This file is indexed.

/usr/share/gap/lib/ringhom.gi is in gap-libs 4r6p5-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
#############################################################################
##
#W  ringhom.gi                   GAP library                  Alexander Hulpke
##
##
#Y  Copyright (C) 2008 The GAP Group
##
##  This file contains methods for ring general mappings and homomorphisms.
##  It is based on alghom.gi
##


#############################################################################
##
#R  IsRingGeneralMappingByImagesDefaultRep
#R  IsSCRingGeneralMappingByImagesDefaultRep
##
##
DeclareRepresentation( "IsRingGeneralMappingByImagesDefaultRep",
    IsRingGeneralMapping and IsAdditiveElementWithInverse
    and IsAttributeStoringRep, [] );

DeclareRepresentation( "IsSCRingGeneralMappingByImagesDefaultRep",
    IsRingGeneralMappingByImagesDefaultRep,[]);


#############################################################################
##
#M  RingGeneralMappingByImages( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( RingGeneralMappingByImages,
    "for two rings and two homogeneous lists",
    [ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ],
function( S, R, gens, imgs )
  local filter,map;        # general mapping from <S> to <R>, result

  # Check the arguments.
  if   Length( gens ) <> Length( imgs )  then
    Error( "<gens> and <imgs> must have the same length" );
  elif not IsSubset( S, gens ) then
    Error( "<gens> must lie in <S>" );
  elif not IsSubset( R, imgs ) then
    Error( "<imgs> must lie in <R>" );
  fi;
  filter:=IsSPGeneralMapping and IsRingGeneralMapping;

  if IsSubringSCRing(S) then
    filter:=filter and IsSCRingGeneralMappingByImagesDefaultRep;
  fi;

  # Make the general mapping.
  map:= Objectify( TypeOfDefaultGeneralMapping( S, R,
			    IsSPGeneralMapping
			and IsRingGeneralMapping
			and IsSCRingGeneralMappingByImagesDefaultRep ),
		    rec(
			) );

    SetMappingGeneratorsImages(map,[Immutable(gens),Immutable(imgs)]);
    # return the general mapping
    return map;
    end );

#############################################################################
##
#M  RingHomomorphismByImagesNC( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( RingHomomorphismByImagesNC,
    "for two rings and two homogeneous lists",
    [ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ],
    function( S, R, gens, imgs )
    local map;        # homomorphism from <source> to <range>, result
    map:= RingGeneralMappingByImages( S, R, gens, imgs );
    SetIsSingleValued( map, true );
    SetIsTotal( map, true );
    return map;
    end );


#############################################################################
##
#F  RingHomomorphismByImages( <S>, <R>, <gens>, <imgs> )
##
InstallGlobalFunction( RingHomomorphismByImages,
    function( S, R, gens, imgs )
    local hom;
    hom:= RingGeneralMappingByImages( S, R, gens, imgs );
    if IsMapping( hom ) then
      return RingHomomorphismByImagesNC( S, R, gens, imgs );
    else
      return fail;
    fi;
end );

#############################################################################
##
#M  ViewObj( <map> )  . . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ViewObj, "for an ring g.m.b.i", true,
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  View(mapi[1]);
  Print(" -> ");
  View(mapi[2]);
end );


#############################################################################
##
#M  PrintObj( <map> ) . . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( PrintObj, "for an ring hom. b.i.", true,
    [     IsMapping
      and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  Print( "RingHomomorphismByImages( ",
	  Source( map ), ", ", Range( map ), ", ",
	  mapi[1], ", ", mapi[2], " )" );
end );

InstallMethod( PrintObj, "for an ring g.m.b.i", true,
    [     IsGeneralMapping
      and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  Print( "RingGeneralMappingByImages( ",
	  Source( map ), ", ", Range( map ), ", ",
	  mapi[1], ", ", mapi[2], " )" );
end );

#############################################################################
##
#M  IsTotal( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsTotal,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local mapi,t;
  mapi:=MappingGeneratorsImages(map);
  if Length(mapi[1])=0 then
    t:=Ring(Zero(Source(map)));
  else
    t:=Ring(mapi[1]);
  fi;
  return Source(map)=t;
end);

#############################################################################
##
#M  MakeSCRingMapping( <map> )
##
BindGlobal( "MakeSCRingMapping",
function(map)
local mapi;
  if not IsBound(map!.stdgens) then
    mapi:=MappingGeneratorsImages(map);
    map!.stdgens:=
      StandardGeneratorsImagesSubringSCRing(FamilyObj(Zero(Source(map))),
	mapi[1],mapi[2]);
  fi;
end);

#############################################################################
##
#M  IsSingleValued( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsSingleValued,
    "for sc ring g.m.b.i.",
    [ IsGeneralMapping and IsSCRingGeneralMappingByImagesDefaultRep ],
function(map)
  local r, moduli, std, stdi, sel, o, elm, elmi, i, j, k;
  r:=Source(map);
  moduli:=FamilyObj(Zero(r))!.moduli;

  MakeSCRingMapping(map);
  std:=map!.stdgens;
  # check additive relations
  for i in [1..Length(std[4])] do
    stdi:=std[1][i];
    sel:=Filtered([1..Length(stdi)],x->stdi[x]<>0);
    if not 0 in moduli{sel} then
      o:=1;
      for j in sel do
        o:=Lcm(o,moduli[j]/Gcd(stdi[j],moduli[j]));
      od;
      if not IsZero(o*std[4][i]) then
	Info(InfoRingHom,2,"Additive order ",o," of generator ",i," failed");
	return false;
      else
	Info(InfoRingHom,3,"Additive order ",o," of generator ",i," OK");
      fi;
    else
      Info(InfoRingHom,3,"Generator ",i,": ",std[1][i]," has order infinity");
    fi;
  od;

  # check multiplicative relations
  for i in [1..Length(std[4])] do
    for j in [1..Length(std[4])] do
      elm:=std[3][i]*std[3][j];
      elm:=SCRingDecompositionStandardGens(std,elm);
      elmi:=Zero(Range(map));
      for k in [1..Length(std[4])] do
        elmi:=elmi+elm[k]*std[4][k];
      od;
      if elmi<>std[4][i]*std[4][j] then
	Info(InfoRingHom,2,"Product ",i," x ",j," failed: ",elm,elmi);
	return false;
      fi;
    od;
  od;
  return true;
end);


#############################################################################
##
#M  ImagesSource( <map> ) . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ImagesSource,
    "for an ring g.m.b.i.",
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  return Subring(Range(map),MappingGeneratorsImages(map)[2]);
end );

#############################################################################
##
#M  PreImagesRange( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( PreImagesRange,
    "for an ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  return Subring(Source(map),MappingGeneratorsImages(map)[1]);
end );

#############################################################################
##
#M  InverseGeneralMapping( <map> ) . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( InverseGeneralMapping,
    "for an ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  return RingGeneralMappingByImages(Range(map),Source(map),mapi[2],mapi[1]);
end );

#############################################################################
##
#M  AdditiveInverseOp( <map> )  . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( AdditiveInverseOp, "for ring g.m.b.i.",
  [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local mapi;
  mapi:=MappingGeneratorsImages(map);
  return RingGeneralMappingByImages(Source(map),Range(map),mapi[1],
    List(mapi[2],AdditiveInverse));
end);

#############################################################################
##
#M  \+( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \+,
    "for ring g.m.b.i. and ring general mapping",
    IsIdenticalObj,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsRingGeneralMapping],
function( map1, map2 )
local mapi,map;
  mapi:=MappingGeneratorsImages(map1);
  map:=RingGeneralMappingByImages(Source(map1),Range(map1),mapi[1],
    List([1..Length(mapi[1])],
	  x->mapi[2][x]+ImagesRepresentative(map2,mapi[1][x])));
  return map;
end );

#############################################################################
##
#M  \+( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \+,
    "for ring ring general mapping and ring g.m.b.i.",
    IsIdenticalObj,
    [ IsRingGeneralMapping,
      IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep],
function( map2, map1 )
local mapi,map;
  mapi:=MappingGeneratorsImages(map1);
  map:=RingGeneralMappingByImages(Source(map1),Range(map1),mapi[1],
    List([1..Length(mapi[1])],
	  x->mapi[2][x]+ImagesRepresentative(map2,mapi[1][x])));
  return map;
end );

#############################################################################
##
#M  \=( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \=,
    "for ring g.m.b.i. and ring general mapping",
    IsIdenticalObj,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsRingGeneralMapping],
function( map1, map2 )
local mapi;
  mapi:=MappingGeneratorsImages(map1);
  return ForAll([1..Length(mapi[1])],
	  x->mapi[2][x]=ImagesRepresentative(map2,mapi[1][x]));
end );

#############################################################################
##
#M  \=( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \=,
    "for ring ring general mapping and ring g.m.b.i.",
    IsIdenticalObj,
    [ IsRingGeneralMapping,
      IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep],
function( map2, map1 )
local mapi;
  mapi:=MappingGeneratorsImages(map1);
  return ForAll([1..Length(mapi[1])],
	  x->mapi[2][x]=ImagesRepresentative(map2,mapi[1][x]));
end );

#############################################################################
##
#M  CoKernelOfAdditiveGeneralMapping( <map> ) . . . . .  for ring g.m.b.i.
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local r, moduli, std, gens, stdi, sel, o, elm, elmi, i, j, k;
  r:=Source(map);
  moduli:=FamilyObj(Zero(r))!.moduli;

  MakeSCRingMapping(map);
  std:=map!.stdgens;
  gens:=[];
  # run through additive relations
  for i in [1..Length(std[4])] do
    stdi:=std[1][i];
    sel:=Filtered([1..Length(stdi)],x->stdi[x]<>0);
    if not 0 in moduli{sel} then
      o:=1;
      for j in sel do
        o:=Lcm(o,moduli[j]/Gcd(stdi[j],moduli[j]));
      od;
      Add(gens,o*std[4][i]);
    fi;
  od;

  # check multiplicative relations
  for i in [1..Length(std[4])] do
    for j in [1..Length(std[4])] do
      elm:=std[3][i]*std[3][j];
      elm:=SCRingDecompositionStandardGens(std,elm);
      elmi:=Zero(Range(map));
      for k in [1..Length(std[4])] do
        elmi:=elmi+elm[k]*std[4][k];
      od;
      Add(gens,elmi-std[4][i]*std[4][j]);
    od;
  od;
  gens:=Filtered(gens,i->not IsZero(i));
  if Length(gens)=0 then Add(gens,Zero(Range(map)));fi;
  return Subring(Range(map),gens);
end);



#############################################################################
##
#M  KernelOfAdditiveGeneralMapping( <map> ) . . . . . .  for ring g.m.b.i.
##
InstallMethod( KernelOfAdditiveGeneralMapping,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  local ker, mapi, i;
  ker:=ShallowCopy(GeneratorsOfRing(
      CoKernelOfAdditiveGeneralMapping(InverseGeneralMapping(map))));
  mapi:=MappingGeneratorsImages(map);
  for i in [1..Length(mapi[1])] do
    if IsZero(mapi[2][i]) then
      Add(ker,mapi[1][i]);
    fi;
  od;
  return Subring(Source(map),ker);
end );


#############################################################################
##
#M  IsInjective( <map> )  . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsInjective,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
  return Size(KernelOfAdditiveGeneralMapping(map))=1;
end);


#############################################################################
##
#M  ImagesRepresentative( <map>, <elm> )  . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ImagesRepresentative, "for SC ring g.m.b.i., and element",
    FamSourceEqFamElm,
    [ IsRingGeneralMapping and IsSCRingGeneralMappingByImagesDefaultRep,
      IsObject ],
function( map, elm )
local std, elmi, k;
  MakeSCRingMapping(map);
  std:=map!.stdgens;
  elm:=SCRingDecompositionStandardGens(std,elm);
  elmi:=Zero(Range(map));
  for k in [1..Length(std[4])] do
    elmi:=elmi+elm[k]*std[4][k];
  od;
  return elmi;
end );

#############################################################################
##
#M  PreImagesRepresentative( <map>, <elm> ) . . . . . .  for ring g.m.b.i.
##
InstallMethod( PreImagesRepresentative,
    "for ring g.m.b.i., and element",
    FamRangeEqFamElm,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsObject ],
function( map, elm )
  return ImagesRepresentative(InverseGeneralMapping(map),elm);
end );

BindGlobal("IsomorphismSCRing",function(R)
local e, z, one, o, sel, g, go, elms, dec, p, cands, m, a, b, nr, hom, i, j;
  if Size(R)>100000 then
    Error("R is too big");
  fi;
  # find generators
  e:=Elements(R);
  z:=Zero(R);
  one:=One(R);
  one:=Position(e,one);
  o:=List(e,i->First([1..Size(R)],x->x*i=z));
  sel:=[1..Length(e)];
  g:=[];
  go:=[];
  elms:=[z];
  dec:=[];
  p:=Position(e,z);
  dec[p]:=[];
  RemoveSet(sel,p);
  cands:=ShallowCopy(sel);
  while Length(cands)>0 do

    # element of maximal order. If possible pick ``one'' to be among the
    # generators
    m:=Maximum(o{cands});
    if one in cands and o[one]=m then
      a:=one;
    else
      a:=First(cands,i->o[i]=m);
    fi;
    RemoveSet(cands,a);
    a:=e[a];
    Add(g,a);
    Add(go,m);
    # all combinations
    for i in [1..Length(elms)] do
      for j in [1..m-1] do
	b:=elms[i]+j*a;
	p:=Position(e,b);
	if p in sel then
	  RemoveSet(sel,p);
	  Add(elms,b);
	  Add(dec,Concatenation(dec[i],[j]));
	fi;
      od;
    od;

    # the remaining candidates must be complements
    for i in ShallowCopy(cands) do
      if ForAny([1..o[i]-1],j->j*e[i] in elms) then
	RemoveSet(cands,i);
      fi;
    od;

    # update dec
    m:=Length(g);
    for i in dec do
      while Length(i)<m do
	Add(i,0);
      od;
    od;
  od;
  m:=EmptySCTable(Length(go),0);
  for i in [1..Length(g)] do
    for j in [1..Length(g)] do
      p:=g[i]*g[j];
      if p<>z then
	p:=Position(elms,p);
	p:=dec[p];
	nr:=[];
	for b in [1..Length(p)] do
	  if p[b]<>0 then
	    Add(nr,p[b]);
	    Add(nr,b);
	  fi;
	od;
	SetEntrySCTable(m,i,j,nr);
      fi;
    od;
  od;
  nr:=RingByStructureConstants(go,m);
  hom:=RingHomomorphismByImages(R,nr,g,GeneratorsOfRing(nr));
  return hom;
end);

#############################################################################
##
#M  NaturalHomomorphismByIdeal(<R>,<I>)
##
InstallMethod( NaturalHomomorphismByIdeal,"sc rings",IsIdenticalObj,
    [ IsSubringSCRing,IsSubringSCRing],
function( R, I )
  local hom, R2, nat, Rgens, std, moduli, newmod, posi, q, t, dec, x, i, j, k;
  if not IsIdeal(R,I) then
    Error("I is not an ideal!");
  fi;
  if not IsWholeFamily(R) then
    hom:=IsomorphismSCRing(R);
    R2:=Range(hom);
    I:=Subring(R2,List(GeneratorsOfRing(I),x->Image(hom,x)));
    nat:=NaturalHomomorphismByIdeal(R2,I);
    return RingHomomorphismByImages(R,Range(nat),GeneratorsOfRing(R),
             List(GeneratorsOfRing(R),x->Image(nat,Image(hom,i))));
  else
    if I=R then
      # catch trivial case
      q:=SmallRing(1,1);
      return RingHomomorphismByImages(R,q,GeneratorsOfRing(R),
        List(GeneratorsOfRing(R),x->Zero(q)));
    fi;
    # R is the full ring. We can read of the factor ring structures from the
    # standard generators of R
    Rgens:=GeneratorsOfRing(R);
    std:=StandardGeneratorsSubringSCRing(I);
    moduli:=FamilyObj(Zero(R))!.moduli;
    newmod:=[];
    posi:=[]; # generator positions
    for i in [1..Length(moduli)] do
      if not IsBound(std[2][i]) then
        # the generator survives as it is
	Add(newmod,moduli[i]);
	Add(posi,i);
      else
        t:=std[1][std[2][i]][i]; 
        if t=1 then
	  # the generator vanishes in the factor
	  Add(newmod,false);
	else
	  # the generator has a smaller order in the factor
	  q:=moduli[i]/t; 
	  Add(newmod,q);
	  Add(posi,i);
	fi;
      fi;
    od;
    # now determine the multiplication
    t:=EmptySCTable(Length(posi),0);
    for i in [1..Length(posi)] do
      for j in [1..Length(posi)] do
	# product of generators
        q:=Rgens[posi[i]]*Rgens[posi[j]];
	q:=q![1]; # the coefficients
	dec:=[];
	for k in [1..Length(posi)] do
	  x:=q[posi[k]] mod newmod[posi[k]];
	  if x<>0 then
	    Add(dec,x);
	    Add(dec,k);
	  fi;
	od;
	if Length(dec)>0 then
	  SetEntrySCTable(t,i,j,dec);
	fi;
      od;
    od;
  fi;
  q:=RingByStructureConstants(newmod{posi},t,"q");
  # image list: Generators are mapped to their images
  x:=List(GeneratorsOfRing(R),x->Zero(q));
  x{posi}:=GeneratorsOfRing(q);
  hom:=RingHomomorphismByImages(R,q,GeneratorsOfRing(R),x);
  SetIsSurjective(hom,true);
  SetKernelOfAdditiveGeneralMapping(hom,I);
  return hom;
end );

InstallOtherMethod( \/,
    "generic method for two rings",
    IsIdenticalObj,
    [ IsRing, IsRing ],
    function( R, I )
    return ImagesSource( NaturalHomomorphismByIdeal( R, I ) );
    end );