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#############################################################################
##
## 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
##
## This file declares the operations for ideals.
##
#############################################################################
##
## <#GAPDoc Label="[1]{ideal}">
## A <E>left ideal</E> in a ring <M>R</M> is a subring of <M>R</M> that
## is closed under multiplication with elements of <M>R</M> from the left.
## <P/>
## A <E>right ideal</E> in a ring <M>R</M> is a subring of <M>R</M> that
## is closed under multiplication with elements of <M>R</M> from the right.
## <P/>
## A <E>two-sided ideal</E> or simply <E>ideal</E> in a ring <M>R</M>
## is both a left ideal and a right ideal in <M>R</M>.
## <P/>
## So being a (left/right/two-sided) ideal is not a property of a domain
## but refers to the acting ring(s).
## Hence we must ask, e. g., <C>IsIdeal( </C><M>R, I</M><C> )</C> if we
## want to know whether the ring <M>I</M> is an ideal in the ring <M>R</M>.
## The property <Ref Prop="IsTwoSidedIdealInParent"/> can be used to store
## whether a ring is an ideal in its parent.
## <P/>
## (Whenever the term <C>"Ideal"</C> occurs in an identifier without a
## specifying prefix <C>"Left"</C> or <C>"Right"</C>,
## this means the same as <C>"TwoSidedIdeal"</C>.
## Conversely, any occurrence of <C>"TwoSidedIdeal"</C> can be substituted
## by <C>"Ideal"</C>.)
## <P/>
## For any of the above kinds of ideals, there is a notion of generators,
## namely <Ref Attr="GeneratorsOfLeftIdeal"/>,
## <Ref Attr="GeneratorsOfRightIdeal"/>, and
## <Ref Attr="GeneratorsOfTwoSidedIdeal"/>.
## The acting rings can be accessed as <Ref Attr="LeftActingRingOfIdeal"/>
## and <Ref Attr="RightActingRingOfIdeal"/>, respectively.
## Note that ideals are detected from known values of these attributes,
## especially it is assumed that whenever a domain has both a left and a
## right acting ring then these two are equal.
## <P/>
## Note that we cannot use <Ref Attr="LeftActingDomain"/> and
## <C>RightActingDomain</C> here,
## since ideals in algebras are themselves vector spaces, and such a space
## can of course also be a module for an action from the right.
## In order to make the usual vector space functionality automatically
## available for ideals, we have to distinguish the left and right module
## structure from the additional closure properties of the ideal.
## <P/>
## Further note that the attributes denoting ideal generators and acting
## ring are used to create ideals if this is explicitly wanted, but the
## ideal relation in the sense of <Ref Oper="IsTwoSidedIdeal"/> is of course
## independent of the presence of the attribute values.
## <P/>
## Ideals are constructed with <Ref Func="LeftIdeal"/>,
## <Ref Func="RightIdeal"/>, <Ref Func="TwoSidedIdeal"/>.
## Principal ideals of the form <M>x * R</M>, <M>R * x</M>, <M>R * x * R</M>
## can also be constructed with a simple multiplication.
## <P/>
## Currently many methods for dealing with ideals need linear algebra to
## work, so they are mainly applicable to ideals in algebras.
## <P/>
## <#/GAPDoc>
#T The sum of two left/right/two-sided ideals with same acting ring can be
#T formed, it is again an ideal.
#T The product of two ideals ...
##
#############################################################################
##
#F TwoSidedIdeal( <R>, <gens>[, "basis"] )
#F Ideal( <R>, <gens>[, "basis"] )
#F LeftIdeal( <R>, <gens>[, "basis"] ) . . left ideal in <R> gen. by <gens>
#F RightIdeal( <R>, <gens>[, "basis"] ) . right ideal in <R> gen. by <gens>
##
## <#GAPDoc Label="TwoSidedIdeal">
## <ManSection>
## <Func Name="TwoSidedIdeal" Arg='R, gens[, "basis"]'/>
## <Func Name="Ideal" Arg='R, gens[, "basis"]'/>
## <Func Name="LeftIdeal" Arg='R, gens[, "basis"]'/>
## <Func Name="RightIdeal" Arg='R, gens[, "basis"]'/>
##
## <Description>
## Let <A>R</A> be a ring, and <A>gens</A> a list of collection of elements
## in <A>R</A>.
## <Ref Func="TwoSidedIdeal"/>, <Ref Func="LeftIdeal"/>,
## and <Ref Func="RightIdeal"/> return the two-sided,
## left, or right ideal, respectively,
## <M>I</M> in <A>R</A> that is generated by <A>gens</A>.
## The ring <A>R</A> can be accessed as <Ref Attr="LeftActingRingOfIdeal"/>
## or <Ref Attr="RightActingRingOfIdeal"/> (or both) of <M>I</M>.
## <P/>
## If <A>R</A> is a left <M>F</M>-module then also <M>I</M> is a left
## <M>F</M>-module,
## in particular the <Ref Attr="LeftActingDomain"/> values of
## <A>R</A> and <M>I</M> are equal.
## <P/>
## If the optional argument <C>"basis"</C> is given then <A>gens</A> are
## assumed to be a list of basis vectors of
## <M>I</M> viewed as a free <M>F</M>-module.
## (This is mainly applicable to ideals in algebras.)
## In this case, it is <E>not</E> checked whether <A>gens</A> really is
## linearly independent and whether <A>gens</A> is a subset of <A>R</A>.
## <P/>
## <Ref Func="Ideal"/> is simply a synonym of <Ref Func="TwoSidedIdeal"/>.
## <P/>
## <Example><![CDATA[
## gap> R:= Integers;;
## gap> I:= Ideal( R, [ 2 ] );
## <two-sided ideal in Integers, (1 generator)>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "TwoSidedIdeal" );
DeclareSynonym( "Ideal", TwoSidedIdeal );
DeclareGlobalFunction( "LeftIdeal" );
DeclareGlobalFunction( "RightIdeal" );
#############################################################################
##
#F TwoSidedIdealNC( <R>, <gens>[, "basis"] )
#F IdealNC( <R>, <gens>[, "basis"] )
#F LeftIdealNC( <R>, <gens>[, "basis"] )
#F RightIdealNC( <R>, <gens>[, "basis"] )
##
## <#GAPDoc Label="TwoSidedIdealNC">
## <ManSection>
## <Func Name="TwoSidedIdealNC" Arg='R, gens[, "basis"]'/>
## <Func Name="IdealNC" Arg='R, gens[, "basis"]'/>
## <Func Name="LeftIdealNC" Arg='R, gens[, "basis"]'/>
## <Func Name="RightIdealNC" Arg='R, gens[, "basis"]'/>
##
## <Description>
## The effects of <Ref Func="TwoSidedIdealNC"/>, <Ref Func="LeftIdealNC"/>,
## and <Ref Func="RightIdealNC"/> are the same as
## <Ref Func="TwoSidedIdeal"/>, <Ref Func="LeftIdeal"/>,
## and <Ref Func="RightIdeal"/>, respectively,
## but they do not check whether all entries of <A>gens</A> lie in <A>R</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "TwoSidedIdealNC" );
DeclareSynonym( "IdealNC", TwoSidedIdealNC );
DeclareGlobalFunction( "LeftIdealNC" );
DeclareGlobalFunction( "RightIdealNC" );
#############################################################################
##
#O IsTwoSidedIdeal( <R>, <I> )
#O IsLeftIdeal( <R>, <I> )
#O IsRightIdeal( <R>, <I> )
#P IsTwoSidedIdealInParent( <I> )
#P IsLeftIdealInParent( <I> )
#P IsRightIdealInParent( <I> )
##
## <#GAPDoc Label="IsTwoSidedIdeal">
## <ManSection>
## <Oper Name="IsTwoSidedIdeal" Arg='R, I'/>
## <Oper Name="IsLeftIdeal" Arg='R, I'/>
## <Oper Name="IsRightIdeal" Arg='R, I'/>
## <Prop Name="IsTwoSidedIdealInParent" Arg='I'/>
## <Prop Name="IsLeftIdealInParent" Arg='I'/>
## <Prop Name="IsRightIdealInParent" Arg='I'/>
##
## <Description>
## The properties <Ref Prop="IsTwoSidedIdealInParent"/> etc., are attributes
## of the ideal, and once known they are stored in the ideal.
## <Example><![CDATA[
## gap> A:= FullMatrixAlgebra( Rationals, 3 );
## ( Rationals^[ 3, 3 ] )
## gap> I:= Ideal( A, [ Random( A ) ] );
## <two-sided ideal in ( Rationals^[ 3, 3 ] ), (1 generator)>
## gap> IsTwoSidedIdeal( A, I );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InParentFOA( "IsTwoSidedIdeal", IsRing, IsRing, DeclareProperty );
InParentFOA( "IsLeftIdeal", IsRing, IsRing, DeclareProperty );
InParentFOA( "IsRightIdeal", IsRing, IsRing, DeclareProperty );
DeclareSynonym( "IsIdeal", IsTwoSidedIdeal );
DeclareSynonym( "IsIdealOp", IsTwoSidedIdealOp );
DeclareSynonymAttr( "IsIdealInParent", IsTwoSidedIdealInParent );
InstallTrueMethod( IsLeftIdealInParent, IsTwoSidedIdealInParent );
InstallTrueMethod( IsRightIdealInParent, IsTwoSidedIdealInParent );
InstallTrueMethod( IsTwoSidedIdealInParent,
IsLeftIdealInParent and IsRightIdealInParent );
#############################################################################
##
#O TwoSidedIdealByGenerators( <R>, <gens> ) . . ideal in <R> gen. by <gens>
#O IdealByGenerators( <R>, <gens> )
##
## <#GAPDoc Label="TwoSidedIdealByGenerators">
## <ManSection>
## <Oper Name="TwoSidedIdealByGenerators" Arg='R, gens'/>
## <Oper Name="IdealByGenerators" Arg='R, gens'/>
##
## <Description>
## <Ref Oper="TwoSidedIdealByGenerators"/> returns the ring that is
## generated by the elements of the collection <A>gens</A> under addition,
## multiplication, and multiplication with elements of the ring <A>R</A>
## from the left and from the right.
## <P/>
## <A>R</A> can be accessed by <Ref Attr="LeftActingRingOfIdeal"/> or
## <Ref Attr="RightActingRingOfIdeal"/>,
## <A>gens</A> can be accessed by <Ref Attr="GeneratorsOfTwoSidedIdeal"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "TwoSidedIdealByGenerators", [ IsRing, IsListOrCollection ] );
DeclareSynonym( "IdealByGenerators", TwoSidedIdealByGenerators );
#############################################################################
##
#O LeftIdealByGenerators( <R>, <gens> )
##
## <#GAPDoc Label="LeftIdealByGenerators">
## <ManSection>
## <Oper Name="LeftIdealByGenerators" Arg='R, gens'/>
##
## <Description>
## <Ref Oper="LeftIdealByGenerators"/> returns the ring that is generated by
## the elements of the collection <A>gens</A> under addition,
## multiplication, and multiplication with elements of the ring <A>R</A>
## from the left.
## <P/>
## <A>R</A> can be accessed by <Ref Attr="LeftActingRingOfIdeal"/>,
## <A>gens</A> can be accessed by <Ref Attr="GeneratorsOfLeftIdeal"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LeftIdealByGenerators", [ IsRing, IsListOrCollection ] );
#############################################################################
##
#O RightIdealByGenerators( <R>, <gens> )
##
## <#GAPDoc Label="RightIdealByGenerators">
## <ManSection>
## <Oper Name="RightIdealByGenerators" Arg='R, gens'/>
##
## <Description>
## <Ref Oper="RightIdealByGenerators"/> returns the ring that is generated
## by the elements of the collection <A>gens</A> under addition,
## multiplication, and multiplication with elements of the ring <A>R</A>
## from the right.
## <P/>
## <A>R</A> can be accessed by <Ref Attr="RightActingRingOfIdeal"/>,
## <A>gens</A> can be accessed by <Ref Attr="GeneratorsOfRightIdeal"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "RightIdealByGenerators", [ IsRing, IsListOrCollection ] );
#############################################################################
##
#A GeneratorsOfTwoSidedIdeal( <I> )
#A GeneratorsOfIdeal( <I> )
##
## <#GAPDoc Label="GeneratorsOfTwoSidedIdeal">
## <ManSection>
## <Attr Name="GeneratorsOfTwoSidedIdeal" Arg='I'/>
## <Attr Name="GeneratorsOfIdeal" Arg='I'/>
##
## <Description>
## is a list of generators for the ideal <A>I</A>, with respect to
## the action of the rings that are stored as the values of
## <Ref Attr="LeftActingRingOfIdeal"/> and
## <Ref Attr="RightActingRingOfIdeal"/>, from the left and from the right,
## respectively.
## <P/>
## <Example><![CDATA[
## gap> A:= FullMatrixAlgebra( Rationals, 3 );;
## gap> I:= Ideal( A, [ One( A ) ] );;
## gap> GeneratorsOfIdeal( I );
## [ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfTwoSidedIdeal", IsRing );
DeclareSynonymAttr( "GeneratorsOfIdeal", GeneratorsOfTwoSidedIdeal );
#############################################################################
##
#A GeneratorsOfLeftIdeal( <I> )
##
## <#GAPDoc Label="GeneratorsOfLeftIdeal">
## <ManSection>
## <Attr Name="GeneratorsOfLeftIdeal" Arg='I'/>
##
## <Description>
## is a list of generators for the left ideal <A>I</A>, with respect to the
## action from the left of the ring that is stored as the value of
## <Ref Attr="LeftActingRingOfIdeal"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfLeftIdeal", IsRing );
#############################################################################
##
#A GeneratorsOfRightIdeal( <I> )
##
## <#GAPDoc Label="GeneratorsOfRightIdeal">
## <ManSection>
## <Attr Name="GeneratorsOfRightIdeal" Arg='I'/>
##
## <Description>
## is a list of generators for the right ideal <A>I</A>, with respect to the
## action from the right of the ring that is stored as the value of
## <Ref Attr="RightActingRingOfIdeal"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfRightIdeal", IsRing );
#############################################################################
##
#A LeftActingRingOfIdeal( <I> )
#A RightActingRingOfIdeal( <I> )
##
## <#GAPDoc Label="LeftActingRingOfIdeal">
## <ManSection>
## <Attr Name="LeftActingRingOfIdeal" Arg='I'/>
## <Attr Name="RightActingRingOfIdeal" Arg='I'/>
##
## <Description>
## returns the left (resp. right) acting ring of an ideal <A>I</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LeftActingRingOfIdeal", IsRing );
DeclareAttribute( "RightActingRingOfIdeal", IsRing );
#############################################################################
##
#O AsLeftIdeal( <R>, <S> )
#O AsRightIdeal( <R>, <S> )
#O AsTwoSidedIdeal( <R>, <S> )
##
## <#GAPDoc Label="AsLeftIdeal">
## <ManSection>
## <Oper Name="AsLeftIdeal" Arg='R, S'/>
## <Oper Name="AsRightIdeal" Arg='R, S'/>
## <Oper Name="AsTwoSidedIdeal" Arg='R, S'/>
##
## <Description>
## Let <A>S</A> be a subring of the ring <A>R</A>.
## <P/>
## If <A>S</A> is a left ideal in <A>R</A> then <Ref Oper="AsLeftIdeal"/>
## returns this left ideal, otherwise <K>fail</K> is returned.
## <P/>
## If <A>S</A> is a right ideal in <A>R</A> then <Ref Oper="AsRightIdeal"/>
## returns this right ideal, otherwise <K>fail</K> is returned.
## <P/>
## If <A>S</A> is a two-sided ideal in <A>R</A> then
## <Ref Oper="AsTwoSidedIdeal"/> returns this two-sided ideal,
## otherwise <K>fail</K> is returned.
## <P/>
## <Example><![CDATA[
## gap> A:= FullMatrixAlgebra( Rationals, 3 );;
## gap> B:= DirectSumOfAlgebras( A, A );
## <algebra over Rationals, with 6 generators>
## gap> C:= Subalgebra( B, Basis( B ){[1..9]} );
## <algebra over Rationals, with 9 generators>
## gap> I:= AsTwoSidedIdeal( B, C );
## <two-sided ideal in <algebra of dimension 18 over Rationals>,
## (9 generators)>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AsLeftIdeal", [ IsRing, IsRing ] );
DeclareOperation( "AsRightIdeal", [ IsRing, IsRing ] );
DeclareOperation( "AsTwoSidedIdeal", [ IsRing, IsRing ] );
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