671:
4692:
5841:
4447:
4585:
5634:
5095:
3793:
1768:
2615:
2417:
4944:
4865:
2174:
5566:
2678:
4071:
2328:
3266:
5020:
4737:
4643:
3637:
2479:
950:
737:
477:
4798:
4344:
4006:
3191:
3098:
2984:
3443:
2779:
4380:
5489:
5382:
5318:
5716:
2849:
5531:
4287:
3540:
5157:
3035:
3826:
3702:
5426:
4757:
4215:
4189:
4167:
3859:
3305:
2091:
1343:
534:
4979:
2716:
4495:
4469:
4400:
1925:
1890:
5742:
5660:
4131:
3753:
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.
5195:
5122:
4892:
4105:
3892:
3567:
2909:
2035:
1798:
1562:
1529:
1500:
1306:
1251:
1216:
1187:
1089:
1028:
977:
814:
419:
577:
5262:
5236:
2004:
1685:
3387:
2929:
2889:
1840:
1818:
1655:
1633:
1608:
1586:
1473:
1443:
1419:
1397:
1275:
1160:
1137:
1115:
1054:
999:
885:
865:
843:
782:
760:
4648:
5384:
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:
4405:
5747:
4503:
3766:
5587:
1693:
2545:
6096:
6091:. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press.
6047:
6017:
5972:
2368:
4897:
5025:
4818:
4195:
under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set
2110:
5536:
4017:
2620:
351:
2270:
3201:
5385:
4984:
4708:
4614:
3586:
2428:
890:
686:
426:
4768:
4314:
3962:
3136:
3048:
3308:
2934:
6009:
3402:
2735:
344:
17:
4353:
545:
is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to
6085:
Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the
Hilbert function and the Samuel function".
5442:
392:
5331:
5267:
3322:. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)
5907:
3925:
213:
2864:
Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the
5938:
2808:
5498:
4263:
3512:
3450:
3319:
2987:
2517:
1452:
5665:
5127:
3008:
5166:
Assuming the gradations of non-identity elements are non-zero, the number of elements of gradation
4698:
3801:
3677:
5409:
4742:
4198:
4172:
4150:
3837:
3274:
2074:
1322:
517:
5428:, the indexing family could be any graded monoid, assuming that the number of elements of degree
546:
304:
4952:
5902:
5878:
3917:
2177:
2060:
2038:
2007:
2689:
4480:
4454:
4385:
3109:
1897:
1862:
666:{\displaystyle R=\bigoplus _{n=0}^{\infty }R_{n}=R_{0}\oplus R_{1}\oplus R_{2}\oplus \cdots }
6058:
5721:
5639:
4110:
3728:
1354:
6121:
6027:
5918:
5173:
5100:
4870:
4083:
3921:
3870:
3545:
2894:
2865:
2352:
2013:
1776:
1540:
1507:
1478:
1284:
1229:
1221:
1194:
1165:
1067:
1006:
955:
792:
397:
291:
283:
255:
250:
241:
198:
140:
5982:
4894:'s, without using the additive part. That is, the set of elements of the graded monoid is
8:
5928:
5241:
5215:
4241:
3909:
3901:
3480:
3129:
3101:
2858:
2727:
2512:: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous
2503:
2499:
1948:
1662:
1317:
1278:
505:
309:
299:
150:
50:
42:
33:
5202:
3913:
3905:
3474:
3458:
3337:
2991:
2914:
2874:
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:
6116:
6092:
6073:
6043:
6013:
5968:
5403:
3862:
3796:
3105:
2212:
2197:
4308:
2787:
is a submodule that is a graded module in own right and such that the set-theoretic
6035:
5978:
5912:
5893:
can be considered as a graded monoid, where the gradation of a word is its length.
4608:
4291:
3761:
2854:
2232:
2101:
376:
135:
4611:
is an example of an anticommutative algebra, graded with respect to the structure
160:
6086:
6023:
5923:
4304:
3832:
3657:
3330:
2252:
2189:
2053:
1943:
227:
221:
208:
188:
179:
145:
82:
4147:
The previously defined notion of "graded ring" now becomes the same thing as an
5933:
4192:
3672:
2513:
677:
388:
269:
6110:
6077:
4687:{\displaystyle \varepsilon \colon \mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} }
3318:
A graded module is said to be finitely generated if the underlying module is
2868:
is the same as to give the structure of a graded algebra to the latter ring.
2788:
2344:
2264:
542:
155:
120:
77:
504:
is defined similarly (see below for the precise definition). It generalizes
5397:
4760:
4260:
4256:
3705:
329:
260:
94:
5967:. Translated by Thomas, Reuben. Cambridge University Press. p. 384.
5886:
4294:. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).
4249:
2331:
2193:
372:
314:
203:
193:
167:
4442:{\displaystyle \varepsilon \colon \Gamma \to \mathbb {Z} /2\mathbb {Z} }
6001:
4245:
569:
69:
3939:
The above definitions have been generalized to rings graded using any
6012:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
5836:{\displaystyle \sum _{p,q\in R \atop p\cdot q=m}s(p)\times _{K}s'(q)}
4580:{\displaystyle xy=(-1)^{\varepsilon (\deg x)\varepsilon (\deg y)}yx,}
4229:
1534:
324:
130:
87:
55:
5492:
125:
1612:. A homogeneous ideal is the direct sum of its homogeneous parts.
5843:. This sum is correctly defined (i.e., finite) because, for each
1092:
482:
4812:
3940:
3716:
3505:
is a field), it is given the trivial grading (every element of
486:
59:
5963:
Sakarovitch, Jacques (2009). "Part II: The power of algebra".
3865:
is also graded, being the direct sum of the cohomology groups
3399:
a finitely generated graded module over it. Then the function
5208:
of the monoid. Therefore the number of elements of gradation
4228:
If we do not require that the ring have an identity element,
2791:
is a morphism of graded modules. Explicitly, a graded module
4311:
of the monoid of the gradation into the additive monoid of
3788:{\displaystyle \textstyle \bigwedge \nolimits ^{\bullet }V}
2052:
be the set of all nonzero homogeneous elements in a graded
5629:{\displaystyle s,s'\in K\langle \langle R\rangle \rangle }
5568:
denotes the semiring of power series with coefficients in
5391:
5320:
else. Indeed, each such element is the product of at most
1763:{\displaystyle R/I=\bigoplus _{n=0}^{\infty }R_{n}/I_{n},}
2610:{\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
5124:
is necessarily 0. Some authors request furthermore that
6059:"Intersection form for quasi-homogeneous singularities"
3653:
Examples of graded algebras are common in mathematics:
2861:
of a morphism of graded modules are graded submodules.
2730:
of the underlying modules that respects grading; i.e.,
847:. By definition of a direct sum, every nonzero element
5334:
5270:
4821:
3770:
2623:
2412:{\displaystyle M=\bigoplus _{i\in \mathbb {N} }M_{i},}
2273:
2113:
1224:, and the direct sum decomposition is a direct sum of
5750:
5724:
5668:
5642:
5590:
5539:
5501:
5445:
5412:
5244:
5218:
5176:
5130:
5103:
5028:
4987:
4955:
4939:{\displaystyle \bigcup _{n\in \mathbb {N} _{0}}R_{n}}
4900:
4873:
4771:
4745:
4711:
4651:
4617:
4506:
4483:
4457:
4408:
4388:
4356:
4317:
4266:
4201:
4175:
4153:
4113:
4086:
4020:
3965:
3873:
3840:
3804:
3769:
3731:
3680:
3589:
3548:
3515:
3405:
3340:
3277:
3204:
3139:
3051:
3011:
2937:
2917:
2897:
2877:
2811:
2738:
2692:
2548:
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:
4860:{\textstyle \bigoplus _{n\in \mathbb {N} _{0}}R_{n}}
4303:
Some graded rings (or algebras) are endowed with an
2169:{\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}}
5090:{\displaystyle \phi (m\cdot m')=\phi (m)+\phi (m')}
2617:is a graded module over the associated graded ring
5835:
5736:
5710:
5654:
5628:
5561:{\displaystyle K\langle \langle R\rangle \rangle }
5560:
5525:
5483:
5420:
5376:
5312:
5256:
5230:
5189:
5151:
5116:
5089:
5014:
4973:
4938:
4886:
4859:
4792:
4751:
4731:
4686:
4637:
4579:
4489:
4463:
4441:
4394:
4374:
4338:
4281:
4209:
4183:
4161:
4125:
4099:
4065:
4000:
3886:
3853:
3820:
3787:
3747:
3696:
3664:are exactly the homogeneous polynomials of degree
3631:
3561:
3534:
3437:
3381:
3299:
3260:
3185:
3112:is an example of such a morphism having degree 1.
3092:
3029:
2978:
2923:
2903:
2883:
2843:
2773:
2710:
2673:{\textstyle \bigoplus _{0}^{\infty }I^{n}/I^{n+1}}
2672:
2609:
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:
5636:is defined pointwise, it is the function sending
4066:{\displaystyle R_{i}R_{j}\subseteq R_{i\cdot j}.}
3179:
3169:
2323:{\textstyle \bigoplus _{i=0}^{\infty }H^{i}(X;R)}
512:. A graded ring could also be viewed as a graded
6108:
4346:, the field with two elements. Specifically, a
3916:. One example is the close relationship between
1423:. (Equivalently, if it is a graded submodule of
5402:These notions allow us to extend the notion of
3580:is also a graded ring, then one requires that
3115:
2330:with the multiplicative structure given by the
3931:
3261:{\displaystyle P(M,t)=\sum \ell (M_{n})t^{n}}
1119:; in particular, the multiplicative identity
352:
5623:
5620:
5614:
5611:
5555:
5552:
5546:
5543:
5962:
5015:{\displaystyle \phi :M\to \mathbb {N} _{0}}
4732:{\displaystyle (\mathbb {Z} ,\varepsilon )}
4638:{\displaystyle (\mathbb {Z} ,\varepsilon )}
4449:is a homomorphism of additive monoids. An
3632:{\displaystyle R_{i}A_{j}\subseteq A_{i+j}}
2520:of a graded module is a homogeneous ideal.
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}}
6056:
5847:, there are only a finite number of pairs
5718:, and the product is the function sending
4793:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
4705:) is the same thing as an anticommutative
4339:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
4001:{\displaystyle R=\bigoplus _{i\in G}R_{i}}
3956:is a ring with a direct sum decomposition
359:
345:
6084:
5414:
5002:
4914:
4835:
4786:
4773:
4716:
4680:
4667:
4659:
4622:
4435:
4422:
4332:
4319:
4269:
4203:
4177:
4155:
3186:{\displaystyle P(M,t)\in \mathbb {Z} \!]}
3162:
3093:{\displaystyle f(M_{n})\subseteq N_{n+d}}
2506:(with the field having trivial grading).
2390:
2079:
522:
6034:
2979:{\displaystyle M(\ell )_{n}=M_{n+\ell }}
2502:is an example of a graded module over a
1139:is a homogeneous element of degree zero.
5406:. Instead of the indexing family being
5392:Power series indexed by a graded monoid
4252:are graded by the corresponding monoid.
3438:{\displaystyle n\mapsto \dim _{k}M_{n}}
2774:{\displaystyle f(N_{i})\subseteq M_{i}}
485:or the set of integers, but can be any
14:
6109:
4949:Formally, a graded monoid is a monoid
4375:{\displaystyle (\Gamma ,\varepsilon )}
5484:{\displaystyle (K,+_{K},\times _{K})}
5377:{\textstyle {\frac {g^{n+1}-1}{g-1}}}
5313:{\textstyle {\frac {g^{n+1}-1}{g-1}}}
3711:. The homogeneous elements of degree
3660:. The homogeneous elements of degree
1687:is also a graded ring, decomposed as
979:is either 0 or homogeneous of degree
6000:
4298:
2799: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
4307:structure. This notion requires a
4244:naturally grades the corresponding
3772:
556:
24:
5756:
5576:. Its elements are functions from
4484:
4458:
4415:
4389:
4360:
4290:-graded algebra. Examples include
3449:. The function coincides with the
3445:is called the Hilbert function of
2634:
2565:
2290:
2130:
1800:is the homogeneous part of degree
1727:
1448:
603:
549:as well; e.g., one can consider a
25:
6133:
4703:skew-commutative associative ring
3900:Graded algebras are much used in
3497:In the usual case where the ring
3468:
2844:{\displaystyle N_{i}=N\cap M_{i}}
1847:
887:can be uniquely written as a sum
5526:{\displaystyle (R,\cdot ,\phi )}
4815:is the subset of a graded ring,
4806:
4282:{\displaystyle \mathbb {Z} _{2}}
3646:to be a graded left module over
3535:{\displaystyle R\subseteq A_{0}}
3501:is not graded (in particular if
2338:
1379:, the homogeneous components of
3124:over a commutative graded ring
3037:is a morphism of modules, then
5956:
5830:
5824:
5803:
5797:
5711:{\displaystyle s(m)+_{K}s'(m)}
5705:
5699:
5678:
5672:
5520:
5502:
5478:
5446:
5152:{\displaystyle \phi (m)\neq 0}
5140:
5134:
5084:
5073:
5064:
5058:
5049:
5032:
4997:
4968:
4956:
4726:
4712:
4663:
4632:
4618:
4563:
4551:
4545:
4533:
4526:
4516:
4418:
4369:
4357:
3409:
3376:
3344:
3307:are finite.) It is called the
3294:
3281:
3245:
3232:
3220:
3208:
3180:
3176:
3170:
3166:
3155:
3143:
3068:
3055:
3030:{\displaystyle f\colon M\to N}
3021:
2948:
2941:
2931:is a graded module defined by
2755:
2742:
2702:
2516:it is a graded submodule. The
2317:
2305:
1993:
1961:
13:
1:
6010:Graduate Texts in Mathematics
5993:
5097:. Note that the gradation of
4763:of the additive structure of
4590:for all homogeneous elements
3821:{\displaystyle S^{\bullet }V}
3697:{\displaystyle T^{\bullet }V}
742:for all nonnegative integers
5949:
5432:is finite, for each integer
5421:{\displaystyle \mathbb {N} }
4981:, with a gradation function
4752:{\displaystyle \varepsilon }
4210:{\displaystyle \mathbb {N} }
4184:{\displaystyle \mathbb {N} }
4162:{\displaystyle \mathbb {N} }
3854:{\displaystyle H^{\bullet }}
3300:{\displaystyle \ell (M_{n})}
3116:Invariants of graded modules
2718: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:
5965:Elements of automata theory
5908:Differential graded algebra
5896:
4601:
3926:Homogeneous coordinate ring
3642:In other words, we require
3576:In the case where the ring
3494:if it is graded as a ring.
2188:; geometrically, it is the
1061:Some basic properties are:
27:Type of algebraic structure
10:
6138:
5939:Differential graded module
5872:
5584:. The sum of two elements
5395:
4974:{\displaystyle (M,\cdot )}
3935:-graded rings and algebras
3472:
2343:The corresponding idea in
568:that is decomposed into a
6072:(2): 211–223 See p. 211.
6038:(1974). "Ch. 1–3, 3 §3".
5388:in such a graded monoid.
3828:are also graded algebras.
3451:integer-valued polynomial
2795:is a graded submodule of
1931:≠0. This is called the
387:such that the underlying
5944:
5201:is the cardinality of a
4699:supercommutative algebra
3128:, one can associate the
2711:{\displaystyle f:N\to M}
2010:: it is a direct sum of
547:non-associative algebras
6088:Commutative Ring Theory
6057:Steenbrink, J. (1977).
4739:-graded algebra, where
4490:{\displaystyle \Gamma }
4477:graded with respect to
4464:{\displaystyle \Gamma }
4395:{\displaystyle \Gamma }
3918:homogeneous polynomials
3509:is of degree 0). Thus,
3309:Hilbert–Poincaré series
3041:is said to have degree
2039:homogeneous polynomials
1920:{\displaystyle R_{i}=0}
1885:{\displaystyle R_{0}=R}
1455:of a homogeneous ideal
6066:Compositio Mathematica
5903:Associated graded ring
5879:formal language theory
5837:
5738:
5737:{\displaystyle m\in R}
5712:
5656:
5655:{\displaystyle m\in R}
5630:
5562:
5533:a graded monoid. Then
5527:
5485:
5422:
5378:
5314:
5258:
5232:
5191:
5153:
5118:
5091:
5016:
4975:
4940:
4888:
4861:
4811:Intuitively, a graded
4794:
4753:
4733:
4688:
4639:
4581:
4491:
4465:
4443:
4396:
4376:
4340:
4283:
4259:is another term for a
4211:
4185:
4163:
4127:
4126:{\displaystyle i\in G}
4101:
4067:
4002:
3888:
3855:
3822:
3789:
3749:
3748:{\displaystyle T^{n}V}
3698:
3633:
3563:
3542:and the graded pieces
3536:
3439:
3383:
3301:
3262:
3187:
3120:Given a graded module
3094:
3031:
3005:be graded modules. If
2988:Serre's twisting sheaf
2980:
2925:
2905:
2885:
2871:Given a graded module
2845:
2775:
2712:
2674:
2638:
2611:
2569:
2530: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:
5838:
5739:
5713:
5657:
5631:
5563:
5528:
5486:
5423:
5379:
5315:
5259:
5233:
5192:
5190:{\displaystyle g^{n}}
5163:is not the identity.
5154:
5119:
5117:{\displaystyle 1_{M}}
5092:
5017:
4976:
4941:
4889:
4887:{\displaystyle R_{n}}
4862:
4795:
4754:
4734:
4689:
4640:
4582:
4492:
4466:
4444:
4397:
4377:
4341:
4284:
4212:
4186:
4164:
4128:
4102:
4100:{\displaystyle R_{i}}
4068:
4003:
3889:
3887:{\displaystyle H^{n}}
3856:
3823:
3790:
3750:
3699:
3634:
3564:
3562:{\displaystyle A_{i}}
3537:
3440:
3384:
3302:
3263:
3188:
3110:differential geometry
3095:
3032:
2981:
2926:
2906:
2904:{\displaystyle \ell }
2886:
2846:
2776:
2713:
2675:
2624:
2612:
2549:
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}}
5919:Graded (mathematics)
5881:, given an alphabet
5748:
5744:to the infinite sum
5722:
5666:
5640:
5588:
5537:
5499:
5443:
5410:
5332:
5268:
5242:
5216:
5174:
5128:
5101:
5026:
4985:
4953:
4898:
4871:
4819:
4769:
4743:
4709:
4701:(sometimes called a
4694:is the quotient map.
4649:
4615:
4504:
4481:
4455:
4406:
4386:
4354:
4315:
4264:
4232:may replace monoids.
4199:
4173:
4169:-graded ring, where
4151:
4111:
4084:
4018:
3963:
3946:as an index set. A
3922:projective varieties
3871:
3838:
3802:
3767:
3729:
3678:
3587:
3546:
3513:
3403:
3338:
3275:
3202:
3137:
3049:
3009:
2935:
2915:
2895:
2875:
2809:
2736:
2690:
2621:
2546:
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
5929:Graded vector space
5439:More formally, let
5257:{\displaystyle g=1}
5231:{\displaystyle n+1}
5212:or less is at most
4867:, generated by the
4350:consists of a pair
3910:homological algebra
3902:commutative algebra
3481:associative algebra
3130:formal power series
3102:exterior derivative
2724:graded homomorphism
2500: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
5915:, a generalization
5833:
5793:
5734:
5708:
5652:
5626:
5558:
5523:
5481:
5418:
5374:
5310:
5254:
5228:
5187:
5149:
5114:
5087:
5012:
4971:
4936:
4925:
4884:
4857:
4846:
4790:
4749:
4729:
4684:
4635:
4577:
4487:
4461:
4439:
4392:
4372:
4336:
4279:
4207:
4181:
4159:
4123:
4097:
4063:
3998:
3987:
3914:algebraic topology
3906:algebraic geometry
3884:
3851:
3818:
3785:
3784:
3745:
3694:
3629:
3559:
3532:
3475:Graded Lie algebra
3459:Hilbert polynomial
3435:
3379:
3320:finitely generated
3297:
3258:
3183:
3106:differential forms
3090:
3027:
2992:algebraic geometry
2976:
2921:
2901:
2881:
2841:
2771:
2708:
2670:
2607:
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:
6098:978-1-107-71712-1
6049:978-3-540-64243-5
6019:978-0-387-95385-4
5974:978-0-521-84425-3
5791:
5751:
5404:power series ring
5372:
5308:
4901:
4822:
4299:Anticommutativity
4292:Clifford algebras
4255:An (associative)
4191:is the monoid of
3972:
3863:cohomology theory
3797:symmetric algebra
3382:{\displaystyle k}
2924:{\displaystyle M}
2884:{\displaystyle M}
2542:, the direct sum
2526:: Given an ideal
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:
16:(Redirected from
6129:
6102:
6081:
6063:
6053:
6030:
5987:
5986:
5960:
5913:Filtered algebra
5868:
5858:
5842:
5840:
5839:
5834:
5823:
5815:
5814:
5792:
5790:
5773:
5743:
5741:
5740:
5735:
5717:
5715:
5714:
5709:
5698:
5690:
5689:
5661:
5659:
5658:
5653:
5635:
5633:
5632:
5627:
5604:
5567:
5565:
5564:
5559:
5532:
5530:
5529:
5524:
5491:be an arbitrary
5490:
5488:
5487:
5482:
5477:
5476:
5464:
5463:
5427:
5425:
5424:
5419:
5417:
5383:
5381:
5380:
5375:
5373:
5371:
5360:
5353:
5352:
5336:
5319:
5317:
5316:
5311:
5309:
5307:
5296:
5289:
5288:
5272:
5263:
5261:
5260:
5255:
5237:
5235:
5234:
5229:
5196:
5194:
5193:
5188:
5186:
5185:
5158:
5156:
5155:
5150:
5123:
5121:
5120:
5115:
5113:
5112:
5096:
5094:
5093:
5088:
5083:
5048:
5021:
5019:
5018:
5013:
5011:
5010:
5005:
4980:
4978:
4977:
4972:
4945:
4943:
4942:
4937:
4935:
4934:
4924:
4923:
4922:
4917:
4893:
4891:
4890:
4885:
4883:
4882:
4866:
4864:
4863:
4858:
4856:
4855:
4845:
4844:
4843:
4838:
4801:
4799:
4797:
4796:
4791:
4789:
4781:
4776:
4758:
4756:
4755:
4750:
4738:
4736:
4735:
4730:
4719:
4693:
4691:
4690:
4685:
4683:
4675:
4670:
4662:
4644:
4642:
4641:
4636:
4625:
4609:exterior algebra
4586:
4584:
4583:
4578:
4567:
4566:
4496:
4494:
4493:
4488:
4470:
4468:
4467:
4462:
4451:anticommutative
4448:
4446:
4445:
4440:
4438:
4430:
4425:
4402:is a monoid and
4401:
4399:
4398:
4393:
4381:
4379:
4378:
4373:
4345:
4343:
4342:
4337:
4335:
4327:
4322:
4288:
4286:
4285:
4280:
4278:
4277:
4272:
4217:with any monoid
4216:
4214:
4213:
4208:
4206:
4190:
4188:
4187:
4182:
4180:
4168:
4166:
4165:
4160:
4158:
4132:
4130:
4129:
4124:
4106:
4104:
4103:
4098:
4096:
4095:
4080:that lie inside
4072:
4070:
4069:
4064:
4059:
4058:
4040:
4039:
4030:
4029:
4007:
4005:
4004:
3999:
3997:
3996:
3986:
3895:
3893:
3891:
3890:
3885:
3883:
3882:
3860:
3858:
3857:
3852:
3850:
3849:
3827:
3825:
3824:
3819:
3814:
3813:
3794:
3792:
3791:
3786:
3780:
3779:
3762:exterior algebra
3756:
3754:
3752:
3751:
3746:
3741:
3740:
3703:
3701:
3700:
3695:
3690:
3689:
3658:Polynomial rings
3638:
3636:
3635:
3630:
3628:
3627:
3609:
3608:
3599:
3598:
3568:
3566:
3565:
3560:
3558:
3557:
3541:
3539:
3538:
3533:
3531:
3530:
3444:
3442:
3441:
3436:
3434:
3433:
3421:
3420:
3390:
3388:
3386:
3385:
3380:
3375:
3374:
3356:
3355:
3306:
3304:
3303:
3298:
3293:
3292:
3267:
3265:
3264:
3259:
3257:
3256:
3244:
3243:
3194:
3192:
3190:
3189:
3184:
3165:
3099:
3097:
3096:
3091:
3089:
3088:
3067:
3066:
3036:
3034:
3033:
3028:
2985:
2983:
2982:
2977:
2975:
2974:
2956:
2955:
2930:
2928:
2927:
2922:
2910:
2908:
2907:
2902:
2890:
2888:
2887:
2882:
2852:
2850:
2848:
2847:
2842:
2840:
2839:
2821:
2820:
2785:graded submodule
2782:
2780:
2778:
2777:
2772:
2770:
2769:
2754:
2753:
2717:
2715:
2714:
2709:
2679:
2677:
2676:
2671:
2669:
2668:
2653:
2648:
2647:
2637:
2632:
2616:
2614:
2613:
2608:
2603:
2602:
2587:
2579:
2578:
2568:
2563:
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:
6137:
6136:
6132:
6131:
6130:
6128:
6127:
6126:
6107:
6106:
6105:
6099:
6061:
6050:
6020:
5996:
5991:
5990:
5975:
5961:
5957:
5952:
5947:
5924:Graded category
5899:
5875:
5860:
5848:
5816:
5810:
5806:
5774:
5757:
5755:
5749:
5746:
5745:
5723:
5720:
5719:
5691:
5685:
5681:
5667:
5664:
5663:
5641:
5638:
5637:
5597:
5589:
5586:
5585:
5538:
5535:
5534:
5500:
5497:
5496:
5472:
5468:
5459:
5455:
5444:
5441:
5440:
5413:
5411:
5408:
5407:
5400:
5394:
5361:
5342:
5338:
5337:
5335:
5333:
5330:
5329:
5297:
5278:
5274:
5273:
5271:
5269:
5266:
5265:
5243:
5240:
5239:
5217:
5214:
5213:
5181:
5177:
5175:
5172:
5171:
5129:
5126:
5125:
5108:
5104:
5102:
5099:
5098:
5076:
5041:
5027:
5024:
5023:
5006:
5001:
5000:
4986:
4983:
4982:
4954:
4951:
4950:
4930:
4926:
4918:
4913:
4912:
4905:
4899:
4896:
4895:
4878:
4874:
4872:
4869:
4868:
4851:
4847:
4839:
4834:
4833:
4826:
4820:
4817:
4816:
4809:
4785:
4777:
4772:
4770:
4767:
4766:
4764:
4744:
4741:
4740:
4715:
4710:
4707:
4706:
4679:
4671:
4666:
4658:
4650:
4647:
4646:
4621:
4616:
4613:
4612:
4604:
4529:
4525:
4505:
4502:
4501:
4482:
4479:
4478:
4456:
4453:
4452:
4434:
4426:
4421:
4407:
4404:
4403:
4387:
4384:
4383:
4355:
4352:
4351:
4331:
4323:
4318:
4316:
4313:
4312:
4305:anticommutative
4301:
4273:
4268:
4267:
4265:
4262:
4261:
4202:
4200:
4197:
4196:
4193:natural numbers
4176:
4174:
4171:
4170:
4154:
4152:
4149:
4148:
4133:are said to be
4112:
4109:
4108:
4091:
4087:
4085:
4082:
4081:
4048:
4044:
4035:
4031:
4025:
4021:
4019:
4016:
4015:
3992:
3988:
3976:
3964:
3961:
3960:
3937:
3878:
3874:
3872:
3869:
3868:
3866:
3845:
3841:
3839:
3836:
3835:
3833:cohomology ring
3809:
3805:
3803:
3800:
3799:
3775:
3771:
3768:
3765:
3764:
3736:
3732:
3730:
3727:
3726:
3724:
3685:
3681:
3679:
3676:
3675:
3617:
3613:
3604:
3600:
3594:
3590:
3588:
3585:
3584:
3553:
3549:
3547:
3544:
3543:
3526:
3522:
3514:
3511:
3510:
3477:
3471:
3429:
3425:
3416:
3412:
3404:
3401:
3400:
3370:
3366:
3351:
3347:
3339:
3336:
3335:
3333:
3331:polynomial ring
3288:
3284:
3276:
3273:
3272:
3252:
3248:
3239:
3235:
3203:
3200:
3199:
3161:
3138:
3135:
3134:
3132:
3118:
3078:
3074:
3062:
3058:
3050:
3047:
3046:
3010:
3007:
3006:
2964:
2960:
2951:
2947:
2936:
2933:
2932:
2916:
2913:
2912:
2896:
2893:
2892:
2876:
2873:
2872:
2835:
2831:
2816:
2812:
2810:
2807:
2806:
2804:
2765:
2761:
2749:
2745:
2737:
2734:
2733:
2731:
2720:graded morphism
2691:
2688:
2687:
2658:
2654:
2649:
2643:
2639:
2633:
2628:
2622:
2619:
2618:
2592:
2588:
2583:
2574:
2570:
2564:
2553:
2547:
2544:
2543:
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:
6135:
6125:
6124:
6119:
6104:
6103:
6097:
6082:
6054:
6048:
6032:
6018:
5997:
5995:
5992:
5989:
5988:
5973:
5954:
5953:
5951:
5948:
5946:
5943:
5942:
5941:
5936:
5934:Tensor algebra
5931:
5926:
5921:
5916:
5910:
5905:
5898:
5895:
5889:of words over
5874:
5871:
5832:
5829:
5826:
5822:
5819:
5813:
5809:
5805:
5802:
5799:
5796:
5789:
5786:
5783:
5780:
5777:
5772:
5769:
5766:
5763:
5760:
5754:
5733:
5730:
5727:
5707:
5704:
5701:
5697:
5694:
5688:
5684:
5680:
5677:
5674:
5671:
5651:
5648:
5645:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5603:
5600:
5596:
5593:
5557:
5554:
5551:
5548:
5545:
5542:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5480:
5475:
5471:
5467:
5462:
5458:
5454:
5451:
5448:
5416:
5393:
5390:
5370:
5367:
5364:
5359:
5356:
5351:
5348:
5345:
5341:
5306:
5303:
5300:
5295:
5292:
5287:
5284:
5281:
5277:
5253:
5250:
5247:
5227:
5224:
5221:
5203:generating set
5184:
5180:
5148:
5145:
5142:
5139:
5136:
5133:
5111:
5107:
5086:
5082:
5079:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5047:
5044:
5040:
5037:
5034:
5031:
5009:
5004:
4999:
4996:
4993:
4990:
4970:
4967:
4964:
4961:
4958:
4933:
4929:
4921:
4916:
4911:
4908:
4904:
4881:
4877:
4854:
4850:
4842:
4837:
4832:
4829:
4825:
4808:
4805:
4804:
4803:
4788:
4784:
4780:
4775:
4748:
4728:
4725:
4722:
4718:
4714:
4695:
4682:
4678:
4674:
4669:
4665:
4661:
4657:
4654:
4634:
4631:
4628:
4624:
4620:
4603:
4600:
4588:
4587:
4576:
4573:
4570:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4528:
4524:
4521:
4518:
4515:
4512:
4509:
4486:
4460:
4437:
4433:
4429:
4424:
4420:
4417:
4414:
4411:
4391:
4371:
4368:
4365:
4362:
4359:
4334:
4330:
4326:
4321:
4300:
4297:
4296:
4295:
4276:
4271:
4253:
4234:
4233:
4205:
4179:
4157:
4122:
4119:
4116:
4094:
4090:
4074:
4073:
4062:
4057:
4054:
4051:
4047:
4043:
4038:
4034:
4028:
4024:
4009:
4008:
3995:
3991:
3985:
3982:
3979:
3975:
3971:
3968:
3936:
3930:
3898:
3897:
3881:
3877:
3848:
3844:
3829:
3817:
3812:
3808:
3783:
3778:
3774:
3758:
3744:
3739:
3735:
3693:
3688:
3684:
3673:tensor algebra
3669:
3640:
3639:
3626:
3623:
3620:
3616:
3612:
3607:
3603:
3597:
3593:
3556:
3552:
3529:
3525:
3521:
3518:
3492:graded algebra
3470:
3469:Graded algebra
3467:
3432:
3428:
3424:
3419:
3415:
3411:
3408:
3378:
3373:
3369:
3365:
3362:
3359:
3354:
3350:
3346:
3343:
3296:
3291:
3287:
3283:
3280:
3269:
3268:
3255:
3251:
3247:
3242:
3238:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3182:
3178:
3175:
3172:
3168:
3164:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3117:
3114:
3087:
3084:
3081:
3077:
3073:
3070:
3065:
3061:
3057:
3054:
3026:
3023:
3020:
3017:
3014:
2973:
2970:
2967:
2963:
2959:
2954:
2950:
2946:
2943:
2940:
2920:
2900:
2880:
2838:
2834:
2830:
2827:
2824:
2819:
2815:
2803:and satisfies
2768:
2764:
2760:
2757:
2752:
2748:
2744:
2741:
2707:
2704:
2701:
2698:
2695:
2667:
2664:
2661:
2657:
2652:
2646:
2642:
2636:
2631:
2627:
2606:
2601:
2598:
2595:
2591:
2586:
2582:
2577:
2573:
2567:
2562:
2559:
2556:
2552:
2514:if and only if
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:
6134:
6123:
6120:
6118:
6115:
6114:
6112:
6100:
6094:
6090:
6089:
6083:
6079:
6075:
6071:
6067:
6060:
6055:
6051:
6045:
6041:
6037:
6033:
6029:
6025:
6021:
6015:
6011:
6007:
6003:
5999:
5998:
5984:
5980:
5976:
5970:
5966:
5959:
5955:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5914:
5911:
5909:
5906:
5904:
5901:
5900:
5894:
5892:
5888:
5884:
5880:
5870:
5867:
5863:
5856:
5852:
5846:
5827:
5820:
5817:
5811:
5807:
5800:
5794:
5787:
5784:
5781:
5778:
5775:
5770:
5767:
5764:
5761:
5758:
5752:
5731:
5728:
5725:
5702:
5695:
5692:
5686:
5682:
5675:
5669:
5649:
5646:
5643:
5617:
5608:
5605:
5601:
5598:
5594:
5591:
5583:
5579:
5575:
5571:
5549:
5540:
5517:
5514:
5511:
5508:
5505:
5494:
5473:
5469:
5465:
5460:
5456:
5452:
5449:
5437:
5435:
5431:
5405:
5399:
5389:
5387:
5368:
5365:
5362:
5357:
5354:
5349:
5346:
5343:
5339:
5327:
5323:
5304:
5301:
5298:
5293:
5290:
5285:
5282:
5279:
5275:
5251:
5248:
5245:
5225:
5222:
5219:
5211:
5207:
5204:
5200:
5182:
5178:
5169:
5164:
5162:
5146:
5143:
5137:
5131:
5109:
5105:
5080:
5077:
5070:
5067:
5061:
5055:
5052:
5045:
5042:
5038:
5035:
5029:
5007:
4994:
4991:
4988:
4965:
4962:
4959:
4947:
4931:
4927:
4919:
4909:
4906:
4902:
4879:
4875:
4852:
4848:
4840:
4830:
4827:
4823:
4814:
4807:Graded monoid
4782:
4778:
4762:
4746:
4723:
4720:
4704:
4700:
4696:
4676:
4672:
4655:
4652:
4629:
4626:
4610:
4606:
4605:
4599:
4597:
4593:
4574:
4571:
4568:
4560:
4557:
4554:
4548:
4542:
4539:
4536:
4530:
4522:
4519:
4513:
4510:
4507:
4500:
4499:
4498:
4476:
4472:
4431:
4427:
4412:
4409:
4366:
4363:
4349:
4348:signed monoid
4328:
4324:
4310:
4306:
4293:
4289:
4274:
4258:
4254:
4251:
4248:; similarly,
4247:
4243:
4239:
4238:
4237:
4231:
4227:
4226:
4225:
4222:
4220:
4194:
4145:
4143:
4140:
4136:
4120:
4117:
4114:
4092:
4088:
4079:
4060:
4055:
4052:
4049:
4045:
4041:
4036:
4032:
4026:
4022:
4014:
4013:
4012:
3993:
3989:
3983:
3980:
3977:
3973:
3969:
3966:
3959:
3958:
3957:
3955:
3952:
3950:
3945:
3942:
3934:
3929:
3927:
3923:
3919:
3915:
3911:
3907:
3903:
3879:
3875:
3864:
3846:
3842:
3834:
3830:
3815:
3810:
3806:
3798:
3781:
3776:
3763:
3759:
3742:
3737:
3733:
3722:
3718:
3714:
3710:
3707:
3691:
3686:
3682:
3674:
3670:
3667:
3663:
3659:
3656:
3655:
3654:
3651:
3649:
3645:
3624:
3621:
3618:
3614:
3610:
3605:
3601:
3595:
3591:
3583:
3582:
3581:
3579:
3574:
3572:
3554:
3550:
3527:
3523:
3519:
3516:
3508:
3504:
3500:
3495:
3493:
3489:
3485:
3482:
3476:
3466:
3464:
3460:
3456:
3452:
3448:
3430:
3426:
3422:
3417:
3413:
3406:
3398:
3395:a field, and
3394:
3371:
3367:
3363:
3360:
3357:
3352:
3348:
3341:
3332:
3328:
3323:
3321:
3316:
3314:
3310:
3289:
3285:
3278:
3253:
3249:
3240:
3236:
3229:
3226:
3223:
3217:
3214:
3211:
3205:
3198:
3197:
3196:
3173:
3158:
3152:
3149:
3146:
3140:
3131:
3127:
3123:
3113:
3111:
3107:
3103:
3085:
3082:
3079:
3075:
3071:
3063:
3059:
3052:
3044:
3040:
3024:
3018:
3015:
3012:
3004:
3000:
2995:
2993:
2989:
2971:
2968:
2965:
2961:
2957:
2952:
2944:
2938:
2918:
2898:
2878:
2869:
2867:
2862:
2860:
2856:
2836:
2832:
2828:
2825:
2822:
2817:
2813:
2802:
2798:
2794:
2790:
2786:
2766:
2762:
2758:
2750:
2746:
2739:
2729:
2725:
2721:
2705:
2699:
2696:
2693:
2686:
2681:
2665:
2662:
2659:
2655:
2650:
2644:
2640:
2629:
2625:
2604:
2599:
2596:
2593:
2589:
2584:
2580:
2575:
2571:
2560:
2557:
2554:
2550:
2541:
2537:
2533:
2529:
2525:
2521:
2519:
2515:
2511:
2507:
2505:
2501:
2497:
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:
18:Graded module
6087:
6069:
6065:
6039:
6036:Bourbaki, N.
6005:
5964:
5958:
5890:
5882:
5876:
5865:
5861:
5854:
5850:
5844:
5581:
5577:
5573:
5569:
5438:
5433:
5429:
5401:
5398:Novikov ring
5325:
5324:elements of
5321:
5209:
5205:
5198:
5167:
5165:
5160:
4948:
4810:
4761:identity map
4702:
4595:
4591:
4589:
4474:
4471:-graded ring
4450:
4347:
4309:homomorphism
4302:
4257:superalgebra
4250:monoid rings
4235:
4223:
4218:
4146:
4141:
4138:
4134:
4077:
4076:Elements of
4075:
4010:
3953:
3951:-graded ring
3948:
3947:
3943:
3938:
3932:
3899:
3720:
3712:
3708:
3706:vector space
3665:
3661:
3652:
3647:
3643:
3641:
3577:
3575:
3570:
3506:
3502:
3498:
3496:
3491:
3487:
3486:over a ring
3483:
3478:
3462:
3454:
3446:
3396:
3392:
3326:
3324:
3317:
3312:
3270:
3125:
3121:
3119:
3042:
3038:
3002:
2998:
2996:
2870:
2863:
2800:
2796:
2792:
2784:
2728:homomorphism
2723:
2719:
2684:
2682:
2539:
2535:
2531:
2527:
2523:
2522:
2509:
2508:
2495:
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: /
6122:Ring theory
6002:Lang, Serge
5887:free monoid
5572:indexed by
5328:, and only
5170:is at most
4497:such that:
4135:homogeneous
3457:called the
2518: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
6111:Categories
5994:References
5983:1188.68177
5859:such that
5396:See also:
5022:such that
4473:is a ring
4246:group ring
4236:Examples:
4230:semigroups
4011:such that
3573:-modules.
3473:See also:
3453:for large
3271:(assuming
2911:-twist of
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
6078:0010-437X
6040:Algebra I
5950:Citations
5808:×
5779:⋅
5768:∈
5753:∑
5729:∈
5647:∈
5624:⟩
5621:⟩
5615:⟨
5612:⟨
5606:∈
5556:⟩
5553:⟩
5547:⟨
5544:⟨
5518:ϕ
5512:⋅
5470:×
5366:−
5355:−
5302:−
5291:−
5144:≠
5132:ϕ
5071:ϕ
5056:ϕ
5039:⋅
5030:ϕ
4998:→
4989:ϕ
4966:⋅
4910:∈
4903:⋃
4831:∈
4824:⨁
4747:ε
4724:ε
4664:→
4656::
4653:ε
4630:ε
4558:
4549:ε
4540:
4531:ε
4520:−
4485:Γ
4459:Γ
4419:→
4416:Γ
4413::
4410:ε
4390:Γ
4367:ε
4361:Γ
4224:Remarks:
4118:∈
4107:for some
4053:⋅
4042:⊆
3981:∈
3974:⨁
3847:∙
3811:∙
3777:∙
3773:⋀
3719:of order
3687:∙
3611:⊆
3520:⊆
3423:
3410:↦
3361:…
3279:ℓ
3230:ℓ
3227:∑
3159:∈
3072:⊆
3022:→
3016::
2972:ℓ
2945:ℓ
2899:ℓ
2829:∩
2789:inclusion
2759:⊆
2703:→
2635:∞
2626:⨁
2566:∞
2551:⨁
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
6117:Algebras
6004:(2002),
5897:See also
5821:′
5696:′
5602:′
5493:semiring
5081:′
5046:′
4602:Examples
3795:and the
3715:are the
3325:Suppose
2857:and the
2685:morphism
2538:-module
1935:on
1310:-algebra
1142:For any
1034:of
1030:are the
483:integers
161:Lie ring
126:Semiring
6028:1878556
6006:Algebra
5873:Example
5386:divisor
4800:
4765:
4759:is the
3894:
3867:
3861:in any
3755:
3725:
3717:tensors
3389:
3334:
3193:
3133:
2851:
2805:
2781:
2732:
2726:, is a
2534:and an
2524:Example
2510:Example
2496:Example
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
6095:
6076:
6046:
6026:
6016:
5981:
5971:
5885:, the
5197:where
4813:monoid
4645:where
4382:where
3941:monoid
3912:, and
2891:, the
2866:center
2855:kernel
2853:. 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
6062:(PDF)
5945:Notes
5264:) or
5238:(for
5159:when
4242:group
4139:grade
3924:(cf.
3704:of a
3490:is a
3329:is a
3100:. An
2986:(cf.
2859:image
2504: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
6093:ISBN
6074:ISSN
6044:ISBN
6014:ISBN
5969:ISBN
5495:and
4594:and
3920:and
3904:and
3831:The
3760:The
3671:The
3569:are
3001:and
2997:Let
2783:. A
2498:: 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
5979:Zbl
5877:In
5662:to
5580:to
4607:An
4555:deg
4537:deg
4137:of
3928:.)
3479:An
3461:of
3414:dim
3311:of
3108:in
3104:of
3045:if
2994:).
2990:in
2722: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
6113::
6070:34
6068:.
6064:.
6042:.
6024:MR
6022:,
6008:,
5977:.
5869:.
5864:=
5862:pq
5853:,
5436:.
4946:.
4697:A
4598:.
4240:A
4221:.
4144:.
3908:,
3723:,
3650:.
3465:.
3391:,
3315:.
3195::
2683:A
2680:.
2492:.
2247:;
2223:;
2215:,
1844:.
1162:,
1058:.
786:.
553:.
500:A
497:.
6101:.
6080:.
6052:.
6031:.
5985:.
5891:A
5883:A
5866:m
5857:)
5855:q
5851:p
5849:(
5845:m
5831:)
5828:q
5825:(
5818:s
5812:K
5804:)
5801:p
5798:(
5795:s
5788:m
5785:=
5782:q
5776:p
5771:R
5765:q
5762:,
5759:p
5732:R
5726:m
5706:)
5703:m
5700:(
5693:s
5687:K
5683:+
5679:)
5676:m
5673:(
5670:s
5650:R
5644:m
5618:R
5609:K
5599:s
5595:,
5592:s
5582:K
5578:R
5574:R
5570:K
5550:R
5541:K
5521:)
5515:,
5509:,
5506:R
5503:(
5479:)
5474:K
5466:,
5461:K
5457:+
5453:,
5450:K
5447:(
5434:n
5430:n
5415:N
5369:1
5363:g
5358:1
5350:1
5347:+
5344:n
5340:g
5326:G
5322:n
5305:1
5299:g
5294:1
5286:1
5283:+
5280:n
5276:g
5252:1
5249:=
5246:g
5226:1
5223:+
5220:n
5210:n
5206:G
5199:g
5183:n
5179:g
5168:n
5161:m
5147:0
5141:)
5138:m
5135:(
5110:M
5106:1
5085:)
5078:m
5074:(
5068:+
5065:)
5062:m
5059:(
5053:=
5050:)
5043:m
5036:m
5033:(
5008:0
5003:N
4995:M
4992::
4969:)
4963:,
4960:M
4957:(
4932:n
4928:R
4920:0
4915:N
4907:n
4880:n
4876:R
4853:n
4849:R
4841:0
4836:N
4828:n
4802:.
4787:Z
4783:2
4779:/
4774:Z
4727:)
4721:,
4717:Z
4713:(
4681:Z
4677:2
4673:/
4668:Z
4660:Z
4633:)
4627:,
4623:Z
4619:(
4596:y
4592:x
4575:,
4572:x
4569:y
4564:)
4561:y
4552:(
4546:)
4543:x
4534:(
4527:)
4523:1
4517:(
4514:=
4511:y
4508:x
4475:A
4436:Z
4432:2
4428:/
4423:Z
4370:)
4364:,
4358:(
4333:Z
4329:2
4325:/
4320:Z
4275:2
4270:Z
4219:G
4204:N
4178:N
4156:N
4142:i
4121:G
4115:i
4093:i
4089:R
4078:R
4061:.
4056:j
4050:i
4046:R
4037:j
4033:R
4027:i
4023:R
3994:i
3990:R
3984:G
3978:i
3970:=
3967:R
3954:R
3949:G
3944:G
3933:G
3896:.
3880:n
3876:H
3843:H
3816:V
3807:S
3782:V
3757:.
3743:V
3738:n
3734:T
3721:n
3713:n
3709:V
3692:V
3683:T
3668:.
3666:n
3662:n
3648:R
3644:A
3625:j
3622:+
3619:i
3615:A
3606:j
3602:A
3596:i
3592:R
3578:R
3571:R
3555:i
3551:A
3528:0
3524:A
3517:R
3507:R
3503:R
3499:R
3488:R
3484:A
3463:M
3455:n
3447:M
3431:n
3427:M
3418:k
3407:n
3397:M
3393:k
3377:]
3372:n
3368:x
3364:,
3358:,
3353:0
3349:x
3345:[
3342:k
3327:R
3313:M
3295:)
3290:n
3286:M
3282:(
3254:n
3250:t
3246:)
3241:n
3237:M
3233:(
3224:=
3221:)
3218:t
3215:,
3212:M
3209:(
3206:P
3181:]
3177:]
3174:t
3171:[
3167:[
3163:Z
3156:)
3153:t
3150:,
3147:M
3144:(
3141:P
3126:R
3122:M
3086:d
3083:+
3080:n
3076:N
3069:)
3064:n
3060:M
3056:(
3053:f
3043:d
3039:f
3025:N
3019:M
3013:f
3003:N
2999:M
2969:+
2966:n
2962:M
2958:=
2953:n
2949:)
2942:(
2939:M
2919:M
2879:M
2837:i
2833:M
2826:N
2823:=
2818:i
2814:N
2801:M
2797:M
2793:N
2767:i
2763:M
2756:)
2751:i
2747:N
2743:(
2740:f
2706:M
2700:N
2697::
2694:f
2666:1
2663:+
2660:n
2656:I
2651:/
2645:n
2641:I
2630:0
2605:M
2600:1
2597:+
2594:n
2590:I
2585:/
2581:M
2576:n
2572:I
2561:0
2558:=
2555:n
2540:M
2536:R
2532:R
2528: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:)
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