1
  
2
  3 Rings
3
  
4
  
5
  3.1 Rings: Category and Representations
6
  
7
  3.1-1 IsHomalgRing
8
  
9
  IsHomalgRing( R )  Category
10
  Returns:  true or false
11
  
12
  The GAP category of homalg rings.
13
  
14
  (It   is   a   subcategory  of  the  GAP  categories  IsStructureObject  and
15
  IsHomalgRingOrModule.)
16
  
17
    Code  
18
    DeclareCategory( "IsHomalgRing",
19
            IsStructureObject and
20
            IsRingWithOne and
21
            IsHomalgRingOrModule );
22
  
23
  
24
  3.1-2 IsPreHomalgRing
25
  
26
  IsPreHomalgRing( R )  Category
27
  Returns:  true or false
28
  
29
  The GAP category of pre homalg rings.
30
  
31
  (It is a subcategory of the GAP category IsHomalgRing.)
32
  These are rings with an incomplete homalgTable. They provide flexibility for
33
  developers  to  support  a  wider  class  of rings, as was necessary for the
34
  development  of  the  LocalizeRingForHomalg package. They are not suited for
35
  direct usage.
36
  
37
    Code  
38
    DeclareCategory( "IsPreHomalgRing",
39
            IsHomalgRing );
40
  
41
  
42
  3.1-3 IsHomalgRingElement
43
  
44
  IsHomalgRingElement( r )  Category
45
  Returns:  true or false
46
  
47
  The GAP category of elements of homalg rings which are not GAP4 built-in.
48
  
49
    Code  
50
    DeclareCategory( "IsHomalgRingElement",
51
            IsExtAElement and
52
            IsExtLElement and
53
            IsExtRElement and
54
            IsAdditiveElementWithInverse and
55
            IsMultiplicativeElementWithInverse and
56
            IsAssociativeElement and
57
            IsAdditivelyCommutativeElement and
58
            ## all the above guarantees IsHomalgRingElement => IsRingElement (in GAP4)
59
            IsAttributeStoringRep );
60
  
61
  
62
  3.1-4 IsHomalgInternalRingRep
63
  
64
  IsHomalgInternalRingRep( R )  Representation
65
  Returns:  true or false
66
  
67
  The internal representation of homalg rings.
68
  
69
  (It is a representation of the GAP category IsHomalgRing.)
70
  
71
  
72
  3.2 Rings: Constructors
73
  
74
  This   section   describes   how   to   construct   rings   for   use   with
75
  MatricesForHomalg,  which exploit the GAP4-built-in abilities to perform the
76
  necessary  ring operations. By this we also mean necessary matrix operations
77
  over  such  rings.  For  the  purposes of MatricesForHomalg only the ring of
78
  integers  is  properly  supported in GAP4. The GAP4 extension packages Gauss
79
  and GaussForHomalg extend these built-in abilities to operations with sparse
80
  matrices over the ring ℤ / p^n for p prime and n positive.
81
  
82
  If a ring R is supported in MatricesForHomalg any of its residue class rings
83
  R/I  is supported as well, provided the ideal I of relations admits a finite
84
  set  of  generators  as  a  left resp. right ideal (--> \/ (3.2-3)). This is
85
  immediate for commutative noetherian rings.
86
  
87
  3.2-1 HomalgRingOfIntegers
88
  
89
  HomalgRingOfIntegers(  )  function
90
  Returns:  a homalg ring
91
  
92
  HomalgRingOfIntegers( c )  function
93
  Returns:  a homalg ring
94
  
95
  The no-argument form returns the ring of integers ℤ for homalg.
96
  
97
  The  one-argument  form  accepts an integer c and returns the ring ℤ / c for
98
  homalg:
99
  
100
      c= 0 defaults to ℤ
101
  
102
      if c is a prime power then the package GaussForHomalg is loaded (if it
103
        fails to load an error is issued)
104
  
105
      otherwise,  the  residue  class ring constructor / (--> \/ (3.2-3)) is
106
        invoked
107
  
108
  The  operation  SetRingProperties  is  automatically invoked to set the ring
109
  properties.
110
  
111
  If  for  some reason you don't want to use the GaussForHomalg package (maybe
112
  because you didn't install it), then use
113
  
114
  HomalgRingOfIntegers( ) / c;
115
  
116
  but note that the computations will then be considerably slower.
117
  
118
  3.2-2 HomalgFieldOfRationals
119
  
120
  HomalgFieldOfRationals(  )  function
121
  Returns:  a homalg ring
122
  
123
  The  package  GaussForHomalg  is  loaded  and  the  field  of rationals ℚ is
124
  returned. If GaussForHomalg fails to load an error is issued.
125
  
126
  The  operation  SetRingProperties  is  automatically invoked to set the ring
127
  properties.
128
  
129
  3.2-3 \/
130
  
131
  \/( R, ring_rel )  operation
132
  Returns:  a homalg ring
133
  
134
  This  is  the homalg constructor for residue class rings R / I, where R is a
135
  homalg  ring and I=ring_rel is the ideal of relations generated by ring_rel.
136
  ring_rel might be:
137
  
138
      a set of ring relations of a left resp. right ideal
139
  
140
      a list of ring elements of R
141
  
142
      a ring element of R
143
  
144
  For noncommutative rings: In the first case the set of ring relations should
145
  generate  the  ideal  of  relations  I  as left resp. right ideal, and their
146
  involutions  should generate I as right resp. left ideal. If ring_rel is not
147
  a set of relations, a left set of relations is constructed.
148
  
149
  The  operation  SetRingProperties  is  automatically invoked to set the ring
150
  properties.
151
  
152
    Example  
153
    gap> ZZ := HomalgRingOfIntegers( );
154
    Z
155
    gap> Display( ZZ );
156
    <An internal ring>
157
    gap> Z256 := ZZ / 2^8;
158
    Z/( 256 )
159
    gap> Display( Z256 );
160
    <A residue class ring>
161
    gap> Z2 := Z256 / 6;
162
    Z/( 256, 6 )
163
    gap> BasisOfRows( MatrixOfRelations( Z2 ) );
164
    <An unevaluated non-zero 1 x 1 matrix over an internal ring>
165
    gap> Z2;
166
    Z/( 2 )
167
    gap> Display( Z2 );
168
    <A residue class ring>
169
  
170
  
171
  
172
  3.3 Rings: Properties
173
  
174
  The  following  properties  are  declared for homalg rings. Note that (apart
175
  from  so-called  true  and immediate methods (--> C.1)) there are no methods
176
  installed  for  ring  properties.  This  means that if the value of the ring
177
  property Prop is not set for a homalg ring R, then
178
  
179
  Prop( R );
180
  
181
  will  cause  an  error. One can use the usual GAP4 mechanism to check if the
182
  value of the property is set or not
183
  
184
  HasProp( R );
185
  
186
  If  you  discover  that  a  specific  property Prop is missing for a certain
187
  homalg ring R you can it add using the usual GAP4 mechanism
188
  
189
  SetProp( R, true );
190
  
191
  or
192
  
193
  SetProp( R, false );
194
  
195
  Be very cautious with setting "missing" properties to homalg objects: If the
196
  value  you  set  is  mathematically  wrong  homalg  will probably draw wrong
197
  conclusions and might return wrong results.
198
  
199
  3.3-1 IsZero
200
  
201
  IsZero( R )  property
202
  Returns:  true or false
203
  
204
  Check if the ring R is a zero, i.e., if One(R)=Zero(R).
205
  
206
  3.3-2 ContainsAField
207
  
208
  ContainsAField( R )  property
209
  Returns:  true or false
210
  
211
  R is a ring for homalg.
212
  
213
  3.3-3 IsRationalsForHomalg
214
  
215
  IsRationalsForHomalg( R )  property
216
  Returns:  true or false
217
  
218
  R is a ring for homalg.
219
  
220
  3.3-4 IsFieldForHomalg
221
  
222
  IsFieldForHomalg( R )  property
223
  Returns:  true or false
224
  
225
  R is a ring for homalg.
226
  
227
  3.3-5 IsDivisionRingForHomalg
228
  
229
  IsDivisionRingForHomalg( R )  property
230
  Returns:  true or false
231
  
232
  R is a ring for homalg.
233
  
234
  3.3-6 IsIntegersForHomalg
235
  
236
  IsIntegersForHomalg( R )  property
237
  Returns:  true or false
238
  
239
  R is a ring for homalg.
240
  
241
  3.3-7 IsResidueClassRingOfTheIntegers
242
  
243
  IsResidueClassRingOfTheIntegers( R )  property
244
  Returns:  true or false
245
  
246
  R is a ring for homalg.
247
  
248
  3.3-8 IsBezoutRing
249
  
250
  IsBezoutRing( R )  property
251
  Returns:  true or false
252
  
253
  R is a ring for homalg.
254
  
255
  3.3-9 IsIntegrallyClosedDomain
256
  
257
  IsIntegrallyClosedDomain( R )  property
258
  Returns:  true or false
259
  
260
  R is a ring for homalg.
261
  
262
  3.3-10 IsUniqueFactorizationDomain
263
  
264
  IsUniqueFactorizationDomain( R )  property
265
  Returns:  true or false
266
  
267
  R is a ring for homalg.
268
  
269
  3.3-11 IsKaplanskyHermite
270
  
271
  IsKaplanskyHermite( R )  property
272
  Returns:  true or false
273
  
274
  R is a ring for homalg.
275
  
276
  3.3-12 IsDedekindDomain
277
  
278
  IsDedekindDomain( R )  property
279
  Returns:  true or false
280
  
281
  R is a ring for homalg.
282
  
283
  3.3-13 IsDiscreteValuationRing
284
  
285
  IsDiscreteValuationRing( R )  property
286
  Returns:  true or false
287
  
288
  R is a ring for homalg.
289
  
290
  3.3-14 IsFreePolynomialRing
291
  
292
  IsFreePolynomialRing( R )  property
293
  Returns:  true or false
294
  
295
  R is a ring for homalg.
296
  
297
  3.3-15 IsWeylRing
298
  
299
  IsWeylRing( R )  property
300
  Returns:  true or false
301
  
302
  R is a ring for homalg.
303
  
304
  3.3-16 IsLocalizedWeylRing
305
  
306
  IsLocalizedWeylRing( R )  property
307
  Returns:  true or false
308
  
309
  R is a ring for homalg.
310
  
311
  3.3-17 IsGlobalDimensionFinite
312
  
313
  IsGlobalDimensionFinite( R )  property
314
  Returns:  true or false
315
  
316
  R is a ring for homalg.
317
  
318
  3.3-18 IsLeftGlobalDimensionFinite
319
  
320
  IsLeftGlobalDimensionFinite( R )  property
321
  Returns:  true or false
322
  
323
  R is a ring for homalg.
324
  
325
  3.3-19 IsRightGlobalDimensionFinite
326
  
327
  IsRightGlobalDimensionFinite( R )  property
328
  Returns:  true or false
329
  
330
  R is a ring for homalg.
331
  
332
  3.3-20 HasInvariantBasisProperty
333
  
334
  HasInvariantBasisProperty( R )  property
335
  Returns:  true or false
336
  
337
  R is a ring for homalg.
338
  
339
  3.3-21 HasLeftInvariantBasisProperty
340
  
341
  HasLeftInvariantBasisProperty( R )  property
342
  Returns:  true or false
343
  
344
  R is a ring for homalg.
345
  
346
  3.3-22 HasRightInvariantBasisProperty
347
  
348
  HasRightInvariantBasisProperty( R )  property
349
  Returns:  true or false
350
  
351
  R is a ring for homalg.
352
  
353
  3.3-23 IsLocal
354
  
355
  IsLocal( R )  property
356
  Returns:  true or false
357
  
358
  R is a ring for homalg.
359
  
360
  3.3-24 IsSemiLocalRing
361
  
362
  IsSemiLocalRing( R )  property
363
  Returns:  true or false
364
  
365
  R is a ring for homalg.
366
  
367
  3.3-25 IsIntegralDomain
368
  
369
  IsIntegralDomain( R )  property
370
  Returns:  true or false
371
  
372
  R is a ring for homalg.
373
  
374
  3.3-26 IsHereditary
375
  
376
  IsHereditary( R )  property
377
  Returns:  true or false
378
  
379
  R is a ring for homalg.
380
  
381
  3.3-27 IsLeftHereditary
382
  
383
  IsLeftHereditary( R )  property
384
  Returns:  true or false
385
  
386
  R is a ring for homalg.
387
  
388
  3.3-28 IsRightHereditary
389
  
390
  IsRightHereditary( R )  property
391
  Returns:  true or false
392
  
393
  R is a ring for homalg.
394
  
395
  3.3-29 IsHermite
396
  
397
  IsHermite( R )  property
398
  Returns:  true or false
399
  
400
  R is a ring for homalg.
401
  
402
  3.3-30 IsLeftHermite
403
  
404
  IsLeftHermite( R )  property
405
  Returns:  true or false
406
  
407
  R is a ring for homalg.
408
  
409
  3.3-31 IsRightHermite
410
  
411
  IsRightHermite( R )  property
412
  Returns:  true or false
413
  
414
  R is a ring for homalg.
415
  
416
  3.3-32 IsNoetherian
417
  
418
  IsNoetherian( R )  property
419
  Returns:  true or false
420
  
421
  R is a ring for homalg.
422
  
423
  3.3-33 IsLeftNoetherian
424
  
425
  IsLeftNoetherian( R )  property
426
  Returns:  true or false
427
  
428
  R is a ring for homalg.
429
  
430
  3.3-34 IsRightNoetherian
431
  
432
  IsRightNoetherian( R )  property
433
  Returns:  true or false
434
  
435
  R is a ring for homalg.
436
  
437
  3.3-35 IsCohenMacaulay
438
  
439
  IsCohenMacaulay( R )  property
440
  Returns:  true or false
441
  
442
  R is a ring for homalg.
443
  
444
  3.3-36 IsGorenstein
445
  
446
  IsGorenstein( R )  property
447
  Returns:  true or false
448
  
449
  R is a ring for homalg.
450
  
451
  3.3-37 IsKoszul
452
  
453
  IsKoszul( R )  property
454
  Returns:  true or false
455
  
456
  R is a ring for homalg.
457
  
458
  3.3-38 IsArtinian
459
  
460
  IsArtinian( R )  property
461
  Returns:  true or false
462
  
463
  R is a ring for homalg.
464
  
465
  3.3-39 IsLeftArtinian
466
  
467
  IsLeftArtinian( R )  property
468
  Returns:  true or false
469
  
470
  R is a ring for homalg.
471
  
472
  3.3-40 IsRightArtinian
473
  
474
  IsRightArtinian( R )  property
475
  Returns:  true or false
476
  
477
  R is a ring for homalg.
478
  
479
  3.3-41 IsOreDomain
480
  
481
  IsOreDomain( R )  property
482
  Returns:  true or false
483
  
484
  R is a ring for homalg.
485
  
486
  3.3-42 IsLeftOreDomain
487
  
488
  IsLeftOreDomain( R )  property
489
  Returns:  true or false
490
  
491
  R is a ring for homalg.
492
  
493
  3.3-43 IsRightOreDomain
494
  
495
  IsRightOreDomain( R )  property
496
  Returns:  true or false
497
  
498
  R is a ring for homalg.
499
  
500
  3.3-44 IsPrincipalIdealRing
501
  
502
  IsPrincipalIdealRing( R )  property
503
  Returns:  true or false
504
  
505
  R is a ring for homalg.
506
  
507
  3.3-45 IsLeftPrincipalIdealRing
508
  
509
  IsLeftPrincipalIdealRing( R )  property
510
  Returns:  true or false
511
  
512
  R is a ring for homalg.
513
  
514
  3.3-46 IsRightPrincipalIdealRing
515
  
516
  IsRightPrincipalIdealRing( R )  property
517
  Returns:  true or false
518
  
519
  R is a ring for homalg.
520
  
521
  3.3-47 IsRegular
522
  
523
  IsRegular( R )  property
524
  Returns:  true or false
525
  
526
  R is a ring for homalg.
527
  
528
  3.3-48 IsFiniteFreePresentationRing
529
  
530
  IsFiniteFreePresentationRing( R )  property
531
  Returns:  true or false
532
  
533
  R is a ring for homalg.
534
  
535
  3.3-49 IsLeftFiniteFreePresentationRing
536
  
537
  IsLeftFiniteFreePresentationRing( R )  property
538
  Returns:  true or false
539
  
540
  R is a ring for homalg.
541
  
542
  3.3-50 IsRightFiniteFreePresentationRing
543
  
544
  IsRightFiniteFreePresentationRing( R )  property
545
  Returns:  true or false
546
  
547
  R is a ring for homalg.
548
  
549
  3.3-51 IsSimpleRing
550
  
551
  IsSimpleRing( R )  property
552
  Returns:  true or false
553
  
554
  R is a ring for homalg.
555
  
556
  3.3-52 IsSemiSimpleRing
557
  
558
  IsSemiSimpleRing( R )  property
559
  Returns:  true or false
560
  
561
  R is a ring for homalg.
562
  
563
  3.3-53 IsSuperCommutative
564
  
565
  IsSuperCommutative( R )  property
566
  Returns:  true or false
567
  
568
  R is a ring for homalg.
569
  
570
  3.3-54 BasisAlgorithmRespectsPrincipalIdeals
571
  
572
  BasisAlgorithmRespectsPrincipalIdeals( R )  property
573
  Returns:  true or false
574
  
575
  R is a ring for homalg.
576
  
577
  3.3-55 AreUnitsCentral
578
  
579
  AreUnitsCentral( R )  property
580
  Returns:  true or false
581
  
582
  R is a ring for homalg.
583
  
584
  3.3-56 IsMinusOne
585
  
586
  IsMinusOne( r )  property
587
  Returns:  true or false
588
  
589
  Check if the ring element r is the additive inverse of one.
590
  
591
  3.3-57 IsMonic
592
  
593
  IsMonic( r )  property
594
  Returns:  true or false
595
  
596
  Check if the homalg ring element r is monic.
597
  
598
  3.3-58 IsMonicUptoUnit
599
  
600
  IsMonicUptoUnit( r )  property
601
  Returns:  true or false
602
  
603
  Check if leading coefficient of the homalg ring element r is a unit.
604
  
605
  3.3-59 IsLeftRegular
606
  
607
  IsLeftRegular( r )  property
608
  Returns:  true or false
609
  
610
  Check if the homalg ring element r is left regular.
611
  
612
  3.3-60 IsRightRegular
613
  
614
  IsRightRegular( r )  property
615
  Returns:  true or false
616
  
617
  Check if the homalg ring element r is right regular.
618
  
619
  3.3-61 IsRegular
620
  
621
  IsRegular( r )  property
622
  Returns:  true or false
623
  
624
  Check if the homalg ring element r is regular, i.e. left and right regular.
625
  
626
  
627
  3.4 Rings: Attributes
628
  
629
  3.4-1 Inverse
630
  
631
  Inverse( r )  attribute
632
  Returns:  a homalg ring element or fail
633
  
634
  The inverse of the homalg ring element r.
635
  
636
    Example  
637
    gap> ZZ := HomalgRingOfIntegers( );;
638
    gap> R := ZZ / 2^8;
639
    Z/( 256 )
640
    gap> r := (1/3*One(R)+1/5)+3/7;
641
    |[ 157 ]|
642
    gap> 1 / r;	## = r^-1;
643
    |[ 181 ]|
644
    gap> s := (1/3*One(R)+2/5)+3/7;
645
    |[ 106 ]|
646
    gap> 1 / s;
647
    fail
648
  
649
  
650
  3.4-2 homalgTable
651
  
652
  homalgTable( R )  attribute
653
  Returns:  a homalg table
654
  
655
  The  homalg  table  of  R  is a ring dictionary, i.e. the translator between
656
  homalg and the (specific implementation of the) ring.
657
  
658
  Every homalg ring has a homalg table.
659
  
660
  3.4-3 RingElementConstructor
661
  
662
  RingElementConstructor( R )  attribute
663
  Returns:  a function
664
  
665
  The constructor of ring elements in the homalg ring R.
666
  
667
  3.4-4 TypeOfHomalgMatrix
668
  
669
  TypeOfHomalgMatrix( R )  attribute
670
  Returns:  a type
671
  
672
  The GAP4-type of homalg matrices over the homalg ring R.
673
  
674
  3.4-5 ConstructorForHomalgMatrices
675
  
676
  ConstructorForHomalgMatrices( R )  attribute
677
  Returns:  a type
678
  
679
  The constructor for homalg matrices over the homalg ring R.
680
  
681
  3.4-6 Zero
682
  
683
  Zero( R )  attribute
684
  Returns:  a homalg ring element
685
  
686
  The zero of the homalg ring R.
687
  
688
  3.4-7 One
689
  
690
  One( R )  attribute
691
  Returns:  a homalg ring element
692
  
693
  The one of the homalg ring R.
694
  
695
  3.4-8 MinusOne
696
  
697
  MinusOne( R )  attribute
698
  Returns:  a homalg ring element
699
  
700
  The minus one of the homalg ring R.
701
  
702
  3.4-9 ProductOfIndeterminates
703
  
704
  ProductOfIndeterminates( R )  attribute
705
  Returns:  a homalg ring element
706
  
707
  The product of indeterminates of the homalg ring R.
708
  
709
  3.4-10 RationalParameters
710
  
711
  RationalParameters( R )  attribute
712
  Returns:  a list of homalg ring elements
713
  
714
  The list of rational parameters of the homalg ring R.
715
  
716
  3.4-11 IndeterminatesOfPolynomialRing
717
  
718
  IndeterminatesOfPolynomialRing( R )  attribute
719
  Returns:  a list of homalg ring elements
720
  
721
  The list of indeterminates of the homalg polynomial ring R.
722
  
723
  3.4-12 RelativeIndeterminatesOfPolynomialRing
724
  
725
  RelativeIndeterminatesOfPolynomialRing( R )  attribute
726
  Returns:  a list of homalg ring elements
727
  
728
  The list of relative indeterminates of the homalg polynomial ring R.
729
  
730
  3.4-13 IndeterminateCoordinatesOfRingOfDerivations
731
  
732
  IndeterminateCoordinatesOfRingOfDerivations( R )  attribute
733
  Returns:  a list of homalg ring elements
734
  
735
  The list of indeterminate coordinates of the homalg Weyl ring R.
736
  
737
  3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations
738
  
739
  RelativeIndeterminateCoordinatesOfRingOfDerivations( R )  attribute
740
  Returns:  a list of homalg ring elements
741
  
742
  The list of relative indeterminate coordinates of the homalg Weyl ring R.
743
  
744
  3.4-15 IndeterminateDerivationsOfRingOfDerivations
745
  
746
  IndeterminateDerivationsOfRingOfDerivations( R )  attribute
747
  Returns:  a list of homalg ring elements
748
  
749
  The list of indeterminate derivations of the homalg Weyl ring R.
750
  
751
  3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations
752
  
753
  RelativeIndeterminateDerivationsOfRingOfDerivations( R )  attribute
754
  Returns:  a list of homalg ring elements
755
  
756
  The list of relative indeterminate derivations of the homalg Weyl ring R.
757
  
758
  3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing
759
  
760
  IndeterminateAntiCommutingVariablesOfExteriorRing( R )  attribute
761
  Returns:  a list of homalg ring elements
762
  
763
  The list of anti-commuting indeterminates of the homalg exterior ring R.
764
  
765
  3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing
766
  
767
  RelativeIndeterminateAntiCommutingVariablesOfExteriorRing( R )  attribute
768
  Returns:  a list of homalg ring elements
769
  
770
  The  list  of  anti-commuting relative indeterminates of the homalg exterior
771
  ring R.
772
  
773
  3.4-19 IndeterminatesOfExteriorRing
774
  
775
  IndeterminatesOfExteriorRing( R )  attribute
776
  Returns:  a list of homalg ring elements
777
  
778
  The  list of all indeterminates (commuting and anti-commuting) of the homalg
779
  exterior ring R.
780
  
781
  3.4-20 CoefficientsRing
782
  
783
  CoefficientsRing( R )  attribute
784
  Returns:  a homalg ring
785
  
786
  The ring of coefficients of the homalg ring R.
787
  
788
  3.4-21 KrullDimension
789
  
790
  KrullDimension( R )  attribute
791
  Returns:  a non-negative integer
792
  
793
  The Krull dimension of the commutative homalg ring R.
794
  
795
  3.4-22 LeftGlobalDimension
796
  
797
  LeftGlobalDimension( R )  attribute
798
  Returns:  a non-negative integer
799
  
800
  The left global dimension of the homalg ring R.
801
  
802
  3.4-23 RightGlobalDimension
803
  
804
  RightGlobalDimension( R )  attribute
805
  Returns:  a non-negative integer
806
  
807
  The right global dimension of the homalg ring R.
808
  
809
  3.4-24 GlobalDimension
810
  
811
  GlobalDimension( R )  attribute
812
  Returns:  a non-negative integer
813
  
814
  The  global dimension of the homalg ring R. The global dimension is defined,
815
  only if the left and right global dimensions coincide.
816
  
817
  3.4-25 GeneralLinearRank
818
  
819
  GeneralLinearRank( R )  attribute
820
  Returns:  a non-negative integer
821
  
822
  The general linear rank of the homalg ring R ([MR01], 11.1.14).
823
  
824
  3.4-26 ElementaryRank
825
  
826
  ElementaryRank( R )  attribute
827
  Returns:  a non-negative integer
828
  
829
  The elementary rank of the homalg ring R ([MR01], 11.3.10).
830
  
831
  3.4-27 StableRank
832
  
833
  StableRank( R )  attribute
834
  Returns:  a non-negative integer
835
  
836
  The stable rank of the homalg ring R ([MR01], 11.3.4).
837
  
838
  3.4-28 AssociatedGradedRing
839
  
840
  AssociatedGradedRing( R )  attribute
841
  Returns:  a homalg ring
842
  
843
  The graded ring associated to the filtered ring R.
844
  
845
  
846
  3.5 Rings: Operations and Functions
847
  
848

849