671:
4688:
5837:
4443:
4581:
5630:
5091:
3789:
1768:
2610:
2417:
4940:
4861:
2174:
5562:
2673:
4067:
2328:
3262:
5016:
4733:
4639:
3633:
2479:
950:
737:
477:
4794:
4340:
4002:
3187:
3094:
2980:
3439:
2775:
4376:
5485:
5378:
5314:
5712:
2845:
5527:
4283:
3536:
5153:
3031:
3822:
3698:
5422:
4753:
4211:
4185:
4163:
3855:
3301:
2091:
1343:
534:
4975:
2712:
4491:
4465:
4396:
1925:
1890:
5738:
5656:
4127:
3749:
1375:
561:
Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.
5191:
5118:
4888:
4101:
3888:
3563:
2905:
2035:
1798:
1562:
1529:
1500:
1306:
1251:
1216:
1187:
1089:
1028:
977:
814:
419:
577:
5258:
5232:
2004:
1685:
3383:
2925:
2885:
1840:
1818:
1655:
1633:
1608:
1586:
1473:
1443:
1419:
1397:
1275:
1160:
1137:
1115:
1054:
999:
885:
865:
843:
782:
760:
4644:
5380:
such products exist. Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unit
358:
4401:
5743:
4499:
3762:
5583:
1693:
2540:
6095:
6090:. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press.
6046:
6016:
5971:
2368:
4893:
5021:
4814:
4191:
under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set
2110:
5532:
4013:
2615:
351:
2270:
17:
3197:
5381:
4980:
4704:
4610:
3582:
2428:
890:
686:
426:
4764:
4310:
3958:
3132:
3044:
3304:
2930:
6008:
3398:
2731:
344:
4349:
545:
is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to
6084:
Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the
Hilbert function and the Samuel function".
5438:
392:
5327:
5263:
3318:. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)
5906:
5878:
3921:
213:
2860:
Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the
5937:
2804:
5494:
4259:
3508:
3446:
3315:
2983:
2515:
1452:
5661:
5123:
3004:
5162:
Assuming the gradations of non-identity elements are non-zero, the number of elements of gradation
4694:
3797:
3673:
5405:
4738:
4194:
4168:
4146:
3833:
3270:
2074:
1322:
517:
5424:, the indexing family could be any graded monoid, assuming that the number of elements of degree
546:
304:
4948:
5901:
5874:
3913:
2177:
2060:
2038:
2007:
2685:
4476:
4450:
4381:
3105:
1897:
1862:
666:{\displaystyle R=\bigoplus _{n=0}^{\infty }R_{n}=R_{0}\oplus R_{1}\oplus R_{2}\oplus \cdots }
6057:
5717:
5635:
4106:
3724:
1354:
6120:
6026:
5917:
5169:
5096:
4866:
4079:
3917:
3866:
3541:
2890:
2861:
2352:
2013:
1776:
1540:
1507:
1478:
1284:
1229:
1221:
1194:
1165:
1067:
1006:
955:
792:
397:
291:
283:
255:
250:
241:
198:
140:
5981:
4890:'s, without using the additive part. That is, the set of elements of the graded monoid is
8:
5927:
5237:
5211:
4237:
3905:
3897:
3476:
3125:
3097:
2854:
2723:
2504:
2500:
1948:
1662:
1317:
1278:
505:
309:
299:
150:
50:
42:
33:
5198:
3909:
3901:
3470:
3454:
3333:
2987:
2910:
2870:
2510:
A graded ring is a graded module over itself. An ideal in a graded ring is homogeneous
1825:
1803:
1640:
1618:
1593:
1571:
1458:
1428:
1404:
1382:
1260:
1145:
1122:
1100:
1039:
984:
870:
850:
828:
767:
745:
565:
550:
384:
115:
106:
64:
6115:
6091:
6072:
6042:
6012:
5967:
5399:
3858:
3792:
3101:
2212:
2197:
4304:
2783:
is a submodule that is a graded module in own right and such that the set-theoretic
6034:
5977:
5911:
5892:
can be considered as a graded monoid, where the gradation of a word is its length.
4604:
4287:
3757:
2850:
2232:
2101:
376:
135:
4607:
is an example of an anticommutative algebra, graded with respect to the structure
160:
6085:
6022:
5922:
4300:
3828:
3653:
3326:
2252:
2189:
2053:
1943:
227:
221:
208:
188:
179:
145:
82:
4143:
The previously defined notion of "graded ring" now becomes the same thing as an
5932:
4188:
3668:
2511:
677:
388:
269:
6109:
6076:
4683:{\displaystyle \varepsilon \colon \mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} }
3314:
A graded module is said to be finitely generated if the underlying module is
2864:
is the same as to give the structure of a graded algebra to the latter ring.
2784:
2344:
2264:
542:
155:
120:
77:
504:
is defined similarly (see below for the precise definition). It generalizes
5393:
4756:
4256:
4252:
3701:
329:
260:
94:
5966:. Translated by Thomas, Reuben. Cambridge University Press. p. 384.
5885:
4290:. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).
4245:
2331:
2193:
372:
314:
203:
193:
167:
4438:{\displaystyle \varepsilon \colon \Gamma \to \mathbb {Z} /2\mathbb {Z} }
6000:
4241:
569:
69:
3935:
The above definitions have been generalized to rings graded using any
6011:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
5832:{\displaystyle \sum _{p,q\in R \atop p\cdot q=m}s(p)\times _{K}s'(q)}
4576:{\displaystyle xy=(-1)^{\varepsilon (\deg x)\varepsilon (\deg y)}yx,}
4225:
1534:
324:
130:
87:
55:
5488:
125:
1612:. A homogeneous ideal is the direct sum of its homogeneous parts.
5839:. This sum is correctly defined (i.e., finite) because, for each
1092:
482:
4808:
3936:
3712:
3501:
is a field), it is given the trivial grading (every element of
486:
59:
5962:
Sakarovitch, Jacques (2009). "Part II: The power of algebra".
3861:
is also graded, being the direct sum of the cohomology groups
3395:
a finitely generated graded module over it. Then the function
5204:
of the monoid. Therefore the number of elements of gradation
4224:
If we do not require that the ring have an identity element,
2787:
is a morphism of graded modules. Explicitly, a graded module
4307:
of the monoid of the gradation into the additive monoid of
3784:{\displaystyle \textstyle \bigwedge \nolimits ^{\bullet }V}
2052:
be the set of all nonzero homogeneous elements in a graded
5625:{\displaystyle s,s'\in K\langle \langle R\rangle \rangle }
5564:
denotes the semiring of power series with coefficients in
5387:
5316:
else. Indeed, each such element is the product of at most
1763:{\displaystyle R/I=\bigoplus _{n=0}^{\infty }R_{n}/I_{n},}
2605:{\displaystyle \bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M}
508:. A graded module that is also a graded ring is called a
489:. The direct sum decomposition is usually referred to as
5120:
is necessarily 0. Some authors request furthermore that
6058:"Intersection form for quasi-homogeneous singularities"
3649:
Examples of graded algebras are common in mathematics:
2857:
of a morphism of graded modules are graded submodules.
2726:
of the underlying modules that respects grading; i.e.,
847:. By definition of a direct sum, every nonzero element
5330:
5266:
4817:
3766:
2618:
2412:{\displaystyle M=\bigoplus _{i\in \mathbb {N} }M_{i},}
2273:
2113:
1224:, and the direct sum decomposition is a direct sum of
5746:
5720:
5664:
5638:
5586:
5535:
5497:
5441:
5408:
5240:
5214:
5172:
5126:
5099:
5024:
4983:
4951:
4935:{\displaystyle \bigcup _{n\in \mathbb {N} _{0}}R_{n}}
4896:
4869:
4767:
4741:
4707:
4647:
4613:
4502:
4479:
4453:
4404:
4384:
4352:
4313:
4262:
4197:
4171:
4149:
4109:
4082:
4016:
3961:
3869:
3836:
3800:
3765:
3727:
3676:
3585:
3544:
3511:
3401:
3336:
3273:
3200:
3135:
3047:
3007:
2933:
2913:
2893:
2873:
2807:
2734:
2688:
2543:
2431:
2371:
2077:
2016:
1951:
1900:
1865:
1828:
1806:
1779:
1696:
1665:
1643:
1621:
1596:
1574:
1543:
1510:
1481:
1461:
1431:
1407:
1385:
1357:
1325:
1287:
1263:
1232:
1197:
1168:
1148:
1125:
1103:
1070:
1042:
1009:
987:
958:
893:
873:
853:
831:
795:
770:
748:
689:
580:
520:
429:
400:
4856:{\textstyle \bigoplus _{n\in \mathbb {N} _{0}}R_{n}}
4299:
Some graded rings (or algebras) are endowed with an
2169:{\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}}
5086:{\displaystyle \phi (m\cdot m')=\phi (m)+\phi (m')}
2612:is a graded module over the associated graded ring
5831:
5732:
5706:
5650:
5624:
5557:{\displaystyle K\langle \langle R\rangle \rangle }
5556:
5521:
5479:
5416:
5372:
5308:
5252:
5226:
5185:
5147:
5112:
5085:
5010:
4969:
4934:
4882:
4855:
4788:
4747:
4727:
4682:
4633:
4575:
4485:
4459:
4437:
4390:
4370:
4334:
4277:
4205:
4179:
4157:
4121:
4095:
4061:
3996:
3882:
3849:
3816:
3783:
3743:
3692:
3660:are exactly the homogeneous polynomials of degree
3627:
3557:
3530:
3433:
3377:
3295:
3256:
3181:
3108:is an example of such a morphism having degree 1.
3088:
3025:
2974:
2919:
2899:
2879:
2839:
2769:
2706:
2668:{\textstyle \bigoplus _{0}^{\infty }I^{n}/I^{n+1}}
2667:
2604:
2473:
2411:
2322:
2168:
2085:
2029:
1998:
1919:
1884:
1834:
1812:
1792:
1762:
1679:
1649:
1627:
1602:
1580:
1556:
1523:
1494:
1467:
1437:
1413:
1391:
1369:
1337:
1300:
1269:
1245:
1210:
1181:
1154:
1131:
1109:
1083:
1048:
1022:
993:
971:
944:
879:
859:
837:
808:
776:
754:
731:
665:
528:
481:. The index set is usually the set of nonnegative
471:
413:
5632:is defined pointwise, it is the function sending
4062:{\displaystyle R_{i}R_{j}\subseteq R_{i\cdot j}.}
3175:
3165:
2323:{\textstyle \bigoplus _{i=0}^{\infty }H^{i}(X;R)}
512:. A graded ring could also be viewed as a graded
6107:
4342:, the field with two elements. Specifically, a
3912:. One example is the close relationship between
1423:. (Equivalently, if it is a graded submodule of
5398:These notions allow us to extend the notion of
3576:is also a graded ring, then one requires that
3111:
2330:with the multiplicative structure given by the
3927:
3257:{\displaystyle P(M,t)=\sum \ell (M_{n})t^{n}}
1119:; in particular, the multiplicative identity
352:
5619:
5616:
5610:
5607:
5551:
5548:
5542:
5539:
5961:
5011:{\displaystyle \phi :M\to \mathbb {N} _{0}}
4728:{\displaystyle (\mathbb {Z} ,\varepsilon )}
4634:{\displaystyle (\mathbb {Z} ,\varepsilon )}
4445:is a homomorphism of additive monoids. An
3628:{\displaystyle R_{i}A_{j}\subseteq A_{i+j}}
2474:{\displaystyle R_{i}M_{j}\subseteq M_{i+j}}
945:{\displaystyle a=a_{0}+a_{1}+\cdots +a_{n}}
732:{\displaystyle R_{m}R_{n}\subseteq R_{m+n}}
472:{\displaystyle R_{i}R_{j}\subseteq R_{i+j}}
6055:
5843:, there are only a finite number of pairs
5714:, and the product is the function sending
4789:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
4701:) is the same thing as an anticommutative
4335:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
3997:{\displaystyle R=\bigoplus _{i\in G}R_{i}}
3952:is a ring with a direct sum decomposition
2518:of a graded module is a homogeneous ideal.
359:
345:
6083:
5410:
4998:
4910:
4831:
4782:
4769:
4712:
4676:
4663:
4655:
4618:
4431:
4418:
4328:
4315:
4265:
4199:
4173:
4151:
3182:{\displaystyle P(M,t)\in \mathbb {Z} \!]}
3158:
3089:{\displaystyle f(M_{n})\subseteq N_{n+d}}
2390:
2079:
522:
6033:
2975:{\displaystyle M(\ell )_{n}=M_{n+\ell }}
2507:(with the field having trivial grading).
2503:is an example of a graded module over a
1139:is a homogeneous element of degree zero.
5402:. Instead of the indexing family being
5388:Power series indexed by a graded monoid
4248:are graded by the corresponding monoid.
3434:{\displaystyle n\mapsto \dim _{k}M_{n}}
2770:{\displaystyle f(N_{i})\subseteq M_{i}}
485:or the set of integers, but can be any
14:
6108:
4945:Formally, a graded monoid is a monoid
4371:{\displaystyle (\Gamma ,\varepsilon )}
5480:{\displaystyle (K,+_{K},\times _{K})}
5373:{\textstyle {\frac {g^{n+1}-1}{g-1}}}
5309:{\textstyle {\frac {g^{n+1}-1}{g-1}}}
3707:. The homogeneous elements of degree
3656:. The homogeneous elements of degree
1687:is also a graded ring, decomposed as
979:is either 0 or homogeneous of degree
5999:
4294:
2795:if and only if it is a submodule of
2263:, is a graded ring whose underlying
1857:can be given a gradation by letting
1635:is a two-sided homogeneous ideal in
4303:structure. This notion requires a
4240:naturally grades the corresponding
3768:
556:
24:
5752:
5572:. Its elements are functions from
4480:
4454:
4411:
4385:
4356:
4286:-graded algebra. Examples include
3445:. The function coincides with the
3441:is called the Hilbert function of
2629:
2560:
2290:
2130:
1800:is the homogeneous part of degree
1727:
1448:
603:
549:as well; e.g., one can consider a
25:
6132:
4699:skew-commutative associative ring
3896:Graded algebras are much used in
3493:In the usual case where the ring
3464:
2840:{\displaystyle N_{i}=N\cap M_{i}}
1847:
887:can be uniquely written as a sum
5522:{\displaystyle (R,\cdot ,\phi )}
4811:is the subset of a graded ring,
4802:
4278:{\displaystyle \mathbb {Z} _{2}}
3642:to be a graded left module over
3531:{\displaystyle R\subseteq A_{0}}
3497:is not graded (in particular if
2338:
1379:, the homogeneous components of
3120:over a commutative graded ring
3033:is a morphism of modules, then
5955:
5868:
5826:
5820:
5799:
5793:
5707:{\displaystyle s(m)+_{K}s'(m)}
5701:
5695:
5674:
5668:
5516:
5498:
5474:
5442:
5148:{\displaystyle \phi (m)\neq 0}
5136:
5130:
5080:
5069:
5060:
5054:
5045:
5028:
4993:
4964:
4952:
4722:
4708:
4659:
4628:
4614:
4559:
4547:
4541:
4529:
4522:
4512:
4414:
4365:
4353:
3405:
3372:
3340:
3303:are finite.) It is called the
3290:
3277:
3241:
3228:
3216:
3204:
3176:
3172:
3166:
3162:
3151:
3139:
3064:
3051:
3026:{\displaystyle f\colon M\to N}
3017:
2944:
2937:
2927:is a graded module defined by
2751:
2738:
2698:
2514:it is a graded submodule. The
2317:
2305:
1993:
1961:
13:
1:
6009:Graduate Texts in Mathematics
5992:
5093:. Note that the gradation of
4759:of the additive structure of
4586:for all homogeneous elements
3817:{\displaystyle S^{\bullet }V}
3693:{\displaystyle T^{\bullet }V}
742:for all nonnegative integers
5948:
5428:is finite, for each integer
5417:{\displaystyle \mathbb {N} }
4977:, with a gradation function
4748:{\displaystyle \varepsilon }
4206:{\displaystyle \mathbb {N} }
4180:{\displaystyle \mathbb {N} }
4158:{\displaystyle \mathbb {N} }
3850:{\displaystyle H^{\bullet }}
3296:{\displaystyle \ell (M_{n})}
3112:Invariants of graded modules
2714:of graded modules, called a
2235:with coefficients in a ring
2176:is a graded ring called the
2086:{\displaystyle \mathbb {Z} }
1338:{\displaystyle I\subseteq R}
529:{\displaystyle \mathbb {Z} }
393:direct sum of abelian groups
7:
5964:Elements of automata theory
5907:Differential graded algebra
5895:
4597:
3922:Homogeneous coordinate ring
3638:In other words, we require
3572:In the case where the ring
3490:if it is graded as a ring.
2188:; geometrically, it is the
1061:Some basic properties are:
27:Type of algebraic structure
10:
6137:
5938:Differential graded module
5580:. The sum of two elements
5391:
4970:{\displaystyle (M,\cdot )}
3931:-graded rings and algebras
3468:
2343:The corresponding idea in
568:that is decomposed into a
6071:(2): 211–223 See p. 211.
6037:(1974). "Ch. 1–3, 3 §3".
5384:in such a graded monoid.
3824:are also graded algebras.
3447:integer-valued polynomial
2791:is a graded submodule of
1931:≠0. This is called the
387:such that the underlying
5943:
5197:is the cardinality of a
4695:supercommutative algebra
3124:, one can associate the
2707:{\displaystyle f:N\to M}
2010:: it is a direct sum of
547:non-associative algebras
6087:Commutative Ring Theory
6056:Steenbrink, J. (1977).
4735:-graded algebra, where
4486:{\displaystyle \Gamma }
4473:graded with respect to
4460:{\displaystyle \Gamma }
4391:{\displaystyle \Gamma }
3914:homogeneous polynomials
3505:is of degree 0). Thus,
3305:Hilbert–Poincaré series
3037:is said to have degree
2039:homogeneous polynomials
1920:{\displaystyle R_{i}=0}
1885:{\displaystyle R_{0}=R}
1455:of a homogeneous ideal
6065:Compositio Mathematica
5902:Associated graded ring
5875:formal language theory
5833:
5734:
5733:{\displaystyle m\in R}
5708:
5652:
5651:{\displaystyle m\in R}
5626:
5558:
5529:a graded monoid. Then
5523:
5481:
5418:
5374:
5310:
5254:
5228:
5187:
5149:
5114:
5087:
5012:
4971:
4936:
4884:
4857:
4807:Intuitively, a graded
4790:
4749:
4729:
4684:
4635:
4577:
4487:
4461:
4439:
4392:
4372:
4336:
4279:
4255:is another term for a
4207:
4181:
4159:
4123:
4122:{\displaystyle i\in G}
4097:
4063:
3998:
3884:
3851:
3818:
3785:
3745:
3744:{\displaystyle T^{n}V}
3694:
3629:
3559:
3538:and the graded pieces
3532:
3435:
3379:
3297:
3258:
3183:
3116:Given a graded module
3090:
3027:
3001:be graded modules. If
2984:Serre's twisting sheaf
2976:
2921:
2901:
2881:
2867:Given a graded module
2841:
2771:
2708:
2669:
2633:
2606:
2564:
2525:in a commutative ring
2475:
2413:
2324:
2294:
2178:associated graded ring
2170:
2134:
2087:
2031:
2000:
1921:
1886:
1853:Any (non-graded) ring
1836:
1814:
1794:
1764:
1731:
1681:
1651:
1629:
1604:
1582:
1558:
1525:
1496:
1469:
1439:
1415:
1393:
1371:
1370:{\displaystyle a\in I}
1339:
1302:
1271:
1247:
1212:
1183:
1156:
1133:
1111:
1085:
1050:
1032:homogeneous components
1024:
995:
973:
946:
881:
861:
839:
810:
778:
756:
733:
667:
607:
530:
473:
415:
5834:
5735:
5709:
5653:
5627:
5559:
5524:
5482:
5419:
5375:
5311:
5255:
5229:
5188:
5186:{\displaystyle g^{n}}
5159:is not the identity.
5150:
5115:
5113:{\displaystyle 1_{M}}
5088:
5013:
4972:
4937:
4885:
4883:{\displaystyle R_{n}}
4858:
4791:
4750:
4730:
4685:
4636:
4578:
4488:
4462:
4440:
4393:
4373:
4337:
4280:
4208:
4182:
4160:
4124:
4098:
4096:{\displaystyle R_{i}}
4064:
3999:
3885:
3883:{\displaystyle H^{n}}
3852:
3819:
3786:
3746:
3695:
3630:
3560:
3558:{\displaystyle A_{i}}
3533:
3436:
3380:
3298:
3259:
3184:
3106:differential geometry
3091:
3028:
2977:
2922:
2902:
2900:{\displaystyle \ell }
2882:
2842:
2772:
2709:
2670:
2619:
2607:
2544:
2476:
2414:
2325:
2274:
2259:with coefficients in
2171:
2114:
2088:
2032:
2030:{\displaystyle R_{i}}
2001:
1922:
1887:
1837:
1815:
1795:
1793:{\displaystyle I_{n}}
1765:
1711:
1682:
1652:
1630:
1605:
1583:
1559:
1557:{\displaystyle R_{n}}
1526:
1524:{\displaystyle R_{0}}
1497:
1495:{\displaystyle R_{n}}
1470:
1440:
1416:
1394:
1372:
1340:
1303:
1301:{\displaystyle R_{0}}
1272:
1248:
1246:{\displaystyle R_{0}}
1213:
1211:{\displaystyle R_{0}}
1184:
1182:{\displaystyle R_{n}}
1157:
1134:
1112:
1086:
1084:{\displaystyle R_{0}}
1051:
1025:
1023:{\displaystyle a_{i}}
996:
974:
972:{\displaystyle a_{i}}
947:
882:
862:
840:
811:
809:{\displaystyle R_{n}}
789:A nonzero element of
779:
757:
734:
668:
587:
531:
474:
416:
414:{\displaystyle R_{i}}
5918:Graded (mathematics)
5744:
5740:to the infinite sum
5718:
5662:
5636:
5584:
5533:
5495:
5439:
5406:
5328:
5264:
5238:
5212:
5170:
5124:
5097:
5022:
4981:
4949:
4894:
4867:
4815:
4765:
4739:
4705:
4697:(sometimes called a
4690:is the quotient map.
4645:
4611:
4500:
4477:
4451:
4402:
4382:
4350:
4311:
4260:
4228:may replace monoids.
4195:
4169:
4165:-graded ring, where
4147:
4107:
4080:
4014:
3959:
3942:as an index set. A
3918:projective varieties
3867:
3834:
3798:
3763:
3725:
3674:
3583:
3542:
3509:
3399:
3334:
3271:
3198:
3133:
3045:
3005:
2931:
2911:
2891:
2871:
2805:
2732:
2686:
2616:
2541:
2429:
2369:
2271:
2111:
2075:
2014:
1949:
1898:
1863:
1826:
1804:
1777:
1694:
1663:
1641:
1619:
1594:
1572:
1541:
1508:
1479:
1459:
1449:§ Graded module
1429:
1405:
1383:
1355:
1323:
1285:
1261:
1230:
1195:
1166:
1146:
1123:
1101:
1068:
1040:
1007:
985:
956:
891:
871:
851:
829:
793:
768:
746:
687:
578:
518:
506:graded vector spaces
427:
398:
256:Group with operators
199:Complemented lattice
34:Algebraic structures
5928:Graded vector space
5435:More formally, let
5253:{\displaystyle g=1}
5227:{\displaystyle n+1}
5208:or less is at most
4863:, generated by the
4346:consists of a pair
3906:homological algebra
3898:commutative algebra
3477:associative algebra
3126:formal power series
3098:exterior derivative
2720:graded homomorphism
2501:graded vector space
2358:over a graded ring
1999:{\displaystyle R=k}
1680:{\displaystyle R/I}
564:A graded ring is a
310:Composition algebra
70:Quasigroup and loop
5914:, a generalization
5829:
5789:
5730:
5704:
5648:
5622:
5554:
5519:
5477:
5414:
5370:
5306:
5250:
5224:
5183:
5145:
5110:
5083:
5008:
4967:
4932:
4921:
4880:
4853:
4842:
4786:
4745:
4725:
4680:
4631:
4573:
4483:
4457:
4435:
4388:
4368:
4332:
4275:
4203:
4177:
4155:
4119:
4093:
4059:
3994:
3983:
3910:algebraic topology
3902:algebraic geometry
3880:
3847:
3814:
3781:
3780:
3741:
3690:
3625:
3555:
3528:
3471:Graded Lie algebra
3455:Hilbert polynomial
3431:
3375:
3316:finitely generated
3293:
3254:
3179:
3102:differential forms
3086:
3023:
2988:algebraic geometry
2972:
2917:
2897:
2877:
2837:
2767:
2704:
2665:
2602:
2471:
2409:
2395:
2320:
2166:
2083:
2027:
1996:
1917:
1882:
1832:
1810:
1790:
1760:
1677:
1647:
1625:
1600:
1578:
1554:
1521:
1492:
1465:
1435:
1411:
1389:
1367:
1335:
1298:
1267:
1243:
1208:
1179:
1152:
1129:
1107:
1081:
1046:
1020:
991:
969:
942:
877:
857:
835:
806:
774:
752:
729:
663:
551:graded Lie algebra
526:
469:
411:
6097:978-1-107-71712-1
6048:978-3-540-64243-5
6018:978-0-387-95385-4
5973:978-0-521-84425-3
5787:
5747:
5400:power series ring
5368:
5304:
4897:
4818:
4295:Anticommutativity
4288:Clifford algebras
4251:An (associative)
4187:is the monoid of
3968:
3859:cohomology theory
3793:symmetric algebra
3378:{\displaystyle k}
2920:{\displaystyle M}
2880:{\displaystyle M}
2537:, the direct sum
2378:
2213:topological space
2100:is an ideal in a
1933:trivial gradation
1835:{\displaystyle I}
1813:{\displaystyle n}
1650:{\displaystyle R}
1628:{\displaystyle I}
1603:{\displaystyle I}
1581:{\displaystyle n}
1468:{\displaystyle I}
1438:{\displaystyle R}
1414:{\displaystyle I}
1392:{\displaystyle a}
1270:{\displaystyle R}
1155:{\displaystyle n}
1132:{\displaystyle 1}
1110:{\displaystyle R}
1049:{\displaystyle a}
994:{\displaystyle i}
880:{\displaystyle R}
860:{\displaystyle a}
838:{\displaystyle n}
777:{\displaystyle n}
755:{\displaystyle m}
369:
368:
18:Homogeneous ideal
16:(Redirected from
6128:
6101:
6080:
6062:
6052:
6029:
5986:
5985:
5959:
5912:Filtered algebra
5864:
5854:
5838:
5836:
5835:
5830:
5819:
5811:
5810:
5788:
5786:
5769:
5739:
5737:
5736:
5731:
5713:
5711:
5710:
5705:
5694:
5686:
5685:
5657:
5655:
5654:
5649:
5631:
5629:
5628:
5623:
5600:
5563:
5561:
5560:
5555:
5528:
5526:
5525:
5520:
5487:be an arbitrary
5486:
5484:
5483:
5478:
5473:
5472:
5460:
5459:
5423:
5421:
5420:
5415:
5413:
5379:
5377:
5376:
5371:
5369:
5367:
5356:
5349:
5348:
5332:
5315:
5313:
5312:
5307:
5305:
5303:
5292:
5285:
5284:
5268:
5259:
5257:
5256:
5251:
5233:
5231:
5230:
5225:
5192:
5190:
5189:
5184:
5182:
5181:
5154:
5152:
5151:
5146:
5119:
5117:
5116:
5111:
5109:
5108:
5092:
5090:
5089:
5084:
5079:
5044:
5017:
5015:
5014:
5009:
5007:
5006:
5001:
4976:
4974:
4973:
4968:
4941:
4939:
4938:
4933:
4931:
4930:
4920:
4919:
4918:
4913:
4889:
4887:
4886:
4881:
4879:
4878:
4862:
4860:
4859:
4854:
4852:
4851:
4841:
4840:
4839:
4834:
4797:
4795:
4793:
4792:
4787:
4785:
4777:
4772:
4754:
4752:
4751:
4746:
4734:
4732:
4731:
4726:
4715:
4689:
4687:
4686:
4681:
4679:
4671:
4666:
4658:
4640:
4638:
4637:
4632:
4621:
4605:exterior algebra
4582:
4580:
4579:
4574:
4563:
4562:
4492:
4490:
4489:
4484:
4466:
4464:
4463:
4458:
4447:anticommutative
4444:
4442:
4441:
4436:
4434:
4426:
4421:
4398:is a monoid and
4397:
4395:
4394:
4389:
4377:
4375:
4374:
4369:
4341:
4339:
4338:
4333:
4331:
4323:
4318:
4284:
4282:
4281:
4276:
4274:
4273:
4268:
4213:with any monoid
4212:
4210:
4209:
4204:
4202:
4186:
4184:
4183:
4178:
4176:
4164:
4162:
4161:
4156:
4154:
4128:
4126:
4125:
4120:
4102:
4100:
4099:
4094:
4092:
4091:
4076:that lie inside
4068:
4066:
4065:
4060:
4055:
4054:
4036:
4035:
4026:
4025:
4003:
4001:
4000:
3995:
3993:
3992:
3982:
3891:
3889:
3887:
3886:
3881:
3879:
3878:
3856:
3854:
3853:
3848:
3846:
3845:
3823:
3821:
3820:
3815:
3810:
3809:
3790:
3788:
3787:
3782:
3776:
3775:
3758:exterior algebra
3752:
3750:
3748:
3747:
3742:
3737:
3736:
3699:
3697:
3696:
3691:
3686:
3685:
3654:Polynomial rings
3634:
3632:
3631:
3626:
3624:
3623:
3605:
3604:
3595:
3594:
3564:
3562:
3561:
3556:
3554:
3553:
3537:
3535:
3534:
3529:
3527:
3526:
3440:
3438:
3437:
3432:
3430:
3429:
3417:
3416:
3386:
3384:
3382:
3381:
3376:
3371:
3370:
3352:
3351:
3302:
3300:
3299:
3294:
3289:
3288:
3263:
3261:
3260:
3255:
3253:
3252:
3240:
3239:
3190:
3188:
3186:
3185:
3180:
3161:
3095:
3093:
3092:
3087:
3085:
3084:
3063:
3062:
3032:
3030:
3029:
3024:
2981:
2979:
2978:
2973:
2971:
2970:
2952:
2951:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2886:
2884:
2883:
2878:
2848:
2846:
2844:
2843:
2838:
2836:
2835:
2817:
2816:
2781:graded submodule
2778:
2776:
2774:
2773:
2768:
2766:
2765:
2750:
2749:
2713:
2711:
2710:
2705:
2674:
2672:
2671:
2666:
2664:
2663:
2648:
2643:
2642:
2632:
2627:
2611:
2609:
2608:
2603:
2598:
2597:
2582:
2574:
2573:
2563:
2558:
2491:
2487:
2480:
2478:
2477:
2472:
2470:
2469:
2451:
2450:
2441:
2440:
2418:
2416:
2415:
2410:
2405:
2404:
2394:
2393:
2351:, namely a left
2329:
2327:
2326:
2321:
2304:
2303:
2293:
2288:
2233:cohomology group
2175:
2173:
2172:
2167:
2165:
2164:
2149:
2144:
2143:
2133:
2128:
2102:commutative ring
2092:
2090:
2089:
2084:
2082:
2067:with respect to
2036:
2034:
2033:
2028:
2026:
2025:
2005:
2003:
2002:
1997:
1992:
1991:
1973:
1972:
1926:
1924:
1923:
1918:
1910:
1909:
1893:
1891:
1889:
1888:
1883:
1875:
1874:
1843:
1841:
1839:
1838:
1833:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1789:
1788:
1769:
1767:
1766:
1761:
1756:
1755:
1746:
1741:
1740:
1730:
1725:
1704:
1686:
1684:
1683:
1678:
1673:
1658:
1656:
1654:
1653:
1648:
1634:
1632:
1631:
1626:
1611:
1609:
1607:
1606:
1601:
1587:
1585:
1584:
1579:
1566:homogeneous part
1563:
1561:
1560:
1555:
1553:
1552:
1532:
1530:
1528:
1527:
1522:
1520:
1519:
1501:
1499:
1498:
1493:
1491:
1490:
1474:
1472:
1471:
1466:
1446:
1444:
1442:
1441:
1436:
1422:
1420:
1418:
1417:
1412:
1398:
1396:
1395:
1390:
1378:
1376:
1374:
1373:
1368:
1344:
1342:
1341:
1336:
1309:
1307:
1305:
1304:
1299:
1297:
1296:
1276:
1274:
1273:
1268:
1254:
1252:
1250:
1249:
1244:
1242:
1241:
1219:
1217:
1215:
1214:
1209:
1207:
1206:
1188:
1186:
1185:
1180:
1178:
1177:
1161:
1159:
1158:
1153:
1138:
1136:
1135:
1130:
1118:
1116:
1114:
1113:
1108:
1090:
1088:
1087:
1082:
1080:
1079:
1057:
1055:
1053:
1052:
1047:
1029:
1027:
1026:
1021:
1019:
1018:
1002:
1000:
998:
997:
992:
978:
976:
975:
970:
968:
967:
951:
949:
948:
943:
941:
940:
922:
921:
909:
908:
886:
884:
883:
878:
866:
864:
863:
858:
846:
844:
842:
841:
836:
815:
813:
812:
807:
805:
804:
785:
783:
781:
780:
775:
761:
759:
758:
753:
738:
736:
735:
730:
728:
727:
709:
708:
699:
698:
672:
670:
669:
664:
656:
655:
643:
642:
630:
629:
617:
616:
606:
601:
557:First properties
537:
535:
533:
532:
527:
525:
480:
478:
476:
475:
470:
468:
467:
449:
448:
439:
438:
420:
418:
417:
412:
410:
409:
377:abstract algebra
375:, in particular
361:
354:
347:
136:Commutative ring
65:Rack and quandle
30:
29:
21:
6136:
6135:
6131:
6130:
6129:
6127:
6126:
6125:
6106:
6105:
6104:
6098:
6060:
6049:
6019:
5995:
5990:
5989:
5974:
5960:
5956:
5951:
5946:
5923:Graded category
5898:
5871:
5856:
5844:
5812:
5806:
5802:
5770:
5753:
5751:
5745:
5742:
5741:
5719:
5716:
5715:
5687:
5681:
5677:
5663:
5660:
5659:
5637:
5634:
5633:
5593:
5585:
5582:
5581:
5534:
5531:
5530:
5496:
5493:
5492:
5468:
5464:
5455:
5451:
5440:
5437:
5436:
5409:
5407:
5404:
5403:
5396:
5390:
5357:
5338:
5334:
5333:
5331:
5329:
5326:
5325:
5293:
5274:
5270:
5269:
5267:
5265:
5262:
5261:
5239:
5236:
5235:
5213:
5210:
5209:
5177:
5173:
5171:
5168:
5167:
5125:
5122:
5121:
5104:
5100:
5098:
5095:
5094:
5072:
5037:
5023:
5020:
5019:
5002:
4997:
4996:
4982:
4979:
4978:
4950:
4947:
4946:
4926:
4922:
4914:
4909:
4908:
4901:
4895:
4892:
4891:
4874:
4870:
4868:
4865:
4864:
4847:
4843:
4835:
4830:
4829:
4822:
4816:
4813:
4812:
4805:
4781:
4773:
4768:
4766:
4763:
4762:
4760:
4740:
4737:
4736:
4711:
4706:
4703:
4702:
4675:
4667:
4662:
4654:
4646:
4643:
4642:
4617:
4612:
4609:
4608:
4600:
4525:
4521:
4501:
4498:
4497:
4478:
4475:
4474:
4452:
4449:
4448:
4430:
4422:
4417:
4403:
4400:
4399:
4383:
4380:
4379:
4351:
4348:
4347:
4327:
4319:
4314:
4312:
4309:
4308:
4301:anticommutative
4297:
4269:
4264:
4263:
4261:
4258:
4257:
4198:
4196:
4193:
4192:
4189:natural numbers
4172:
4170:
4167:
4166:
4150:
4148:
4145:
4144:
4129:are said to be
4108:
4105:
4104:
4087:
4083:
4081:
4078:
4077:
4044:
4040:
4031:
4027:
4021:
4017:
4015:
4012:
4011:
3988:
3984:
3972:
3960:
3957:
3956:
3933:
3874:
3870:
3868:
3865:
3864:
3862:
3841:
3837:
3835:
3832:
3831:
3829:cohomology ring
3805:
3801:
3799:
3796:
3795:
3771:
3767:
3764:
3761:
3760:
3732:
3728:
3726:
3723:
3722:
3720:
3681:
3677:
3675:
3672:
3671:
3613:
3609:
3600:
3596:
3590:
3586:
3584:
3581:
3580:
3549:
3545:
3543:
3540:
3539:
3522:
3518:
3510:
3507:
3506:
3473:
3467:
3425:
3421:
3412:
3408:
3400:
3397:
3396:
3366:
3362:
3347:
3343:
3335:
3332:
3331:
3329:
3327:polynomial ring
3284:
3280:
3272:
3269:
3268:
3248:
3244:
3235:
3231:
3199:
3196:
3195:
3157:
3134:
3131:
3130:
3128:
3114:
3074:
3070:
3058:
3054:
3046:
3043:
3042:
3006:
3003:
3002:
2960:
2956:
2947:
2943:
2932:
2929:
2928:
2912:
2909:
2908:
2892:
2889:
2888:
2872:
2869:
2868:
2831:
2827:
2812:
2808:
2806:
2803:
2802:
2800:
2761:
2757:
2745:
2741:
2733:
2730:
2729:
2727:
2716:graded morphism
2687:
2684:
2683:
2653:
2649:
2644:
2638:
2634:
2628:
2623:
2617:
2614:
2613:
2587:
2583:
2578:
2569:
2565:
2559:
2548:
2542:
2539:
2538:
2521:Given an ideal
2489:
2485:
2459:
2455:
2446:
2442:
2436:
2432:
2430:
2427:
2426:
2400:
2396:
2389:
2382:
2370:
2367:
2366:
2341:
2299:
2295:
2289:
2278:
2272:
2269:
2268:
2253:cohomology ring
2190:coordinate ring
2154:
2150:
2145:
2139:
2135:
2129:
2118:
2112:
2109:
2108:
2078:
2076:
2073:
2072:
2054:integral domain
2021:
2017:
2015:
2012:
2011:
1987:
1983:
1968:
1964:
1950:
1947:
1946:
1944:polynomial ring
1905:
1901:
1899:
1896:
1895:
1870:
1866:
1864:
1861:
1860:
1858:
1850:
1827:
1824:
1823:
1821:
1805:
1802:
1801:
1784:
1780:
1778:
1775:
1774:
1751:
1747:
1742:
1736:
1732:
1726:
1715:
1700:
1695:
1692:
1691:
1669:
1664:
1661:
1660:
1642:
1639:
1638:
1636:
1620:
1617:
1616:
1595:
1592:
1591:
1589:
1573:
1570:
1569:
1548:
1544:
1542:
1539:
1538:
1515:
1511:
1509:
1506:
1505:
1503:
1486:
1482:
1480:
1477:
1476:
1460:
1457:
1456:
1430:
1427:
1426:
1424:
1406:
1403:
1402:
1400:
1399:also belong to
1384:
1381:
1380:
1356:
1353:
1352:
1350:
1349:, if for every
1324:
1321:
1320:
1292:
1288:
1286:
1283:
1282:
1280:
1262:
1259:
1258:
1237:
1233:
1231:
1228:
1227:
1225:
1202:
1198:
1196:
1193:
1192:
1190:
1189:is a two-sided
1173:
1169:
1167:
1164:
1163:
1147:
1144:
1143:
1124:
1121:
1120:
1102:
1099:
1098:
1096:
1075:
1071:
1069:
1066:
1065:
1041:
1038:
1037:
1035:
1014:
1010:
1008:
1005:
1004:
986:
983:
982:
980:
963:
959:
957:
954:
953:
936:
932:
917:
913:
904:
900:
892:
889:
888:
872:
869:
868:
852:
849:
848:
830:
827:
826:
824:
800:
796:
794:
791:
790:
769:
766:
765:
763:
747:
744:
743:
717:
713:
704:
700:
694:
690:
688:
685:
684:
678:additive groups
651:
647:
638:
634:
625:
621:
612:
608:
602:
591:
579:
576:
575:
559:
521:
519:
516:
515:
513:
457:
453:
444:
440:
434:
430:
428:
425:
424:
422:
405:
401:
399:
396:
395:
365:
336:
335:
334:
305:Non-associative
287:
276:
275:
265:
245:
234:
233:
222:Map of lattices
218:
214:Boolean algebra
209:Heyting algebra
183:
172:
171:
165:
146:Integral domain
110:
99:
98:
92:
46:
28:
23:
22:
15:
12:
11:
5:
6134:
6124:
6123:
6118:
6103:
6102:
6096:
6081:
6053:
6047:
6031:
6017:
5996:
5994:
5991:
5988:
5987:
5972:
5953:
5952:
5950:
5947:
5945:
5942:
5941:
5940:
5935:
5933:Tensor algebra
5930:
5925:
5920:
5915:
5909:
5904:
5897:
5894:
5888:of words over
5870:
5867:
5828:
5825:
5822:
5818:
5815:
5809:
5805:
5801:
5798:
5795:
5792:
5785:
5782:
5779:
5776:
5773:
5768:
5765:
5762:
5759:
5756:
5750:
5729:
5726:
5723:
5703:
5700:
5697:
5693:
5690:
5684:
5680:
5676:
5673:
5670:
5667:
5647:
5644:
5641:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5599:
5596:
5592:
5589:
5553:
5550:
5547:
5544:
5541:
5538:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5476:
5471:
5467:
5463:
5458:
5454:
5450:
5447:
5444:
5412:
5389:
5386:
5366:
5363:
5360:
5355:
5352:
5347:
5344:
5341:
5337:
5302:
5299:
5296:
5291:
5288:
5283:
5280:
5277:
5273:
5249:
5246:
5243:
5223:
5220:
5217:
5199:generating set
5180:
5176:
5144:
5141:
5138:
5135:
5132:
5129:
5107:
5103:
5082:
5078:
5075:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5043:
5040:
5036:
5033:
5030:
5027:
5005:
5000:
4995:
4992:
4989:
4986:
4966:
4963:
4960:
4957:
4954:
4929:
4925:
4917:
4912:
4907:
4904:
4900:
4877:
4873:
4850:
4846:
4838:
4833:
4828:
4825:
4821:
4804:
4801:
4800:
4799:
4784:
4780:
4776:
4771:
4744:
4724:
4721:
4718:
4714:
4710:
4691:
4678:
4674:
4670:
4665:
4661:
4657:
4653:
4650:
4630:
4627:
4624:
4620:
4616:
4599:
4596:
4584:
4583:
4572:
4569:
4566:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4524:
4520:
4517:
4514:
4511:
4508:
4505:
4482:
4456:
4433:
4429:
4425:
4420:
4416:
4413:
4410:
4407:
4387:
4367:
4364:
4361:
4358:
4355:
4330:
4326:
4322:
4317:
4296:
4293:
4292:
4291:
4272:
4267:
4249:
4230:
4229:
4201:
4175:
4153:
4118:
4115:
4112:
4090:
4086:
4070:
4069:
4058:
4053:
4050:
4047:
4043:
4039:
4034:
4030:
4024:
4020:
4005:
4004:
3991:
3987:
3981:
3978:
3975:
3971:
3967:
3964:
3932:
3926:
3894:
3893:
3877:
3873:
3844:
3840:
3825:
3813:
3808:
3804:
3779:
3774:
3770:
3754:
3740:
3735:
3731:
3689:
3684:
3680:
3669:tensor algebra
3665:
3636:
3635:
3622:
3619:
3616:
3612:
3608:
3603:
3599:
3593:
3589:
3552:
3548:
3525:
3521:
3517:
3514:
3488:graded algebra
3466:
3465:Graded algebra
3463:
3428:
3424:
3420:
3415:
3411:
3407:
3404:
3374:
3369:
3365:
3361:
3358:
3355:
3350:
3346:
3342:
3339:
3292:
3287:
3283:
3279:
3276:
3265:
3264:
3251:
3247:
3243:
3238:
3234:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3178:
3174:
3171:
3168:
3164:
3160:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3113:
3110:
3083:
3080:
3077:
3073:
3069:
3066:
3061:
3057:
3053:
3050:
3022:
3019:
3016:
3013:
3010:
2969:
2966:
2963:
2959:
2955:
2950:
2946:
2942:
2939:
2936:
2916:
2896:
2876:
2834:
2830:
2826:
2823:
2820:
2815:
2811:
2799:and satisfies
2764:
2760:
2756:
2753:
2748:
2744:
2740:
2737:
2703:
2700:
2697:
2694:
2691:
2677:
2676:
2662:
2659:
2656:
2652:
2647:
2641:
2637:
2631:
2626:
2622:
2601:
2596:
2593:
2590:
2586:
2581:
2577:
2572:
2568:
2562:
2557:
2554:
2551:
2547:
2519:
2512:if and only if
2508:
2482:
2481:
2468:
2465:
2462:
2458:
2454:
2449:
2445:
2439:
2435:
2420:
2419:
2408:
2403:
2399:
2392:
2388:
2385:
2381:
2377:
2374:
2340:
2337:
2336:
2335:
2319:
2316:
2313:
2310:
2307:
2302:
2298:
2292:
2287:
2284:
2281:
2277:
2205:
2163:
2160:
2157:
2153:
2148:
2142:
2138:
2132:
2127:
2124:
2121:
2117:
2094:
2081:
2046:
2037:consisting of
2024:
2020:
1995:
1990:
1986:
1982:
1979:
1976:
1971:
1967:
1963:
1960:
1957:
1954:
1940:
1916:
1913:
1908:
1904:
1881:
1878:
1873:
1869:
1849:
1848:Basic examples
1846:
1831:
1809:
1787:
1783:
1771:
1770:
1759:
1754:
1750:
1745:
1739:
1735:
1729:
1724:
1721:
1718:
1714:
1710:
1707:
1703:
1699:
1676:
1672:
1668:
1646:
1624:
1599:
1577:
1551:
1547:
1518:
1514:
1489:
1485:
1464:
1434:
1410:
1388:
1366:
1363:
1360:
1334:
1331:
1328:
1314:
1313:
1295:
1291:
1266:
1256:
1240:
1236:
1205:
1201:
1176:
1172:
1151:
1140:
1128:
1106:
1078:
1074:
1045:
1017:
1013:
1003:. The nonzero
990:
966:
962:
939:
935:
931:
928:
925:
920:
916:
912:
907:
903:
899:
896:
876:
856:
834:
816:is said to be
803:
799:
773:
751:
740:
739:
726:
723:
720:
716:
712:
707:
703:
697:
693:
674:
673:
662:
659:
654:
650:
646:
641:
637:
633:
628:
624:
620:
615:
611:
605:
600:
597:
594:
590:
586:
583:
558:
555:
524:
510:graded algebra
466:
463:
460:
456:
452:
447:
443:
437:
433:
408:
404:
389:additive group
367:
366:
364:
363:
356:
349:
341:
338:
337:
333:
332:
327:
322:
317:
312:
307:
302:
296:
295:
294:
288:
282:
281:
278:
277:
274:
273:
270:Linear algebra
264:
263:
258:
253:
247:
246:
240:
239:
236:
235:
232:
231:
228:Lattice theory
224:
217:
216:
211:
206:
201:
196:
191:
185:
184:
178:
177:
174:
173:
164:
163:
158:
153:
148:
143:
138:
133:
128:
123:
118:
112:
111:
105:
104:
101:
100:
91:
90:
85:
80:
74:
73:
72:
67:
62:
53:
47:
41:
40:
37:
36:
26:
9:
6:
4:
3:
2:
6133:
6122:
6119:
6117:
6114:
6113:
6111:
6099:
6093:
6089:
6088:
6082:
6078:
6074:
6070:
6066:
6059:
6054:
6050:
6044:
6040:
6036:
6032:
6028:
6024:
6020:
6014:
6010:
6006:
6002:
5998:
5997:
5983:
5979:
5975:
5969:
5965:
5958:
5954:
5939:
5936:
5934:
5931:
5929:
5926:
5924:
5921:
5919:
5916:
5913:
5910:
5908:
5905:
5903:
5900:
5899:
5893:
5891:
5887:
5883:
5880:
5876:
5866:
5863:
5859:
5852:
5848:
5842:
5823:
5816:
5813:
5807:
5803:
5796:
5790:
5783:
5780:
5777:
5774:
5771:
5766:
5763:
5760:
5757:
5754:
5748:
5727:
5724:
5721:
5698:
5691:
5688:
5682:
5678:
5671:
5665:
5645:
5642:
5639:
5613:
5604:
5601:
5597:
5594:
5590:
5587:
5579:
5575:
5571:
5567:
5545:
5536:
5513:
5510:
5507:
5504:
5501:
5490:
5469:
5465:
5461:
5456:
5452:
5448:
5445:
5433:
5431:
5427:
5401:
5395:
5385:
5383:
5364:
5361:
5358:
5353:
5350:
5345:
5342:
5339:
5335:
5323:
5319:
5300:
5297:
5294:
5289:
5286:
5281:
5278:
5275:
5271:
5247:
5244:
5241:
5221:
5218:
5215:
5207:
5203:
5200:
5196:
5178:
5174:
5165:
5160:
5158:
5142:
5139:
5133:
5127:
5105:
5101:
5076:
5073:
5066:
5063:
5057:
5051:
5048:
5041:
5038:
5034:
5031:
5025:
5003:
4990:
4987:
4984:
4961:
4958:
4955:
4943:
4927:
4923:
4915:
4905:
4902:
4898:
4875:
4871:
4848:
4844:
4836:
4826:
4823:
4819:
4810:
4803:Graded monoid
4778:
4774:
4758:
4742:
4719:
4716:
4700:
4696:
4692:
4672:
4668:
4651:
4648:
4625:
4622:
4606:
4602:
4601:
4595:
4593:
4589:
4570:
4567:
4564:
4556:
4553:
4550:
4544:
4538:
4535:
4532:
4526:
4518:
4515:
4509:
4506:
4503:
4496:
4495:
4494:
4472:
4468:
4427:
4423:
4408:
4405:
4362:
4359:
4345:
4344:signed monoid
4324:
4320:
4306:
4302:
4289:
4285:
4270:
4254:
4250:
4247:
4244:; similarly,
4243:
4239:
4235:
4234:
4233:
4227:
4223:
4222:
4221:
4218:
4216:
4190:
4141:
4139:
4136:
4132:
4116:
4113:
4110:
4088:
4084:
4075:
4056:
4051:
4048:
4045:
4041:
4037:
4032:
4028:
4022:
4018:
4010:
4009:
4008:
3989:
3985:
3979:
3976:
3973:
3969:
3965:
3962:
3955:
3954:
3953:
3951:
3948:
3946:
3941:
3938:
3930:
3925:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3875:
3871:
3860:
3842:
3838:
3830:
3826:
3811:
3806:
3802:
3794:
3777:
3772:
3759:
3755:
3738:
3733:
3729:
3718:
3714:
3710:
3706:
3703:
3687:
3682:
3678:
3670:
3666:
3663:
3659:
3655:
3652:
3651:
3650:
3647:
3645:
3641:
3620:
3617:
3614:
3610:
3606:
3601:
3597:
3591:
3587:
3579:
3578:
3577:
3575:
3570:
3568:
3550:
3546:
3523:
3519:
3515:
3512:
3504:
3500:
3496:
3491:
3489:
3485:
3481:
3478:
3472:
3462:
3460:
3456:
3452:
3448:
3444:
3426:
3422:
3418:
3413:
3409:
3402:
3394:
3391:a field, and
3390:
3367:
3363:
3359:
3356:
3353:
3348:
3344:
3337:
3328:
3324:
3319:
3317:
3312:
3310:
3306:
3285:
3281:
3274:
3249:
3245:
3236:
3232:
3225:
3222:
3219:
3213:
3210:
3207:
3201:
3194:
3193:
3192:
3169:
3154:
3148:
3145:
3142:
3136:
3127:
3123:
3119:
3109:
3107:
3103:
3099:
3081:
3078:
3075:
3071:
3067:
3059:
3055:
3048:
3040:
3036:
3020:
3014:
3011:
3008:
3000:
2996:
2991:
2989:
2985:
2967:
2964:
2961:
2957:
2953:
2948:
2940:
2934:
2914:
2894:
2874:
2865:
2863:
2858:
2856:
2852:
2832:
2828:
2824:
2821:
2818:
2813:
2809:
2798:
2794:
2790:
2786:
2782:
2762:
2758:
2754:
2746:
2742:
2735:
2725:
2721:
2717:
2701:
2695:
2692:
2689:
2682:
2660:
2657:
2654:
2650:
2645:
2639:
2635:
2624:
2620:
2599:
2594:
2591:
2588:
2584:
2579:
2575:
2570:
2566:
2555:
2552:
2549:
2545:
2536:
2532:
2528:
2524:
2520:
2517:
2513:
2509:
2506:
2502:
2498:
2497:
2496:
2493:
2466:
2463:
2460:
2456:
2452:
2447:
2443:
2437:
2433:
2425:
2424:
2423:
2406:
2401:
2397:
2386:
2383:
2379:
2375:
2372:
2365:
2364:
2363:
2361:
2357:
2354:
2350:
2349:graded module
2347:is that of a
2346:
2345:module theory
2339:Graded module
2333:
2314:
2311:
2308:
2300:
2296:
2285:
2282:
2279:
2275:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2161:
2158:
2155:
2151:
2146:
2140:
2136:
2125:
2122:
2119:
2115:
2106:
2103:
2099:
2095:
2093:-graded ring.
2070:
2066:
2062:
2058:
2055:
2051:
2047:
2044:
2040:
2022:
2018:
2009:
2006:is graded by
1988:
1984:
1980:
1977:
1974:
1969:
1965:
1958:
1955:
1952:
1945:
1941:
1938:
1934:
1930:
1914:
1911:
1906:
1902:
1879:
1876:
1871:
1867:
1856:
1852:
1851:
1845:
1829:
1807:
1785:
1781:
1757:
1752:
1748:
1743:
1737:
1733:
1722:
1719:
1716:
1712:
1708:
1705:
1701:
1697:
1690:
1689:
1688:
1674:
1670:
1666:
1644:
1622:
1613:
1597:
1575:
1567:
1549:
1545:
1536:
1516:
1512:
1487:
1483:
1462:
1454:
1450:
1432:
1408:
1386:
1364:
1361:
1358:
1348:
1332:
1329:
1326:
1319:
1311:
1293:
1289:
1264:
1257:
1238:
1234:
1223:
1203:
1199:
1174:
1170:
1149:
1141:
1126:
1104:
1094:
1076:
1072:
1064:
1063:
1062:
1059:
1043:
1033:
1015:
1011:
988:
964:
960:
937:
933:
929:
926:
923:
918:
914:
910:
905:
901:
897:
894:
874:
854:
832:
823:
819:
801:
797:
787:
771:
749:
724:
721:
718:
714:
710:
705:
701:
695:
691:
683:
682:
681:
680:, such that
679:
660:
657:
652:
648:
644:
639:
635:
631:
626:
622:
618:
613:
609:
598:
595:
592:
588:
584:
581:
574:
573:
572:
571:
567:
562:
554:
552:
548:
544:
543:associativity
539:
511:
507:
503:
502:graded module
498:
496:
492:
488:
484:
464:
461:
458:
454:
450:
445:
441:
435:
431:
406:
402:
394:
390:
386:
382:
378:
374:
362:
357:
355:
350:
348:
343:
342:
340:
339:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
293:
290:
289:
285:
280:
279:
272:
271:
267:
266:
262:
259:
257:
254:
252:
249:
248:
243:
238:
237:
230:
229:
225:
223:
220:
219:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
181:
176:
175:
170:
169:
162:
159:
157:
156:Division ring
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
113:
108:
103:
102:
97:
96:
89:
86:
84:
81:
79:
78:Abelian group
76:
75:
71:
68:
66:
63:
61:
57:
54:
52:
49:
48:
44:
39:
38:
35:
32:
31:
19:
6086:
6068:
6064:
6038:
6035:Bourbaki, N.
6004:
5963:
5957:
5889:
5881:
5872:
5861:
5857:
5850:
5846:
5840:
5577:
5573:
5569:
5565:
5434:
5429:
5425:
5397:
5394:Novikov ring
5321:
5320:elements of
5317:
5205:
5201:
5194:
5163:
5161:
5156:
4944:
4806:
4757:identity map
4698:
4591:
4587:
4585:
4470:
4467:-graded ring
4446:
4343:
4305:homomorphism
4298:
4253:superalgebra
4246:monoid rings
4231:
4219:
4214:
4142:
4137:
4134:
4130:
4073:
4072:Elements of
4071:
4006:
3949:
3947:-graded ring
3944:
3943:
3939:
3934:
3928:
3895:
3716:
3708:
3704:
3702:vector space
3661:
3657:
3648:
3643:
3639:
3637:
3573:
3571:
3566:
3502:
3498:
3494:
3492:
3487:
3483:
3482:over a ring
3479:
3474:
3458:
3450:
3442:
3392:
3388:
3322:
3320:
3313:
3308:
3266:
3121:
3117:
3115:
3038:
3034:
2998:
2994:
2992:
2866:
2859:
2796:
2792:
2788:
2780:
2724:homomorphism
2719:
2715:
2680:
2678:
2534:
2530:
2526:
2522:
2494:
2483:
2421:
2359:
2355:
2348:
2342:
2260:
2256:
2248:
2244:
2240:
2236:
2228:
2224:
2220:
2216:
2208:
2201:
2185:
2181:
2104:
2097:
2068:
2064:
2061:localization
2056:
2049:
2042:
1936:
1932:
1928:
1854:
1772:
1614:
1565:
1453:intersection
1346:
1315:
1279:associative
1060:
1031:
821:
817:
788:
741:
675:
563:
560:
540:
509:
501:
499:
494:
490:
380:
370:
330:Hopf algebra
319:
268:
261:Vector space
226:
166:
95:Group theory
93:
58: /
6121:Ring theory
6001:Lang, Serge
5886:free monoid
5877:, given an
5869:Free monoid
5568:indexed by
5324:, and only
5166:is at most
4493:such that:
4131:homogeneous
3453:called the
2516:annihilator
2332:cup product
2200:defined by
2194:normal cone
2059:. Then the
1564:called the
1347:homogeneous
952:where each
818:homogeneous
381:graded ring
373:mathematics
315:Lie algebra
300:Associative
204:Total order
194:Semilattice
168:Ring theory
6110:Categories
5993:References
5982:1188.68177
5855:such that
5392:See also:
5018:such that
4469:is a ring
4242:group ring
4232:Examples:
4226:semigroups
4007:such that
3569:-modules.
3469:See also:
3449:for large
3267:(assuming
2907:-twist of
2495:Examples:
2484:for every
2362:such that
2198:subvariety
2196:along the
2041:of degree
1568:of degree
570:direct sum
538:-algebra.
421:such that
6077:0010-437X
6039:Algebra I
5949:Citations
5804:×
5775:⋅
5764:∈
5749:∑
5725:∈
5643:∈
5620:⟩
5617:⟩
5611:⟨
5608:⟨
5602:∈
5552:⟩
5549:⟩
5543:⟨
5540:⟨
5514:ϕ
5508:⋅
5466:×
5362:−
5351:−
5298:−
5287:−
5140:≠
5128:ϕ
5067:ϕ
5052:ϕ
5035:⋅
5026:ϕ
4994:→
4985:ϕ
4962:⋅
4906:∈
4899:⋃
4827:∈
4820:⨁
4743:ε
4720:ε
4660:→
4652::
4649:ε
4626:ε
4554:
4545:ε
4536:
4527:ε
4516:−
4481:Γ
4455:Γ
4415:→
4412:Γ
4409::
4406:ε
4386:Γ
4363:ε
4357:Γ
4220:Remarks:
4114:∈
4103:for some
4049:⋅
4038:⊆
3977:∈
3970:⨁
3843:∙
3807:∙
3773:∙
3769:⋀
3715:of order
3683:∙
3607:⊆
3516:⊆
3419:
3406:↦
3357:…
3275:ℓ
3226:ℓ
3223:∑
3155:∈
3068:⊆
3018:→
3012::
2968:ℓ
2941:ℓ
2895:ℓ
2825:∩
2785:inclusion
2755:⊆
2699:→
2630:∞
2621:⨁
2561:∞
2546:⨁
2453:⊆
2387:∈
2380:⨁
2291:∞
2276:⨁
2131:∞
2116:⨁
1978:…
1728:∞
1713:⨁
1535:submodule
1362:∈
1330:⊆
1255:-modules.
927:⋯
711:⊆
661:⋯
658:⊕
645:⊕
632:⊕
604:∞
589:⨁
491:gradation
451:⊆
325:Bialgebra
131:Near-ring
88:Lie group
56:Semigroup
6116:Algebras
6003:(2002),
5896:See also
5879:alphabet
5817:′
5692:′
5598:′
5489:semiring
5077:′
5042:′
4598:Examples
3791:and the
3711:are the
3321:Suppose
2853:and the
2681:morphism
2533:-module
1935:on
1310:-algebra
1142:For any
1034:of
1030:are the
483:integers
161:Lie ring
126:Semiring
6027:1878556
6005:Algebra
5382:divisor
4796:
4761:
4755:is the
3890:
3863:
3857:in any
3751:
3721:
3713:tensors
3385:
3330:
3189:
3129:
2847:
2801:
2777:
2728:
2722:, is a
2529:and an
2251:), the
2239:. Then
2192:of the
2107:, then
1892:
1859:
1842:
1822:
1659:, then
1657:
1637:
1610:
1590:
1531:
1504:
1451:.) The
1445:
1425:
1421:
1401:
1377:
1351:
1308:
1281:
1253:
1226:
1218:
1191:
1117:
1097:
1093:subring
1056:
1036:
1001:
981:
845:
825:
784:
764:
536:
514:
495:grading
479:
423:
292:Algebra
284:Algebra
189:Lattice
180:Lattice
6094:
6075:
6045:
6025:
6015:
5980:
5970:
5884:, the
5193:where
4809:monoid
4641:where
4378:where
3937:monoid
3908:, and
2887:, the
2862:center
2851:kernel
2849:. The
2353:module
2227:) the
2184:along
2008:degree
1894:, and
1773:where
1502:is an
1447:; see
1277:is an
1222:module
822:degree
487:monoid
320:Graded
251:Module
242:Module
141:Domain
60:Monoid
6061:(PDF)
5944:Notes
5260:) or
5234:(for
5155:when
4238:group
4135:grade
3920:(cf.
3700:of a
3486:is a
3325:is a
3096:. An
2982:(cf.
2855:image
2505:field
2265:group
2211:be a
2071:is a
1475:with
1318:ideal
1091:is a
391:is a
383:is a
286:-like
244:-like
182:-like
151:Field
109:-like
83:Magma
51:Group
45:-like
43:Group
6092:ISBN
6073:ISSN
6043:ISBN
6013:ISBN
5968:ISBN
5491:and
4590:and
3916:and
3900:and
3827:The
3756:The
3667:The
3565:are
2997:and
2993:Let
2779:. A
2488:and
2422:and
2207:Let
2048:Let
1942:The
1927:for
762:and
676:of
566:ring
541:The
385:ring
379:, a
116:Ring
107:Ring
5978:Zbl
5873:In
5658:to
5576:to
4603:An
4551:deg
4533:deg
4133:of
3924:.)
3475:An
3457:of
3410:dim
3307:of
3104:in
3100:of
3041:if
2990:).
2986:in
2718:or
2267:is
2255:of
2231:th
2180:of
2096:If
2063:of
1820:of
1615:If
1588:of
1537:of
1345:is
1316:An
1095:of
867:of
820:of
493:or
371:In
121:Rng
6112::
6069:34
6067:.
6063:.
6041:.
6023:MR
6021:,
6007:,
5976:.
5865:.
5860:=
5858:pq
5849:,
5432:.
4942:.
4693:A
4594:.
4236:A
4217:.
4140:.
3904:,
3719:,
3646:.
3461:.
3387:,
3311:.
3191::
2679:A
2499:A
2492:.
2247:;
2223:;
2215:,
1844:.
1162:,
1058:.
786:.
553:.
500:A
497:.
6100:.
6079:.
6051:.
6030:.
5984:.
5890:A
5882:A
5862:m
5853:)
5851:q
5847:p
5845:(
5841:m
5827:)
5824:q
5821:(
5814:s
5808:K
5800:)
5797:p
5794:(
5791:s
5784:m
5781:=
5778:q
5772:p
5767:R
5761:q
5758:,
5755:p
5728:R
5722:m
5702:)
5699:m
5696:(
5689:s
5683:K
5679:+
5675:)
5672:m
5669:(
5666:s
5646:R
5640:m
5614:R
5605:K
5595:s
5591:,
5588:s
5578:K
5574:R
5570:R
5566:K
5546:R
5537:K
5517:)
5511:,
5505:,
5502:R
5499:(
5475:)
5470:K
5462:,
5457:K
5453:+
5449:,
5446:K
5443:(
5430:n
5426:n
5411:N
5365:1
5359:g
5354:1
5346:1
5343:+
5340:n
5336:g
5322:G
5318:n
5301:1
5295:g
5290:1
5282:1
5279:+
5276:n
5272:g
5248:1
5245:=
5242:g
5222:1
5219:+
5216:n
5206:n
5202:G
5195:g
5179:n
5175:g
5164:n
5157:m
5143:0
5137:)
5134:m
5131:(
5106:M
5102:1
5081:)
5074:m
5070:(
5064:+
5061:)
5058:m
5055:(
5049:=
5046:)
5039:m
5032:m
5029:(
5004:0
4999:N
4991:M
4988::
4965:)
4959:,
4956:M
4953:(
4928:n
4924:R
4916:0
4911:N
4903:n
4876:n
4872:R
4849:n
4845:R
4837:0
4832:N
4824:n
4798:.
4783:Z
4779:2
4775:/
4770:Z
4723:)
4717:,
4713:Z
4709:(
4677:Z
4673:2
4669:/
4664:Z
4656:Z
4629:)
4623:,
4619:Z
4615:(
4592:y
4588:x
4571:,
4568:x
4565:y
4560:)
4557:y
4548:(
4542:)
4539:x
4530:(
4523:)
4519:1
4513:(
4510:=
4507:y
4504:x
4471:A
4432:Z
4428:2
4424:/
4419:Z
4366:)
4360:,
4354:(
4329:Z
4325:2
4321:/
4316:Z
4271:2
4266:Z
4215:G
4200:N
4174:N
4152:N
4138:i
4117:G
4111:i
4089:i
4085:R
4074:R
4057:.
4052:j
4046:i
4042:R
4033:j
4029:R
4023:i
4019:R
3990:i
3986:R
3980:G
3974:i
3966:=
3963:R
3950:R
3945:G
3940:G
3929:G
3892:.
3876:n
3872:H
3839:H
3812:V
3803:S
3778:V
3753:.
3739:V
3734:n
3730:T
3717:n
3709:n
3705:V
3688:V
3679:T
3664:.
3662:n
3658:n
3644:R
3640:A
3621:j
3618:+
3615:i
3611:A
3602:j
3598:A
3592:i
3588:R
3574:R
3567:R
3551:i
3547:A
3524:0
3520:A
3513:R
3503:R
3499:R
3495:R
3484:R
3480:A
3459:M
3451:n
3443:M
3427:n
3423:M
3414:k
3403:n
3393:M
3389:k
3373:]
3368:n
3364:x
3360:,
3354:,
3349:0
3345:x
3341:[
3338:k
3323:R
3309:M
3291:)
3286:n
3282:M
3278:(
3250:n
3246:t
3242:)
3237:n
3233:M
3229:(
3220:=
3217:)
3214:t
3211:,
3208:M
3205:(
3202:P
3177:]
3173:]
3170:t
3167:[
3163:[
3159:Z
3152:)
3149:t
3146:,
3143:M
3140:(
3137:P
3122:R
3118:M
3082:d
3079:+
3076:n
3072:N
3065:)
3060:n
3056:M
3052:(
3049:f
3039:d
3035:f
3021:N
3015:M
3009:f
2999:N
2995:M
2965:+
2962:n
2958:M
2954:=
2949:n
2945:)
2938:(
2935:M
2915:M
2875:M
2833:i
2829:M
2822:N
2819:=
2814:i
2810:N
2797:M
2793:M
2789:N
2763:i
2759:M
2752:)
2747:i
2743:N
2739:(
2736:f
2702:M
2696:N
2693::
2690:f
2675:.
2661:1
2658:+
2655:n
2651:I
2646:/
2640:n
2636:I
2625:0
2600:M
2595:1
2592:+
2589:n
2585:I
2580:/
2576:M
2571:n
2567:I
2556:0
2553:=
2550:n
2535:M
2531:R
2527:R
2523:I
2490:j
2486:i
2467:j
2464:+
2461:i
2457:M
2448:j
2444:M
2438:i
2434:R
2407:,
2402:i
2398:M
2391:N
2384:i
2376:=
2373:M
2360:R
2356:M
2334:.
2318:)
2315:R
2312:;
2309:X
2306:(
2301:i
2297:H
2286:0
2283:=
2280:i
2261:R
2257:X
2249:R
2245:X
2243:(
2241:H
2237:R
2229:i
2225:R
2221:X
2219:(
2217:H
2209:X
2204:.
2202:I
2186:I
2182:R
2162:1
2159:+
2156:n
2152:I
2147:/
2141:n
2137:I
2126:0
2123:=
2120:n
2105:R
2098:I
2080:Z
2069:S
2065:R
2057:R
2050:S
2045:.
2043:i
2023:i
2019:R
1994:]
1989:n
1985:t
1981:,
1975:,
1970:1
1966:t
1962:[
1959:k
1956:=
1953:R
1939:.
1937:R
1929:i
1915:0
1912:=
1907:i
1903:R
1880:R
1877:=
1872:0
1868:R
1855:R
1830:I
1808:n
1786:n
1782:I
1758:,
1753:n
1749:I
1744:/
1738:n
1734:R
1723:0
1720:=
1717:n
1709:=
1706:I
1702:/
1698:R
1675:I
1671:/
1667:R
1645:R
1623:I
1598:I
1576:n
1550:n
1546:R
1533:-
1517:0
1513:R
1488:n
1484:R
1463:I
1433:R
1409:I
1387:a
1365:I
1359:a
1333:R
1327:I
1312:.
1294:0
1290:R
1265:R
1239:0
1235:R
1220:-
1204:0
1200:R
1175:n
1171:R
1150:n
1127:1
1105:R
1077:0
1073:R
1044:a
1016:i
1012:a
989:i
965:i
961:a
938:n
934:a
930:+
924:+
919:1
915:a
911:+
906:0
902:a
898:=
895:a
875:R
855:a
833:n
802:n
798:R
772:n
750:m
725:n
722:+
719:m
715:R
706:n
702:R
696:m
692:R
653:2
649:R
640:1
636:R
627:0
623:R
619:=
614:n
610:R
599:0
596:=
593:n
585:=
582:R
523:Z
465:j
462:+
459:i
455:R
446:j
442:R
436:i
432:R
407:i
403:R
360:e
353:t
346:v
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.