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##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## 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 contains declarations of operations for ring general mappings
## and homomorphisms. It is based on alghom.gd
##
##
DeclareInfoClass("InfoRingHom");
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##
#O RingGeneralMappingByImages( <R>, <S>, <gens>, <imgs> )
##
## <#GAPDoc Label="RingGeneralMappingByImages">
## <ManSection>
## <Oper Name="RingGeneralMappingByImages" Arg='R, S, gens, imgs'/>
##
## <Description>
## is a general mapping from the ring <A>A</A> to the ring <A>S</A>.
## This general mapping is defined by mapping the entries in the list
## <A>gens</A> (elements of <A>R</A>) to the entries in the list <A>imgs</A>
## (elements of <A>S</A>),
## and taking the additive and multiplicative closure.
## <P/>
## <A>gens</A> need not generate <A>R</A> as a ring,
## and if the specification does not define an additive and multiplicative
## mapping then the result will be multivalued.
## Hence, in general it is not a mapping.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "RingGeneralMappingByImages",
[ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ] );
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##
#F RingHomomorphismByImages( <R>, <S>, <gens>, <imgs> )
##
## <#GAPDoc Label="RingHomomorphismByImages">
## <ManSection>
## <Func Name="RingHomomorphismByImages" Arg='R, S, gens, imgs'/>
##
## <Description>
## <Ref Func="RingHomomorphismByImages"/> returns the ring homomorphism with
## source <A>R</A> and range <A>S</A> that is defined by mapping the list
## <A>gens</A> of generators of <A>R</A> to the list <A>imgs</A> of images
## in <A>S</A>.
## <P/>
## If <A>gens</A> does not generate <A>R</A> or if the homomorphism does not
## exist (i.e., if mapping the generators describes only a multi-valued
## mapping) then <K>fail</K> is returned.
## <P/>
## One can avoid the checks by calling
## <Ref Oper="RingHomomorphismByImagesNC"/>,
## and one can construct multi-valued mappings with
## <Ref Oper="RingGeneralMappingByImages"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RingHomomorphismByImages" );
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##
#O RingHomomorphismByImagesNC( <R>, <S>, <gens>, <imgs> )
##
## <#GAPDoc Label="RingHomomorphismByImagesNC">
## <ManSection>
## <Oper Name="RingHomomorphismByImagesNC" Arg='R, S, gens, imgs'/>
##
## <Description>
## <Ref Oper="RingHomomorphismByImagesNC"/> is the operation that is called
## by the function <Ref Func="RingHomomorphismByImages"/>.
## Its methods may assume that <A>gens</A> generates <A>R</A> as a ring
## and that the mapping of <A>gens</A> to <A>imgs</A> defines a ring
## homomorphism.
## Results are unpredictable if these conditions do not hold.
## <P/>
## For creating a possibly multi-valued mapping from <A>R</A> to <A>S</A>
## that respects addition and multiplication,
## <Ref Oper="RingGeneralMappingByImages"/> can be used.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "RingHomomorphismByImagesNC",
[ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ] );
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##
#O NaturalHomomorphismByIdeal( <R>, <I> ) . . . . . . . map onto factor ring
##
## <#GAPDoc Label="NaturalHomomorphismByIdeal">
## <ManSection>
## <Oper Name="NaturalHomomorphismByIdeal" Arg='R, I'/>
##
## <Description>
## is the homomorphism of rings provided by the natural
## projection map of <A>R</A> onto the quotient ring <A>R</A>/<A>I</A>.
## This map can be used to take pre-images in the original ring from
## elements in the quotient.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "NaturalHomomorphismByIdeal",
[ IsRing, IsRing ] );
