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Implement StandardAssociate(Unit) for ZmodnZ, tweak documentation for IsEuclideanRing #1990

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Dec 18, 2017
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Implement StandardAssociate(Unit) for ZmodnZ, tweak documentation for IsEuclideanRing #1990

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7e94d89
Tweak documentation for IsEuclideanRing
fingolfin Dec 18, 2017
618940c
Implement StandardAssociate(Unit) for ZmodnZ
fingolfin Dec 18, 2017
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33 changes: 33 additions & 0 deletions hpcgap/lib/zmodnz.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -655,6 +655,39 @@ InstallMethod(ZOp,
end);


#############################################################################
##
#M StandardAssociate( <r> )
##
InstallMethod( StandardAssociate,
"for full ring Z/nZ and an element in Z/nZ",
IsCollsElms,
[ IsZmodnZObjNonprimeCollection and IsWholeFamily and IsRing, IsZmodnZObj and IsModulusRep ],
function ( R, r )
local m, n;
m := ModulusOfZmodnZObj( r );
n := GcdInt( r![1], m );
return ZmodnZObj( FamilyObj( r ), n );
end );

#############################################################################
##
#M StandardAssociateUnit( <r> )
##
InstallMethod( StandardAssociateUnit,
"for full ring Z/nZ and an element in Z/nZ",
IsCollsElms,
[ IsZmodnZObjNonprimeCollection and IsWholeFamily and IsRing, IsZmodnZObj and IsModulusRep ],
function ( R, r )
local m, n;
m := ModulusOfZmodnZObj( r );
if r![1] = 0 then
n := 1;
else
n := QuotientMod(GcdInt( r![1], m ), r![1], m);
fi;
return ZmodnZObj( FamilyObj( r ), n );
end );


#############################################################################
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2 changes: 1 addition & 1 deletion lib/ring.gd
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Expand Up @@ -546,7 +546,7 @@ DeclareCategory( "IsUniqueFactorizationRing", IsRing );
## <Description>
## A ring <M>R</M> is called a Euclidean ring if it is an integral ring and
## there exists a function <M>\delta</M>, called the Euclidean degree, from
## <M>R-\{0_R\}</M> to the nonnegative integers,
## <M>R-\{0_R\}</M> into a well-ordered set (such as the the nonnegative integers),
## such that for every pair <M>r \in R</M> and <M>s \in R-\{0_R\}</M> there
## exists an element <M>q</M> such that either
## <M>r - q s = 0_R</M> or <M>\delta(r - q s) &lt; \delta( s )</M>.
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33 changes: 33 additions & 0 deletions lib/zmodnz.gi
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Expand Up @@ -655,6 +655,39 @@ InstallMethod(ZOp,
end);


#############################################################################
##
#M StandardAssociate( <r> )
##
InstallMethod( StandardAssociate,
"for full ring Z/nZ and an element in Z/nZ",
IsCollsElms,
[ IsZmodnZObjNonprimeCollection and IsWholeFamily and IsRing, IsZmodnZObj and IsModulusRep ],
function ( R, r )
local m, n;
m := ModulusOfZmodnZObj( r );
n := GcdInt( r![1], m );
return ZmodnZObj( FamilyObj( r ), n );
end );

#############################################################################
##
#M StandardAssociateUnit( <r> )
##
InstallMethod( StandardAssociateUnit,
"for full ring Z/nZ and an element in Z/nZ",
IsCollsElms,
[ IsZmodnZObjNonprimeCollection and IsWholeFamily and IsRing, IsZmodnZObj and IsModulusRep ],
function ( R, r )
local m, n;
m := ModulusOfZmodnZObj( r );
if r![1] = 0 then
n := 1;
else
n := QuotientMod(GcdInt( r![1], m ), r![1], m);
fi;
return ZmodnZObj( FamilyObj( r ), n );
end );


#############################################################################
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41 changes: 41 additions & 0 deletions tst/testinstall/zmodnz.tst
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Expand Up @@ -467,6 +467,47 @@ ZmodnZObj( 0, 10 )
gap> 3/y;
fail

# test StandardAssociate
gap> checkCompatible := R -> ForAll(R, r -> StandardAssociateUnit(R,r) * r = StandardAssociate(R,r));;

#
gap> R := Integers mod 4;;
gap> List(Elements(R), x -> Int(StandardAssociate(R, x)));
[ 0, 1, 2, 1 ]
gap> List(Elements(R), x -> Int(StandardAssociateUnit(R, x)));
[ 1, 1, 1, 3 ]

#
gap> R := Integers mod 5;;
gap> List(Elements(R), x -> Int(StandardAssociate(R, x)));
[ 0, 1, 1, 1, 1 ]
gap> List(Elements(R), x -> Int(StandardAssociateUnit(R, x)));
[ 1, 1, 3, 4, 2 ]
gap> ForAll(R, r -> StandardAssociateUnit(R,r) * r = StandardAssociate(R,r));
true

#
gap> R := Integers mod 6;;
gap> List(Elements(R), x -> Int(StandardAssociate(R, x)));
[ 0, 1, 2, 3, 2, 1 ]
gap> List(Elements(R), x -> Int(StandardAssociateUnit(R, x)));
[ 1, 1, 1, 1, 2, 5 ]
gap> ForAll(R, r -> StandardAssociateUnit(R,r) * r = StandardAssociate(R,r));
true

#
gap> R := Integers mod 9;;
gap> List(Elements(R), x -> Int(StandardAssociate(R, x)));
[ 0, 1, 1, 3, 1, 1, 3, 1, 1 ]
gap> List(Elements(R), x -> Int(StandardAssociateUnit(R, x)));
[ 1, 1, 5, 1, 7, 2, 2, 4, 8 ]
gap> ForAll(R, r -> StandardAssociateUnit(R,r) * r = StandardAssociate(R,r));
true

#
gap> ForAll([1..100], m -> checkCompatible(Integers mod m));
true

#
gap> STOP_TEST( "zmodnz.tst", 1);

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