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