R ) then
z:= Zero( R );
if z * r = z then
return R;
else
return [ z * r ];
fi;
elif IsFinite( R ) then
return Set( Enumerator( R ), elm -> elm * r );
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M \*( <I>, <R> ) . . . . . . . . . . . . . . . construct a two-sided ideal
#M \*( <R>, <I> ) . . . . . . . . . . . . . . . construct a two-sided ideal
##
InstallMethod( \*,
"for left ideal and ring (construct a two-sided ideal)",
IsIdenticalObj,
[ IsRing and HasLeftActingRingOfIdeal, IsRing ], 0,
function( I, R )
if HasRightActingRingOfIdeal( I ) then
if IsSubset( RightActingRingOfIdeal( I ), R ) and One(R) <> fail then
return I;
fi;
elif LeftActingRingOfIdeal( I ) = R then
return TwoSidedIdealByGenerators( R, GeneratorsOfLeftIdeal( I ) );
else
TryNextMethod();
fi;
end );
InstallMethod( \*,
"for ring and right ideal (construct a two-sided ideal)",
IsCollsElms,
[ IsRing, IsRing and HasRightActingRingOfIdeal ], 0,
function( R, I )
if HasLeftActingRingOfIdeal( I ) then
if IsSubset( LeftActingRingOfIdeal( I ), R ) and One(R) <> fail then
return I;
fi;
elif RightActingRingOfIdeal( I ) = R then
return TwoSidedIdealByGenerators( R, GeneratorsOfRightIdeal( I ) );
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M AsLeftIdeal( <R>, <S> ) . . . . . . . . . . . . . . . . . . for two rings
#M AsRightIdeal( <R>, <S> ) . . . . . . . . . . . . . . . . . for two rings
#M AsTwoSidedIdeal( <R>, <S> ) . . . . . . . . . . . . . . . . for two rings
##
InstallMethod( AsLeftIdeal,
"for two rings",
IsIdenticalObj,
[ IsRing, IsRing ], 0,
function( R, S )
local I, gens;
if not IsLeftIdeal( R, S ) then
I:= fail;
else
gens:= GeneratorsOfRing( S );
I:= LeftIdealByGenerators( R, gens );
SetGeneratorsOfRing( I, gens );
fi;
return I;
end );
InstallMethod( AsRightIdeal,
"for two rings",
IsIdenticalObj,
[ IsRing, IsRing ], 0,
function( R, S )
local I, gens;
if not IsRightIdeal( R, S ) then
I:= fail;
else
gens:= GeneratorsOfRing( S );
I:= RightIdealByGenerators( R, gens );
SetGeneratorsOfRing( I, gens );
fi;
return I;
end );
InstallMethod( AsTwoSidedIdeal,
"for two rings",
IsIdenticalObj,
[ IsRing, IsRing ], 0,
function( R, S )
local I, gens;
if not IsTwoSidedIdeal( R, S ) then
I:= fail;
else
gens:= GeneratorsOfRing( S );
I:= TwoSidedIdealByGenerators( R, gens );
SetGeneratorsOfRing( I, gens );
fi;
return I;
end );
InstallMethod(IsSubset,"2-sided ideal in ring, naive",IsIdenticalObj,
[IsRing,IsRing and HasRightActingRingOfIdeal and HasLeftActingRingOfIdeal],
2*SIZE_FLAGS(FLAGS_FILTER(IsFLMLOR)),
function(R,I)
if IsIdenticalObj(R,RightActingRingOfIdeal(I)) and
IsIdenticalObj(R,LeftActingRingOfIdeal(I)) then
return IsSubset(R,GeneratorsOfTwoSidedIdeal(I));
fi;
TryNextMethod();
end);
InstallMethod(IsLeftIdeal,"left ideal in ring, naive",IsIdenticalObj,
[IsRing,IsRing and HasLeftActingRingOfIdeal],
2*SIZE_FLAGS(FLAGS_FILTER(IsFLMLOR)),
function(R,I)
if IsIdenticalObj(LeftActingRingOfIdeal(I),R) then
return true;
else
TryNextMethod();
fi;
end);
[ zur Elbe Produktseite wechseln0.41Quellennavigators
Analyse erneut starten
]
|
