3975:
describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them. For Lie algebras, such enveloping algebras have a universal property, which does not hold, in general, for non-associative algebras. The best-known example is, perhaps the
3029:(an algebra over the reals), and generalizations of the octonions over other fields. All associative algebras are alternative. Up to isomorphism, the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions.
3149:
a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and
3252:
There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a
778:
means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for
3877:
2797:
1286:
Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy
3642:
3480:
251:
3966:
543:
2512:
739:
operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (
496:
459:
3035:, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras over a field other than
931:
909:
205:
4687:
Some
Aspects of Ring Theory: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna (Como), Italy, August 23-31, 1965
2922:
2896:
3103:, in which multiplicative inverses exist. The finite-dimensional alternative division algebras over the field of real numbers have been classified. They are the
2559:
4560:
4829:
1268:
648:
3793:
3119:(dimension 8). The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions.
885:-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any
3688:
Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:
2996:
as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.
2715:
2943:
is an example of an algebra which is anticommutative and not associative. The cross product also satisfies the Jacobi identity.
17:
3561:
4911:
4727:
4671:
3402:
105:
4524:. American Mathematical Society Colloquium Publ. Vol. 24 (Corrected reprint of the revised 1961 ed.). New York:
3015:)/2. In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called
3281:
is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of
5096:
3998:
1261:
30:
This article is about a particular structure known as a non-associative algebra. For non-associativity in general, see
5125:
5106:
5073:
5049:
5016:
4987:
4805:
4772:
4699:
4533:
641:
593:
215:
3889:
510:
3025:
are algebras satisfying the alternative property. The most important examples of alternative algebras are the
834:
may be studied by associating it with other associative algebras which are subalgebras of the full algebra of
5041:
4943:
4838:
4764:
4719:
4525:
3384:
2455:
1254:
3226:. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.
3972:
634:
501:
1123:
351:
4971:
3154:
111:
472:
435:
126:
3200:
3032:
3984:
that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.
3000:
1963:
586:
389:
339:
914:
892:
188:
2957:
398:
132:
91:
4681:
3223:
1288:
555:
406:
357:
138:
4620:
4615:
4551:
4515:
3047:
678:
4921:
4869:
4737:
4599:
3208:
2940:
2901:
1291:, or properties, which simplify multiplication somewhat. These include the following ones.
1201:
1193:
1165:
1160:
1151:
1108:
1050:
875:
670:
279:
153:
5083:
5026:
5011:. Mathematics and its Applications. Vol. 393. Dordrecht: Kluwer Academic Publishers.
4997:
4929:
4877:
4815:
4782:
4607:
4543:
2809:
8:
4746:
3189:
3074:
3022:
1558:
1219:
1209:
1060:
960:
952:
943:
780:
689:
682:
561:
369:
320:
265:
159:
145:
73:
41:
5059:
4754:
4718:. American Mathematical Society Colloquium Publications, Vol. XXXIX. Providence, R.I.:
4650:
4637:
4587:
4498:
3377:
3305:
3059:
2544:
1025:
1016:
974:
852:
574:
60:
867:
More generally, some authors consider the concept of a non-associative algebra over a
5102:
5069:
5045:
5012:
4983:
4907:
4801:
4768:
4723:
4695:
4667:
4579:
4529:
3122:
3082:
2976:
1835:
1716:
615:
412:
177:
118:
31:
4889:
4763:. Colloquium Publications. Vol. 44. With a preface by J. Tits. Providence, RI:
4713:
4657:
4294:
5092:
5079:
5022:
4993:
4975:
4952:
4925:
4899:
4873:
4842:
4811:
4778:
4629:
4603:
4569:
4539:
3993:
3510:
3242:
3100:
3094:
3086:
1604:
1045:
868:
797:
674:
621:
607:
421:
363:
326:
99:
85:
4933:
1070:
5063:
5035:
5006:
4965:
4917:
4895:
4885:
4865:
4797:
4791:
4758:
4733:
4709:
4691:
4685:
4595:
4519:
3229:
3219:
3055:
2933:
2450:
1818:
Power commutative: the subalgebra generated by any element is commutative, i.e.,
1509:
1448:
1432:
1137:
1131:
1118:
1098:
1089:
1055:
992:
383:
333:
171:
2561:, and can be used to conveniently express some possible identities satisfied by
4856:
Kurosh, A.G. (1947). "Non-associative algebras and free products of algebras".
4330:
3981:
3977:
3108:
3078:
3070:
2986:
2969:
1356:
1179:
788:
728:
427:
27:
Algebra over a field where binary multiplication is not necessarily associative
5119:
4979:
4583:
3235:
3003:
other than 2 gives rise to a Jordan algebra by defining a new multiplication
1416:
1390:
1065:
1030:
987:
839:
736:
568:
464:
79:
3667:
is the centraliser of the enveloping algebra in the endomorphism algebra End
3659:
is the associative algebra generated by the left and right linear maps. The
3254:
3090:
2953:
1239:
1170:
1004:
860:, the latter being in a sense "the smallest associative algebra containing
708:
696:
600:
375:
271:
4970:. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2.
4750:
4212:
4210:
3528:
3514:
3104:
2946:
1917:
1229:
1224:
1113:
1103:
1077:
824:
724:
580:
291:
165:
47:
3301:). The algebra is unital if one takes the empty product as a monomial.
4956:
4641:
4591:
3175:
3112:
2993:
2989:
are algebras which satisfy the commutative law and the Jordan identity.
2432:
979:
345:
4967:
Introduction to
Octonion and Other Non-Associative Algebras in Physics
4318:
4207:
1319:. Let powers to positive (non-zero) integer be recursively defined by
5090:
4941:
Mikheev, I.M. (1976). "Right nilpotency in right alternative rings".
4663:
4300:
3026:
2705:
is the set of elements that commute and associate with everything in
1234:
1040:
997:
965:
305:
210:
4847:
4824:
4633:
4574:
4555:
3872:{\displaystyle Q(a):x\mapsto 2a\cdot (a\cdot x)-(a\cdot a)\cdot x\ }
2666:
is the set of elements that associate with all others: that is, the
5091:
Zhevlakov, Konstantin A.; Slin'ko, Arkadii M.; Shestakov, Ivan P.;
4903:
3261:
have a two-sided inverse, but some of them are also zero divisors.
3258:
3215:-algebras, they thus include Cayley-Dickson algebras and many more.
3196:
3185:
3116:
3040:
2992:
Every associative algebra gives rise to a Lie algebra by using the
1035:
820:
732:
299:
285:
2949:
are algebras satisfying anticommutativity and the Jacobi identity.
3328:-vector space and so one can consider the associative algebra End
3308:
that every subalgebra of a free non-associative algebra is free.
183:
67:
1719:: the subalgebra generated by any element is associative, i.e.,
2626:
It implies that permuting the extremal terms changes the sign:
969:
4790:
Koecher, Max (1999). Krieg, Aloys; Walcher, Sebastian (eds.).
4793:
The
Minnesota notes on Jordan algebras and their applications
4354:
4342:
3036:
2171:
4690:. C.I.M.E. Summer Schools. Vol. 37 (reprint ed.).
4649:
Bremner, Murray; Murakami, Lúcia; Shestakov, Ivan (2013) .
4012:
3232:
are non-associative algebras used in mathematical genetics.
3054:, which was an experimental algebra before the adoption of
2792:{\displaystyle C(A)=\{n\in A\ |\ nr=rn\,\forall r\in A\,\}}
2602:
It implies that permuting any two terms changes the sign:
735:, and three-dimensional Euclidean space equipped with the
4405:
1849:
elements, in any association, vanishes, but not for some
4796:. Lecture Notes in Mathematics. Vol. 1710. Berlin:
4648:
4371:
4369:
4336:
4324:
4159:
4071:
4069:
4041:
4039:
3637:{\displaystyle L(a):x\mapsto ax;\ \ R(a):x\mapsto xa\ .}
3199:, and the infinite sequence of Cayley-Dickson algebras (
4744:
4453:
4381:
4222:
4216:
4056:
4054:
4306:
4282:
4270:
4258:
4234:
4086:
4084:
3513:
of two derivations is again a derivation, so that the
3475:{\displaystyle D(x\cdot y)=D(x)\cdot y+x\cdot D(y)\ .}
723:
which may or may not be associative. Examples include
4465:
4441:
4366:
4246:
4171:
4125:
4123:
4066:
4036:
3892:
3796:
3564:
3405:
2904:
2812:
2718:
2547:
2458:
917:
895:
513:
475:
438:
218:
191:
4417:
4393:
4135:
4108:
4051:
3073:. These include most of the algebras of interest to
2802:
with the nucleus. It turns out that for elements of
4429:
4195:
4183:
4147:
4081:
4024:
911:-algebra, so some authors refer to non-associative
4120:
4096:
3960:
3871:
3636:
3474:
2916:
2890:
2791:
2553:
2506:
925:
903:
537:
490:
453:
245:
199:
4561:Transactions of the American Mathematical Society
3285:retaining parentheses. The product of monomials
1281:
5117:
4830:Proceedings of the American Mathematical Society
4715:Structure and representations of Jordan algebras
3346:. We can associate to the algebra structure on
4825:"Power-associative rings of characteristic two"
3264:
1953:
2541:It measures the degree of nonassociativity of
4001:, which give rise to non-associative algebras
1262:
642:
3257:. For example, all non-zero elements of the
2911:
2905:
2786:
2734:
246:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
5065:An Introduction to Nonassociative Algebras
4618:(1948b). "On right alternative algebras".
3961:{\displaystyle Q(a)=2L^{2}(a)-L(a^{2})\ .}
2999:Every associative algebra over a field of
2580:denote arbitrary elements of the algebra.
1269:
1255:
649:
635:
5004:
4884:
4846:
4573:
4459:
4387:
4228:
4165:
2785:
2772:
2685:The nucleus is an associative subring of
1311:denote arbitrary elements of the algebra
919:
897:
538:{\displaystyle \mathbb {Z} (p^{\infty })}
515:
478:
441:
239:
226:
193:
4708:
4680:
4493:
4491:
4240:
3171:(a commutative and associative algebra);
3093:. Graded algebras can be generalized to
830:The nonassociative algebra structure of
5058:
4940:
4822:
4789:
4614:
4550:
4471:
4423:
4375:
4360:
4348:
4312:
4288:
4276:
4264:
4252:
4177:
4141:
4114:
4090:
4075:
4060:
4045:
4030:
4018:
2924:for the third to also be the zero set.
2507:{\displaystyle :A\times A\times A\to A}
14:
5118:
4855:
4514:
4447:
4411:
4337:Bremner, Murakami & Shestakov 2013
4325:Bremner, Murakami & Shestakov 2013
3311:
5037:Graduate Algebra: Noncommutative View
5033:
4963:
4651:"Chapter 86: Nonassociative Algebras"
4488:
4435:
4399:
4201:
4189:
4153:
4129:
4102:
3534:
3371:
3342:-linear vector space endomorphism of
2391:(one of the two identities defining
2351:(one of the two identities defining
1980:Any two out of the three properties
1352:(left powers) depending on authors.
771:)) may all yield different answers.
688:is a non-associative algebra over a
106:Free product of associative algebras
5040:. Graduate studies in mathematics.
3685:-scalar multiples of the identity.
1294:
24:
4894:. Universitext. Berlin, New York:
3999:Commutative non-associative magmas
3730:) commutes with the corresponding
3211:are all finite-dimensional unital
2806:it is enough that two of the sets
2773:
889:-algebra), then it is naturally a
527:
25:
5137:
5098:Rings that are nearly associative
2939:with multiplication given by the
850:-vector space. Two such are the
819:in the algebra. For example, the
594:Noncommutative algebraic geometry
5101:. Translated by Smith, Harry F.
3700:) is equal to the corresponding
3681:if its centroid consists of the
3366:(associative) enveloping algebra
3039:(see previous section), and the
858:(associative) enveloping algebra
711:binary multiplication operation
491:{\displaystyle \mathbb {Q} _{p}}
454:{\displaystyle \mathbb {Z} _{p}}
2709:, that is the intersection of
675:binary multiplication operation
3949:
3936:
3927:
3921:
3902:
3896:
3857:
3845:
3839:
3827:
3815:
3806:
3800:
3776:) and similarly for the right.
3649:associative enveloping algebra
3619:
3610:
3604:
3583:
3574:
3568:
3463:
3457:
3436:
3430:
3421:
3409:
2885:
2882:
2864:
2858:
2840:
2834:
2816:
2813:
2750:
2728:
2722:
2498:
2477:
2459:
1282:Algebras satisfying identities
532:
519:
13:
1:
5042:American Mathematical Society
4944:Siberian Mathematical Journal
4839:American Mathematical Society
4765:American Mathematical Society
4720:American Mathematical Society
4526:American Mathematical Society
4508:
3973:universal enveloping algebras
3247:
2630:; the converse holds only if
2606:; the converse holds only if
2426:
1377:; in that case we can define
4005:
3271:free non-associative algebra
3265:Free non-associative algebra
1954:Relations between properties
926:{\displaystyle \mathbb {Z} }
904:{\displaystyle \mathbb {Z} }
200:{\displaystyle \mathbb {Z} }
7:
5034:Rowen, Louis Halle (2008).
4656:. In Hogben, Leslie (ed.).
3987:
2927:
1938:and there exist an element
1914:for a specific association.
352:Unique factorization domain
10:
5142:
4972:Cambridge University Press
4891:A taste of Jordan algebras
4823:Kokoris, Louis A. (1955).
4659:Handbook of Linear Algebra
3375:
3240:
3201:power-associative algebras
3066:More classes of algebras:
3033:Power-associative algebras
2430:
1796:Fourth power commutative:
1739:th power commutative with
1687:Fourth power associative:
1633:th power associative with
112:Tensor product of algebras
29:
5005:Rosenfeld, Boris (1997).
4556:"Power-associative rings"
4339:, pp. 18–19, fact 6.
3547:attached to each element
3181:(an associative algebra);
3145:in the ground field, and
2692:
2654:Third power associative:
1780:Third power commutative:
1671:Third power associative:
1359:: there exist an element
5126:Non-associative algebras
4980:10.1017/CBO9780511524479
4481:
3782:quadratic representation
2393:fourth power associative
2353:fourth power associative
2063:second power commutative
2059:Second power associative
1811:fourth power associative
1709:fourth power commutative
390:Formal power series ring
340:Integrally closed domain
4964:Okubo, Susumu (2005) .
4760:The book of involutions
4363:, p. 554, lemma 3.
4351:, p. 554, lemma 4.
3155:Cayley–Dickson algebras
3115:(dimension 4), and the
2958:differentiable manifold
2595:(left alternative) and
2376:Third power associative
2073:third power commutative
2069:Third power associative
2053:third power associative
1992:, imply the third one.
1581:(left alternative) and
703:and is equipped with a
663:non-associative algebra
399:Algebraic number theory
92:Total ring of fractions
5008:Geometry of Lie groups
3962:
3873:
3653:multiplication algebra
3638:
3539:There are linear maps
3476:
3350:two subalgebras of End
3224:geometric quantization
2918:
2892:
2793:
2555:
2508:
927:
905:
556:Noncommutative algebra
539:
492:
455:
407:Algebraic number field
358:Principal ideal domain
247:
201:
139:Frobenius endomorphism
18:Nonassociative algebra
4621:Annals of Mathematics
4521:Structure of algebras
4301:Zhevlakov et al. 1982
3963:
3874:
3639:
3477:
3209:Hypercomplex algebras
3165:), which begin with:
3125:, which require that
3048:hyperbolic quaternion
2919:
2917:{\displaystyle \{0\}}
2893:
2794:
2651:depending on authors.
2599:(right alternative).
2556:
2509:
1845:: the product of any
1555:depending on authors.
1506:depending on authors.
935:non-associative rings
928:
906:
677:is not assumed to be
540:
493:
456:
248:
202:
5093:Shirshov, Anatoly I.
4747:Merkurjev, Alexander
4497:It follows from the
3890:
3794:
3562:
3403:
3137:, for some elements
3023:Alternative algebras
2941:vector cross product
2902:
2891:{\displaystyle (,,)}
2810:
2716:
2545:
2456:
2274:nilpotent of index 2
2115:Nilpotent of index 3
2089:th power commutative
2082:th power associative
1823:th power commutative
1724:th power associative
1601:(right alternative).
1166:Group with operators
1109:Complemented lattice
944:Algebraic structures
915:
893:
781:noncommutative rings
671:algebra over a field
667:distributive algebra
562:Noncommutative rings
511:
473:
436:
280:Non-associative ring
216:
189:
146:Algebraic structures
5060:Schafer, Richard D.
4755:Tignol, Jean-Pierre
4414:, pp. 237–262.
3760:Alternative: every
3489:form a subspace Der
3485:The derivations on
3324:is in particular a
3312:Associated algebras
3190:alternative algebra
3111:(dimension 2), the
3107:(dimension 1), the
3075:multilinear algebra
2125:Nilpotent of index
1220:Composition algebra
980:Quasigroup and loop
683:algebraic structure
321:Commutative algebra
160:Associative algebra
42:Algebraic structure
4957:10.1007/BF00969304
4745:Knus, Max-Albert;
3958:
3869:
3715:commutes with any
3692:Commutative: each
3677:). An algebra is
3634:
3535:Enveloping algebra
3472:
3396:with the property
3378:Derivation algebra
3372:Derivation algebra
3362:derivation algebra
3222:are considered in
3123:Quadratic algebras
3060:special relativity
2914:
2888:
2789:
2551:
2504:
1342:(right powers) or
923:
901:
853:derivation algebra
774:While this use of
575:Semiprimitive ring
535:
488:
451:
259:Related structures
243:
197:
133:Inner automorphism
119:Ring homomorphisms
4913:978-0-387-95447-9
4729:978-0-821-84640-7
4673:978-1-498-78560-0
4616:Albert, A. Adrian
4552:Albert, A. Adrian
4516:Albert, A. Adrian
3980:, an exceptional
3954:
3883:or equivalently,
3868:
3711:Associative: any
3630:
3600:
3597:
3527:) a structure of
3468:
3101:Division algebras
3095:filtered algebras
3083:symmetric algebra
2977:algebraic variety
2756:
2748:
2643:Jordan identity:
2554:{\displaystyle A}
2397:power associative
2395:) together imply
2357:power associative
2355:) together imply
2319:are incompatible.
2296:are incompatible.
2260:power associative
2230:power associative
2219:power associative
2215:Right alternative
2196:power associative
2161:are incompatible.
2043:power associative
1986:right alternative
1929:power associative
1888:elements so that
1762:for all integers
1717:Power associative
1653:for all integers
1279:
1278:
881:equipped with an
659:
658:
616:Geometric algebra
327:Commutative rings
178:Category of rings
32:Non-associativity
16:(Redirected from
5133:
5112:
5087:
5055:
5030:
5001:
4960:
4937:
4886:McCrimmon, Kevin
4881:
4852:
4850:
4819:
4786:
4741:
4710:Jacobson, Nathan
4705:
4677:
4662:(2nd ed.).
4655:
4645:
4611:
4577:
4547:
4502:
4495:
4475:
4469:
4463:
4457:
4451:
4445:
4439:
4433:
4427:
4421:
4415:
4409:
4403:
4397:
4391:
4385:
4379:
4373:
4364:
4358:
4352:
4346:
4340:
4334:
4328:
4322:
4316:
4310:
4304:
4298:
4292:
4286:
4280:
4274:
4268:
4262:
4256:
4250:
4244:
4238:
4232:
4226:
4220:
4217:Knus et al. 1998
4214:
4205:
4199:
4193:
4187:
4181:
4175:
4169:
4163:
4157:
4151:
4145:
4139:
4133:
4127:
4118:
4112:
4106:
4100:
4094:
4088:
4079:
4073:
4064:
4058:
4049:
4043:
4034:
4028:
4022:
4016:
3994:List of algebras
3967:
3965:
3964:
3959:
3952:
3948:
3947:
3920:
3919:
3878:
3876:
3875:
3870:
3866:
3749:) commutes with
3722:Flexible: every
3643:
3641:
3640:
3635:
3628:
3598:
3595:
3481:
3479:
3478:
3473:
3466:
3243:list of algebras
3230:Genetic algebras
3220:Poisson algebras
3087:exterior algebra
2923:
2921:
2920:
2915:
2897:
2895:
2894:
2889:
2798:
2796:
2795:
2790:
2754:
2753:
2746:
2680:
2669:
2657:
2650:
2646:
2637:
2629:
2622:
2613:
2605:
2598:
2594:
2587:
2579:
2575:
2571:
2560:
2558:
2557:
2552:
2536:
2513:
2511:
2510:
2505:
2410:
2390:
2370:
2350:
2330:
2307:
2226:left alternative
2209:
2182:
2174:
2159:
2147:
2135:
2128:
2088:
2081:
2065:are always true.
1982:left alternative
1961:
1948:
1941:
1937:
1926:
1913:
1887:
1881:and there exist
1880:
1855:
1848:
1844:
1831:
1822:
1808:
1792:
1776:
1765:
1761:
1745:
1738:
1732:
1723:
1706:
1683:
1667:
1656:
1652:
1639:
1632:
1626:
1600:
1580:
1554:
1534:
1505:
1478:
1444:
1428:
1412:
1386:
1376:
1362:
1351:
1341:
1328:
1318:
1314:
1310:
1306:
1302:
1295:Usual properties
1271:
1264:
1257:
1046:Commutative ring
975:Rack and quandle
940:
939:
932:
930:
929:
924:
922:
910:
908:
907:
902:
900:
869:commutative ring
823:are unital, but
798:identity element
651:
644:
637:
622:Operator algebra
608:Clifford algebra
544:
542:
541:
536:
531:
530:
518:
497:
495:
494:
489:
487:
486:
481:
460:
458:
457:
452:
450:
449:
444:
422:Ring of integers
416:
413:Integers modulo
364:Euclidean domain
252:
250:
249:
244:
242:
234:
229:
206:
204:
203:
198:
196:
100:Product of rings
86:Fractional ideal
45:
37:
36:
21:
5141:
5140:
5136:
5135:
5134:
5132:
5131:
5130:
5116:
5115:
5109:
5076:
5052:
5019:
4990:
4914:
4896:Springer-Verlag
4848:10.2307/2032920
4808:
4798:Springer-Verlag
4775:
4730:
4702:
4692:Springer-Verlag
4684:, ed. (2011) .
4682:Herstein, I. N.
4674:
4653:
4634:10.2307/1969457
4575:10.2307/1990399
4536:
4511:
4506:
4505:
4499:Artin's theorem
4496:
4489:
4484:
4479:
4478:
4470:
4466:
4458:
4454:
4446:
4442:
4434:
4430:
4422:
4418:
4410:
4406:
4398:
4394:
4386:
4382:
4374:
4367:
4359:
4355:
4347:
4343:
4335:
4331:
4323:
4319:
4311:
4307:
4299:
4295:
4287:
4283:
4275:
4271:
4263:
4259:
4251:
4247:
4239:
4235:
4227:
4223:
4215:
4208:
4200:
4196:
4188:
4184:
4176:
4172:
4164:
4160:
4152:
4148:
4140:
4136:
4128:
4121:
4113:
4109:
4101:
4097:
4089:
4082:
4074:
4067:
4059:
4052:
4044:
4037:
4029:
4025:
4017:
4013:
4008:
3990:
3971:The article on
3943:
3939:
3915:
3911:
3891:
3888:
3887:
3795:
3792:
3791:
3672:
3563:
3560:
3559:
3537:
3522:
3504:
3494:
3404:
3401:
3400:
3380:
3374:
3355:
3333:
3314:
3267:
3250:
3245:
3109:complex numbers
3071:Graded algebras
3056:Minkowski space
2987:Jordan algebras
2970:complex numbers
2934:Euclidean space
2930:
2903:
2900:
2899:
2811:
2808:
2807:
2749:
2717:
2714:
2713:
2695:
2678:
2667:
2655:
2648:
2644:
2631:
2627:
2620:
2607:
2603:
2596:
2592:
2585:
2577:
2573:
2569:
2546:
2543:
2542:
2518:
2457:
2454:
2453:
2451:multilinear map
2435:
2429:
2422:are equivalent.
2420:anticommutative
2404:
2379:
2364:
2339:
2324:
2317:Jacobi identity
2301:
2294:anticommutative
2280:Anticommutative
2272:together imply
2270:anticommutative
2258:together imply
2252:Jordan identity
2244:together imply
2242:Jordan identity
2203:
2194:together imply
2188:Jordan identity
2176:
2166:
2157:
2138:
2133:
2126:
2119:Jacobi identity
2109:Jordan identity
2099:anticommutative
2086:
2079:
2075:are equivalent.
2029:Anticommutative
2013:Jordan identity
1959:
1956:
1943:
1939:
1932:
1921:
1911:
1901:
1895:
1889:
1882:
1878:
1869:
1863:
1857:
1850:
1846:
1839:
1826:
1820:
1797:
1781:
1767:
1763:
1747:
1740:
1736:
1727:
1721:
1688:
1672:
1658:
1654:
1641:
1634:
1630:
1608:
1582:
1562:
1536:
1513:
1510:Jordan identity
1480:
1452:
1449:Jacobi identity
1436:
1433:Anticommutative
1420:
1394:
1378:
1364:
1360:
1343:
1330:
1320:
1316:
1315:over the field
1312:
1308:
1304:
1300:
1297:
1284:
1275:
1246:
1245:
1244:
1215:Non-associative
1197:
1186:
1185:
1175:
1155:
1144:
1143:
1132:Map of lattices
1128:
1124:Boolean algebra
1119:Heyting algebra
1093:
1082:
1081:
1075:
1056:Integral domain
1020:
1009:
1008:
1002:
956:
918:
916:
913:
912:
896:
894:
891:
890:
776:non-associative
729:Jordan algebras
655:
626:
625:
558:
548:
547:
526:
522:
514:
512:
509:
508:
482:
477:
476:
474:
471:
470:
445:
440:
439:
437:
434:
433:
414:
384:Polynomial ring
334:Integral domain
323:
313:
312:
238:
230:
225:
217:
214:
213:
192:
190:
187:
186:
172:Involutive ring
57:
46:
40:
35:
28:
23:
22:
15:
12:
11:
5:
5139:
5129:
5128:
5114:
5113:
5107:
5088:
5074:
5056:
5050:
5031:
5017:
5002:
4988:
4961:
4951:(1): 178–180.
4938:
4912:
4904:10.1007/b97489
4882:
4853:
4820:
4806:
4787:
4773:
4742:
4728:
4706:
4700:
4678:
4672:
4646:
4612:
4548:
4534:
4510:
4507:
4504:
4503:
4486:
4485:
4483:
4480:
4477:
4476:
4464:
4460:McCrimmon 2004
4452:
4450:, p. 113.
4440:
4428:
4416:
4404:
4402:, p. 321.
4392:
4388:McCrimmon 2004
4380:
4365:
4353:
4341:
4329:
4317:
4315:, p. 148.
4305:
4303:, p. 343.
4293:
4291:, p. 179.
4281:
4279:, p. 319.
4269:
4267:, p. 710.
4257:
4245:
4233:
4229:Rosenfeld 1997
4221:
4219:, p. 451.
4206:
4194:
4182:
4170:
4168:, p. 153.
4166:McCrimmon 2004
4158:
4146:
4134:
4119:
4107:
4095:
4080:
4078:, p. 128.
4065:
4050:
4048:, p. 553.
4035:
4023:
4010:
4009:
4007:
4004:
4003:
4002:
3996:
3989:
3986:
3982:Jordan algebra
3978:Albert algebra
3969:
3968:
3957:
3951:
3946:
3942:
3938:
3935:
3932:
3929:
3926:
3923:
3918:
3914:
3910:
3907:
3904:
3901:
3898:
3895:
3881:
3880:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3787:is defined by
3778:
3777:
3758:
3741:Jordan: every
3739:
3720:
3709:
3668:
3645:
3644:
3633:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3551:of an algebra
3536:
3533:
3518:
3500:
3490:
3483:
3482:
3471:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3376:Main article:
3373:
3370:
3351:
3329:
3313:
3310:
3266:
3263:
3249:
3246:
3239:
3238:
3236:Triple systems
3233:
3227:
3216:
3206:
3205:
3204:
3193:
3182:
3172:
3151:
3120:
3098:
3079:tensor algebra
3077:, such as the
3064:
3063:
3044:
3030:
3020:
3001:characteristic
2997:
2990:
2984:
2950:
2944:
2929:
2926:
2913:
2910:
2907:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2800:
2799:
2788:
2784:
2781:
2778:
2775:
2771:
2768:
2765:
2762:
2759:
2752:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2694:
2691:
2683:
2682:
2660:
2659:
2652:
2641:
2640:
2639:
2617:
2616:
2615:
2589:
2550:
2539:
2538:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2431:Main article:
2428:
2425:
2424:
2423:
2401:
2400:
2361:
2360:
2321:
2320:
2298:
2297:
2287:
2284:nil of index 2
2277:
2263:
2249:
2235:
2234:
2233:
2200:
2199:
2163:
2162:
2149:
2122:
2112:
2105:Nil of index 2
2102:
2095:Nil of index 2
2092:
2076:
2066:
2056:
2046:
2036:
2026:
2016:
2006:
2005:
2004:
1978:
1964:characteristic
1955:
1952:
1951:
1950:
1915:
1906:
1899:
1893:
1874:
1867:
1861:
1833:
1816:
1815:
1814:
1809:(compare with
1794:
1734:
1714:
1713:
1712:
1707:(compare with
1685:
1628:
1602:
1556:
1507:
1446:
1430:
1414:
1388:
1296:
1293:
1283:
1280:
1277:
1276:
1274:
1273:
1266:
1259:
1251:
1248:
1247:
1243:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1206:
1205:
1204:
1198:
1192:
1191:
1188:
1187:
1184:
1183:
1180:Linear algebra
1174:
1173:
1168:
1163:
1157:
1156:
1150:
1149:
1146:
1145:
1142:
1141:
1138:Lattice theory
1134:
1127:
1126:
1121:
1116:
1111:
1106:
1101:
1095:
1094:
1088:
1087:
1084:
1083:
1074:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1038:
1033:
1028:
1022:
1021:
1015:
1014:
1011:
1010:
1001:
1000:
995:
990:
984:
983:
982:
977:
972:
963:
957:
951:
950:
947:
946:
921:
899:
786:An algebra is
681:. That is, an
657:
656:
654:
653:
646:
639:
631:
628:
627:
619:
618:
590:
589:
583:
577:
571:
559:
554:
553:
550:
549:
546:
545:
534:
529:
525:
521:
517:
498:
485:
480:
461:
448:
443:
431:-adic integers
424:
418:
409:
395:
394:
393:
392:
386:
380:
379:
378:
366:
360:
354:
348:
342:
324:
319:
318:
315:
314:
311:
310:
309:
308:
296:
295:
294:
288:
276:
275:
274:
256:
255:
254:
253:
241:
237:
233:
228:
224:
221:
207:
195:
174:
168:
162:
156:
142:
141:
135:
129:
115:
114:
108:
102:
96:
95:
94:
88:
76:
70:
58:
56:Basic concepts
55:
54:
51:
50:
26:
9:
6:
4:
3:
2:
5138:
5127:
5124:
5123:
5121:
5110:
5108:0-12-779850-1
5104:
5100:
5099:
5094:
5089:
5085:
5081:
5077:
5075:0-486-68813-5
5071:
5067:
5066:
5061:
5057:
5053:
5051:0-8218-8408-5
5047:
5043:
5039:
5038:
5032:
5028:
5024:
5020:
5018:0-7923-4390-5
5014:
5010:
5009:
5003:
4999:
4995:
4991:
4989:0-521-01792-0
4985:
4981:
4977:
4973:
4969:
4968:
4962:
4958:
4954:
4950:
4946:
4945:
4939:
4935:
4931:
4927:
4923:
4919:
4915:
4909:
4905:
4901:
4897:
4893:
4892:
4887:
4883:
4879:
4875:
4871:
4867:
4863:
4859:
4854:
4849:
4844:
4840:
4836:
4832:
4831:
4826:
4821:
4817:
4813:
4809:
4807:3-540-66360-6
4803:
4799:
4795:
4794:
4788:
4784:
4780:
4776:
4774:0-8218-0904-0
4770:
4766:
4762:
4761:
4756:
4752:
4748:
4743:
4739:
4735:
4731:
4725:
4721:
4717:
4716:
4711:
4707:
4703:
4701:3-6421-1036-3
4697:
4693:
4689:
4688:
4683:
4679:
4675:
4669:
4665:
4661:
4660:
4652:
4647:
4643:
4639:
4635:
4631:
4627:
4623:
4622:
4617:
4613:
4609:
4605:
4601:
4597:
4593:
4589:
4585:
4581:
4576:
4571:
4567:
4563:
4562:
4557:
4553:
4549:
4545:
4541:
4537:
4535:0-8218-1024-3
4531:
4527:
4523:
4522:
4517:
4513:
4512:
4500:
4494:
4492:
4487:
4474:, p. 57.
4473:
4468:
4462:, p. 57.
4461:
4456:
4449:
4444:
4438:, p. 24.
4437:
4432:
4425:
4420:
4413:
4408:
4401:
4396:
4390:, p. 56.
4389:
4384:
4378:, p. 14.
4377:
4372:
4370:
4362:
4357:
4350:
4345:
4338:
4333:
4327:, p. 18.
4326:
4321:
4314:
4309:
4302:
4297:
4290:
4285:
4278:
4273:
4266:
4261:
4255:, p. 92.
4254:
4249:
4243:, p. 36.
4242:
4241:Jacobson 1968
4237:
4231:, p. 91.
4230:
4225:
4218:
4213:
4211:
4204:, p. 17.
4203:
4198:
4192:, p. 16.
4191:
4186:
4180:, p. 28.
4179:
4174:
4167:
4162:
4156:, p. 18.
4155:
4150:
4143:
4138:
4132:, p. 13.
4131:
4126:
4124:
4117:, p. 91.
4116:
4111:
4105:, p. 12.
4104:
4099:
4092:
4087:
4085:
4077:
4072:
4070:
4063:, p. 30.
4062:
4057:
4055:
4047:
4042:
4040:
4032:
4027:
4020:
4015:
4011:
4000:
3997:
3995:
3992:
3991:
3985:
3983:
3979:
3974:
3955:
3944:
3940:
3933:
3930:
3924:
3916:
3912:
3908:
3905:
3899:
3893:
3886:
3885:
3884:
3863:
3860:
3854:
3851:
3848:
3842:
3836:
3833:
3830:
3824:
3821:
3818:
3812:
3809:
3803:
3797:
3790:
3789:
3788:
3786:
3783:
3775:
3771:
3767:
3763:
3759:
3756:
3752:
3748:
3744:
3740:
3737:
3733:
3729:
3725:
3721:
3718:
3714:
3710:
3707:
3703:
3699:
3695:
3691:
3690:
3689:
3686:
3684:
3680:
3676:
3671:
3666:
3662:
3658:
3654:
3650:
3631:
3625:
3622:
3616:
3613:
3607:
3601:
3592:
3589:
3586:
3580:
3577:
3571:
3565:
3558:
3557:
3556:
3554:
3550:
3546:
3542:
3532:
3530:
3526:
3521:
3516:
3512:
3508:
3503:
3498:
3493:
3488:
3469:
3460:
3454:
3451:
3448:
3445:
3442:
3439:
3433:
3427:
3424:
3418:
3415:
3412:
3406:
3399:
3398:
3397:
3395:
3391:
3387:
3386:
3379:
3369:
3367:
3363:
3359:
3354:
3349:
3345:
3341:
3337:
3332:
3327:
3323:
3320:over a field
3319:
3309:
3307:
3302:
3300:
3296:
3292:
3288:
3284:
3280:
3277:over a field
3276:
3272:
3262:
3260:
3256:
3244:
3237:
3234:
3231:
3228:
3225:
3221:
3217:
3214:
3210:
3207:
3202:
3198:
3194:
3191:
3187:
3183:
3180:
3177:
3173:
3170:
3167:
3166:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3121:
3118:
3114:
3110:
3106:
3102:
3099:
3096:
3092:
3089:over a given
3088:
3084:
3080:
3076:
3072:
3069:
3068:
3067:
3061:
3057:
3053:
3050:algebra over
3049:
3045:
3042:
3038:
3034:
3031:
3028:
3024:
3021:
3018:
3014:
3010:
3006:
3002:
2998:
2995:
2991:
2988:
2985:
2982:
2979:(for general
2978:
2974:
2971:
2967:
2963:
2959:
2955:
2954:vector fields
2951:
2948:
2945:
2942:
2938:
2935:
2932:
2931:
2925:
2908:
2879:
2876:
2873:
2870:
2867:
2861:
2855:
2852:
2849:
2846:
2843:
2837:
2831:
2828:
2825:
2822:
2819:
2805:
2782:
2779:
2776:
2769:
2766:
2763:
2760:
2757:
2743:
2740:
2737:
2731:
2725:
2719:
2712:
2711:
2710:
2708:
2704:
2700:
2690:
2688:
2677:
2676:
2675:
2673:
2665:
2653:
2642:
2635:
2625:
2624:
2618:
2611:
2601:
2600:
2591:Alternative:
2590:
2584:Associative:
2583:
2582:
2581:
2566:
2564:
2548:
2534:
2530:
2526:
2522:
2517:
2516:
2515:
2501:
2495:
2492:
2489:
2486:
2483:
2480:
2474:
2471:
2468:
2465:
2462:
2452:
2448:
2444:
2440:
2434:
2421:
2417:
2414:
2413:
2412:
2408:
2398:
2394:
2389:
2386:
2382:
2377:
2374:
2373:
2372:
2368:
2358:
2354:
2349:
2346:
2342:
2337:
2334:
2333:
2332:
2328:
2318:
2314:
2311:
2310:
2309:
2305:
2295:
2291:
2288:
2285:
2281:
2278:
2275:
2271:
2267:
2264:
2261:
2257:
2253:
2250:
2247:
2243:
2239:
2236:
2231:
2227:
2223:
2222:
2220:
2216:
2213:
2212:
2211:
2207:
2197:
2193:
2189:
2186:
2185:
2184:
2180:
2173:
2169:
2160:
2156:nil of index
2153:
2150:
2146:
2142:
2136:
2132:nil of index
2129:
2123:
2120:
2116:
2113:
2110:
2106:
2103:
2100:
2096:
2093:
2090:
2083:
2077:
2074:
2070:
2067:
2064:
2060:
2057:
2054:
2050:
2047:
2044:
2040:
2037:
2034:
2030:
2027:
2024:
2020:
2017:
2014:
2010:
2007:
2002:
1998:
1994:
1993:
1991:
1987:
1983:
1979:
1976:
1972:
1969:
1968:
1967:
1965:
1946:
1935:
1930:
1924:
1919:
1916:
1909:
1905:
1898:
1892:
1885:
1877:
1873:
1866:
1860:
1853:
1842:
1837:
1834:
1829:
1824:
1817:
1812:
1807:
1803:
1800:
1795:
1791:
1787:
1784:
1779:
1778:
1775:
1771:
1760:
1757:
1753:
1750:
1743:
1735:
1730:
1725:
1718:
1715:
1710:
1705:
1701:
1698:
1694:
1691:
1686:
1682:
1678:
1675:
1670:
1669:
1666:
1662:
1651:
1647:
1644:
1637:
1629:
1624:
1620:
1616:
1612:
1606:
1603:
1598:
1594:
1590:
1586:
1578:
1574:
1570:
1566:
1560:
1557:
1552:
1548:
1544:
1540:
1532:
1528:
1524:
1520:
1517:
1511:
1508:
1503:
1499:
1495:
1491:
1487:
1483:
1476:
1472:
1468:
1464:
1460:
1456:
1450:
1447:
1443:
1439:
1434:
1431:
1427:
1423:
1418:
1415:
1410:
1406:
1402:
1398:
1392:
1389:
1385:
1381:
1375:
1371:
1367:
1358:
1355:
1354:
1353:
1350:
1346:
1340:
1337:
1333:
1327:
1323:
1292:
1290:
1272:
1267:
1265:
1260:
1258:
1253:
1252:
1250:
1249:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1207:
1203:
1200:
1199:
1195:
1190:
1189:
1182:
1181:
1177:
1176:
1172:
1169:
1167:
1164:
1162:
1159:
1158:
1153:
1148:
1147:
1140:
1139:
1135:
1133:
1130:
1129:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1096:
1091:
1086:
1085:
1080:
1079:
1072:
1069:
1067:
1066:Division ring
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1027:
1024:
1023:
1018:
1013:
1012:
1007:
1006:
999:
996:
994:
991:
989:
988:Abelian group
986:
985:
981:
978:
976:
973:
971:
967:
964:
962:
959:
958:
954:
949:
948:
945:
942:
941:
938:
936:
933:-algebras as
888:
884:
880:
878:
873:
870:
865:
863:
859:
855:
854:
849:
845:
841:
840:endomorphisms
837:
833:
828:
826:
822:
818:
814:
810:
806:
802:
799:
796:if it has an
795:
791:
790:
784:
782:
777:
772:
770:
766:
762:
758:
754:
750:
746:
742:
738:
737:cross product
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
691:
687:
684:
680:
676:
672:
668:
664:
652:
647:
645:
640:
638:
633:
632:
630:
629:
624:
623:
617:
613:
612:
611:
610:
609:
604:
603:
602:
597:
596:
595:
588:
584:
582:
578:
576:
572:
570:
569:Division ring
566:
565:
564:
563:
557:
552:
551:
523:
507:
505:
499:
483:
469:
468:-adic numbers
467:
462:
446:
432:
430:
425:
423:
419:
417:
410:
408:
404:
403:
402:
401:
400:
391:
387:
385:
381:
377:
373:
372:
371:
367:
365:
361:
359:
355:
353:
349:
347:
343:
341:
337:
336:
335:
331:
330:
329:
328:
322:
317:
316:
307:
303:
302:
301:
297:
293:
289:
287:
283:
282:
281:
277:
273:
269:
268:
267:
263:
262:
261:
260:
235:
231:
222:
219:
212:
211:Terminal ring
208:
185:
181:
180:
179:
175:
173:
169:
167:
163:
161:
157:
155:
151:
150:
149:
148:
147:
140:
136:
134:
130:
128:
124:
123:
122:
121:
120:
113:
109:
107:
103:
101:
97:
93:
89:
87:
83:
82:
81:
80:Quotient ring
77:
75:
71:
69:
65:
64:
63:
62:
53:
52:
49:
44:→ Ring theory
43:
39:
38:
33:
19:
5097:
5064:
5036:
5007:
4966:
4948:
4942:
4890:
4861:
4858:Mat. Sbornik
4857:
4834:
4828:
4792:
4759:
4751:Rost, Markus
4714:
4686:
4658:
4625:
4619:
4565:
4559:
4520:
4472:Koecher 1999
4467:
4455:
4443:
4431:
4426:, p. 4.
4424:Schafer 1995
4419:
4407:
4395:
4383:
4376:Schafer 1995
4361:Albert 1948a
4356:
4349:Albert 1948a
4344:
4332:
4320:
4313:Schafer 1995
4308:
4296:
4289:Mikheev 1976
4284:
4277:Albert 1948b
4272:
4265:Kokoris 1955
4260:
4253:Schafer 1995
4248:
4236:
4224:
4197:
4185:
4178:Schafer 1995
4173:
4161:
4149:
4144:, p. 5.
4142:Schafer 1995
4137:
4115:Schafer 1995
4110:
4098:
4093:, p. 3.
4091:Schafer 1995
4076:Schafer 1995
4061:Schafer 1995
4046:Albert 1948a
4033:, p. 1.
4031:Schafer 1995
4026:
4021:, Chapter 1.
4019:Schafer 1995
4014:
3970:
3882:
3784:
3781:
3779:
3773:
3769:
3765:
3761:
3754:
3750:
3746:
3742:
3735:
3731:
3727:
3723:
3716:
3712:
3705:
3701:
3697:
3693:
3687:
3682:
3678:
3674:
3669:
3664:
3660:
3656:
3652:
3648:
3646:
3552:
3548:
3544:
3540:
3538:
3524:
3519:
3506:
3501:
3496:
3491:
3486:
3484:
3393:
3389:
3383:
3381:
3365:
3361:
3357:
3352:
3347:
3343:
3339:
3335:
3330:
3325:
3321:
3317:
3315:
3303:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3268:
3255:zero divisor
3251:
3212:
3178:
3168:
3162:
3158:
3146:
3142:
3138:
3134:
3130:
3126:
3105:real numbers
3091:vector space
3065:
3051:
3016:
3012:
3008:
3004:
2980:
2972:
2965:
2961:
2952:Algebras of
2947:Lie algebras
2936:
2803:
2801:
2706:
2702:
2698:
2696:
2686:
2684:
2671:
2663:
2661:
2633:
2609:
2567:
2562:
2540:
2532:
2528:
2524:
2520:
2446:
2442:
2438:
2436:
2419:
2415:
2406:
2402:
2396:
2392:
2387:
2384:
2380:
2375:
2366:
2362:
2356:
2352:
2347:
2344:
2340:
2335:
2326:
2322:
2316:
2312:
2303:
2299:
2293:
2289:
2283:
2279:
2273:
2269:
2265:
2259:
2255:
2251:
2245:
2241:
2237:
2229:
2225:
2218:
2214:
2205:
2201:
2195:
2191:
2187:
2178:
2167:
2164:
2155:
2151:
2144:
2140:
2131:
2124:
2118:
2114:
2108:
2104:
2098:
2094:
2085:
2078:
2072:
2068:
2062:
2058:
2052:
2048:
2042:
2038:
2032:
2028:
2022:
2018:
2012:
2008:
2000:
1996:
1989:
1985:
1981:
1974:
1970:
1957:
1944:
1933:
1928:
1922:
1907:
1903:
1896:
1890:
1883:
1875:
1871:
1864:
1858:
1851:
1840:
1827:
1819:
1810:
1805:
1801:
1798:
1789:
1785:
1782:
1773:
1769:
1758:
1755:
1751:
1748:
1741:
1728:
1720:
1708:
1703:
1699:
1696:
1692:
1689:
1680:
1676:
1673:
1664:
1660:
1649:
1645:
1642:
1635:
1622:
1618:
1614:
1610:
1596:
1592:
1588:
1584:
1576:
1572:
1568:
1564:
1550:
1546:
1542:
1538:
1530:
1526:
1522:
1518:
1515:
1501:
1497:
1493:
1489:
1485:
1481:
1474:
1470:
1466:
1462:
1458:
1454:
1441:
1437:
1425:
1421:
1408:
1404:
1400:
1396:
1383:
1379:
1373:
1369:
1365:
1348:
1344:
1338:
1335:
1331:
1325:
1321:
1298:
1285:
1240:Hopf algebra
1214:
1178:
1171:Vector space
1136:
1076:
1005:Group theory
1003:
968: /
934:
886:
882:
876:
871:
866:
861:
857:
851:
847:
843:
835:
831:
829:
825:Lie algebras
816:
812:
808:
804:
800:
793:
787:
785:
775:
773:
768:
764:
760:
756:
752:
748:
744:
740:
725:Lie algebras
720:
716:
712:
704:
700:
697:vector space
692:
685:
666:
662:
660:
620:
606:
605:
601:Free algebra
599:
598:
592:
591:
560:
503:
465:
428:
397:
396:
376:Finite field
325:
272:Finite field
258:
257:
184:Initial ring
144:
143:
117:
116:
59:
4841:: 705–710.
4628:: 318–328.
4568:: 552–593.
4448:Albert 2003
4412:Kurosh 1947
3529:Lie algebra
3515:Lie bracket
3316:An algebra
3176:quaternions
3113:quaternions
2679:= = = {0}
2604:= − = − = −
2416:Commutative
2336:Commutative
2266:Commutative
2224:Similarly,
2192:commutative
2039:Alternative
2019:Commutative
2009:Alternative
1997:alternative
1975:alternative
1971:Associative
1559:Alternative
1417:Commutative
1391:Associative
1329:and either
1225:Lie algebra
1210:Associative
1114:Total order
1104:Semilattice
1078:Ring theory
827:never are.
695:if it is a
679:associative
581:Simple ring
292:Jordan ring
166:Graded ring
48:Ring theory
5084:0145.25601
5027:0867.53002
4998:0841.17001
4930:1044.17001
4878:0041.16803
4816:1072.17513
4783:0955.16001
4608:0033.15402
4544:0023.19901
4509:References
4436:Okubo 2005
4400:Rowen 2008
4202:Okubo 2005
4190:Okubo 2005
4154:Okubo 2005
4130:Okubo 2005
4103:Okubo 2005
3511:commutator
3385:derivation
3248:Properties
3241:See also:
3150:octonions.
2994:commutator
2674:such that
2619:Flexible:
2439:associator
2433:Associator
2427:Associator
2329:) ∉ {2,3,5
1856:elements:
1289:identities
673:where the
587:Commutator
346:GCD domain
5095:(1982) .
5068:. Dover.
5062:(1995) .
4664:CRC Press
4584:0002-9947
4554:(1948a).
4518:(2003) .
4006:Citations
3931:−
3861:⋅
3852:⋅
3843:−
3834:⋅
3825:⋅
3816:↦
3620:↦
3584:↦
3517:gives Der
3452:⋅
3440:⋅
3416:⋅
3392:is a map
3293:is just (
3273:on a set
3259:sedenions
3197:sedenions
3186:octonions
3117:octonions
3041:sedenions
3027:octonions
2780:∈
2774:∀
2741:∈
2514:given by
2499:→
2493:×
2487:×
2475:⋅
2469:⋅
2463:⋅
1920:of index
1838:of index
1836:Nilpotent
1235:Bialgebra
1041:Near-ring
998:Lie group
966:Semigroup
821:octonions
733:octonions
528:∞
306:Semifield
5120:Category
4888:(2004).
4757:(1998).
4712:(1968).
3988:See also
3661:centroid
3509:). The
3499:) in End
3364:and the
2975:) or an
2928:Examples
2282:implies
2256:flexible
2246:flexible
2228:implies
2217:implies
2130:implies
2117:implies
2107:implies
2097:implies
2084:implies
2051:implies
2049:Flexible
2041:implies
2033:flexible
2031:implies
2023:flexible
2021:implies
2011:implies
2001:flexible
1999:implies
1990:flexible
1973:implies
1942:so that
1825:for all
1766:so that
1726:for all
1657:so that
1605:Flexible
1363:so that
1071:Lie ring
1036:Semiring
856:and the
815:for all
709:bilinear
669:) is an
300:Semiring
286:Lie ring
68:Subrings
4922:2014924
4870:0020986
4738:0251099
4642:1969457
4600:0027750
4592:1990399
3679:central
3360:), the
3304:Kurosh
3157:(where
3017:special
2968:or the
2664:nucleus
2445:is the
1962:of any
1813:above).
1768:0 <
1711:below).
1659:0 <
1202:Algebra
1194:Algebra
1099:Lattice
1090:Lattice
879:-module
794:unitary
502:Prüfer
104:•
5105:
5082:
5072:
5048:
5025:
5015:
4996:
4986:
4934:Errata
4928:
4920:
4910:
4876:
4868:
4864:(62).
4814:
4804:
4781:
4771:
4736:
4726:
4698:
4670:
4640:
4606:
4598:
4590:
4582:
4542:
4532:
3953:
3867:
3629:
3599:
3596:
3467:
3306:proved
3085:, and
2755:
2747:
2699:center
2693:Center
2313:Unital
2290:Unital
2238:Unital
2152:Unital
1995:Thus,
1988:, and
1357:Unital
1230:Graded
1161:Module
1152:Module
1051:Domain
970:Monoid
789:unital
731:, the
154:Module
127:Kernel
4837:(5).
4654:(PDF)
4638:JSTOR
4588:JSTOR
4482:Notes
3338:) of
3037:GF(2)
2956:on a
2636:) ≠ 2
2632:char(
2612:) ≠ 2
2608:char(
2409:) = 2
2405:char(
2369:) = 0
2365:char(
2325:char(
2306:) ≠ 3
2302:char(
2208:) ≠ 2
2204:char(
2181:) ≤ 3
2172:GF(2)
2137:with
1772:<
1663:<
1504:) = 0
1196:-like
1154:-like
1092:-like
1061:Field
1019:-like
993:Magma
961:Group
955:-like
953:Group
874:: An
846:as a
803:with
699:over
690:field
506:-ring
370:Field
266:Field
74:Ideal
61:Rings
5103:ISBN
5070:ISBN
5046:ISBN
5013:ISBN
4984:ISBN
4908:ISBN
4802:ISBN
4769:ISBN
4724:ISBN
4696:ISBN
4668:ISBN
4580:ISSN
4530:ISBN
3780:The
3768:) =
3647:The
3543:and
3269:The
3218:The
3195:the
3188:(an
3184:the
3174:the
3153:The
3141:and
3058:for
3046:The
2960:(if
2898:are
2804:C(A)
2697:The
2662:The
2576:and
2568:Let
2437:The
2418:and
2378:and
2338:and
2315:and
2292:and
2268:and
2254:and
2240:and
2190:and
2177:dim(
2154:and
2139:2 ≤
2071:and
2061:and
1958:For
1931:and
1496:) +
1488:) +
1307:and
1299:Let
1026:Ring
1017:Ring
759:and
747:), (
665:(or
5080:Zbl
5023:Zbl
4994:Zbl
4976:doi
4953:doi
4926:Zbl
4900:doi
4874:Zbl
4843:doi
4812:Zbl
4779:Zbl
4630:doi
4604:Zbl
4570:doi
4540:Zbl
3663:of
3655:of
3651:or
3388:on
3161:is
3007:= (
3005:x*y
2964:is
2701:of
2670:in
2656:= 0
2649:= 0
2647:or
2645:= 0
2628:= −
2621:= 0
2597:= 0
2593:= 0
2586:= 0
2519:= (
2441:on
2403:If
2363:If
2331:}:
2323:If
2300:If
2202:If
2175:or
2165:If
1947:≠ 0
1936:= 0
1925:≥ 2
1918:Nil
1912:≠ 0
1879:= 0
1843:≥ 2
1830:≥ 2
1744:≥ 2
1731:≥ 2
1638:≥ 2
1535:or
1479:or
1477:= 0
1469:+ (
1461:+ (
1440:= −
1031:Rng
864:".
842:of
792:or
5122::
5078:.
5044:.
5021:.
4992:.
4982:.
4974:.
4949:17
4947:.
4932:.
4924:.
4918:MR
4916:.
4906:.
4898:.
4872:.
4866:MR
4862:20
4860:.
4833:.
4827:.
4810:.
4800:.
4777:.
4767:.
4753:;
4749:;
4734:MR
4732:.
4722:.
4694:.
4666:.
4636:.
4626:50
4624:.
4602:.
4596:MR
4594:.
4586:.
4578:.
4566:64
4564:.
4558:.
4538:.
4528:.
4490:^
4368:^
4209:^
4122:^
4083:^
4068:^
4053:^
4038:^
3757:);
3738:);
3708:);
3555::
3531:.
3382:A
3368:.
3297:)(
3289:,
3203:).
3192:);
3135:sx
3133:+
3131:re
3129:=
3127:xx
3081:,
3013:yx
3009:xy
2983:);
2689:.
2623:.
2572:,
2565:.
2533:yz
2527:−
2521:xy
2411::
2383:=
2371::
2343:=
2308::
2221:.
2210::
2183::
2170:≠
2143:≤
1984:,
1966::
1927::
1910:−1
1886:−1
1854:−1
1806:xx
1804:=
1790:xx
1788:=
1777:.
1754:=
1746::
1704:xx
1702:=
1695:=
1681:xx
1679:=
1668:.
1648:=
1640::
1623:yx
1617:=
1611:xy
1607::
1597:xx
1591:=
1585:yx
1577:xy
1571:=
1565:xx
1561::
1551:yx
1545:=
1539:xy
1531:yx
1525:=
1512::
1502:xy
1494:zx
1486:yz
1471:zx
1463:yz
1455:xy
1451::
1442:yx
1438:xy
1435::
1426:yx
1424:=
1422:xy
1419::
1409:yz
1403:=
1397:xy
1393::
1382:≝
1374:xe
1372:=
1368:=
1366:ex
1349:xx
1347:≝
1334:≝
1324:≝
1303:,
937:.
813:xe
811:=
807:=
805:ex
783:.
769:cd
755:))
753:bc
745:cd
743:)(
741:ab
727:,
719:→
715:×
661:A
614:•
585:•
579:•
573:•
567:•
500:•
463:•
426:•
420:•
411:•
405:•
388:•
382:•
374:•
368:•
362:•
356:•
350:•
344:•
338:•
332:•
304:•
298:•
290:•
284:•
278:•
270:•
264:•
209:•
182:•
176:•
170:•
164:•
158:•
152:•
137:•
131:•
125:•
110:•
98:•
90:•
84:•
78:•
72:•
66:•
5111:.
5086:.
5054:.
5029:.
5000:.
4978::
4959:.
4955::
4936:.
4902::
4880:.
4851:.
4845::
4835:6
4818:.
4785:.
4740:.
4704:.
4676:.
4644:.
4632::
4610:.
4572::
4546:.
4501:.
3956:.
3950:)
3945:2
3941:a
3937:(
3934:L
3928:)
3925:a
3922:(
3917:2
3913:L
3909:2
3906:=
3903:)
3900:a
3897:(
3894:Q
3879:,
3864:x
3858:)
3855:a
3849:a
3846:(
3840:)
3837:x
3831:a
3828:(
3822:a
3819:2
3813:x
3810::
3807:)
3804:a
3801:(
3798:Q
3785:Q
3774:a
3772:(
3770:L
3766:a
3764:(
3762:L
3755:a
3753:(
3751:R
3747:a
3745:(
3743:L
3736:a
3734:(
3732:R
3728:a
3726:(
3724:L
3719:;
3717:R
3713:L
3706:a
3704:(
3702:R
3698:a
3696:(
3694:L
3683:K
3675:A
3673:(
3670:K
3665:A
3657:A
3632:.
3626:a
3623:x
3617:x
3614::
3611:)
3608:a
3605:(
3602:R
3593:;
3590:x
3587:a
3581:x
3578::
3575:)
3572:a
3569:(
3566:L
3553:A
3549:a
3545:R
3541:L
3525:A
3523:(
3520:K
3507:A
3505:(
3502:K
3497:A
3495:(
3492:K
3487:A
3470:.
3464:)
3461:y
3458:(
3455:D
3449:x
3446:+
3443:y
3437:)
3434:x
3431:(
3428:D
3425:=
3422:)
3419:y
3413:x
3410:(
3407:D
3394:D
3390:A
3358:A
3356:(
3353:K
3348:A
3344:A
3340:K
3336:A
3334:(
3331:K
3326:K
3322:K
3318:A
3299:v
3295:u
3291:v
3287:u
3283:X
3279:K
3275:X
3213:R
3179:H
3169:C
3163:R
3159:K
3147:e
3143:s
3139:r
3097:.
3062:.
3052:R
3043:.
3019:.
3011:+
2981:K
2973:C
2966:R
2962:K
2937:R
2912:}
2909:0
2906:{
2886:)
2883:]
2880:n
2877:,
2874:A
2871:,
2868:A
2865:[
2862:,
2859:]
2856:A
2853:,
2850:n
2847:,
2844:A
2841:[
2838:,
2835:]
2832:A
2829:,
2826:A
2823:,
2820:n
2817:[
2814:(
2787:}
2783:A
2777:r
2770:n
2767:r
2764:=
2761:r
2758:n
2751:|
2744:A
2738:n
2735:{
2732:=
2729:)
2726:A
2723:(
2720:C
2707:A
2703:A
2687:A
2681:.
2672:A
2668:n
2658:.
2638:.
2634:K
2614:.
2610:K
2588:.
2578:z
2574:y
2570:x
2563:A
2549:A
2537:.
2535:)
2531:(
2529:x
2525:z
2523:)
2502:A
2496:A
2490:A
2484:A
2481::
2478:]
2472:,
2466:,
2460:[
2449:-
2447:K
2443:A
2407:K
2399:.
2388:x
2385:x
2381:x
2367:K
2359:.
2348:x
2345:x
2341:x
2327:K
2304:K
2286:.
2276:.
2262:.
2248:.
2232:.
2206:K
2198:.
2179:A
2168:K
2158:n
2148:.
2145:n
2141:N
2134:N
2127:n
2121:.
2111:.
2101:.
2091:.
2087:n
2080:n
2055:.
2045:.
2035:.
2025:.
2015:.
2003:.
1977:.
1960:K
1949:.
1945:y
1940:y
1934:x
1923:n
1908:n
1904:y
1902:…
1900:2
1897:y
1894:1
1891:y
1884:n
1876:n
1872:x
1870:…
1868:2
1865:x
1862:1
1859:x
1852:n
1847:n
1841:n
1832:.
1828:n
1821:n
1802:x
1799:x
1793:.
1786:x
1783:x
1774:n
1770:k
1764:k
1759:x
1756:x
1752:x
1749:x
1742:n
1737:n
1733:.
1729:n
1722:n
1700:x
1697:x
1693:x
1690:x
1684:.
1677:x
1674:x
1665:n
1661:k
1655:k
1650:x
1646:x
1643:x
1636:n
1631:n
1627:.
1625:)
1621:(
1619:x
1615:x
1613:)
1609:(
1599:)
1595:(
1593:y
1589:x
1587:)
1583:(
1579:)
1575:(
1573:x
1569:y
1567:)
1563:(
1553:)
1549:(
1547:x
1543:x
1541:)
1537:(
1533:)
1529:(
1527:x
1523:x
1521:)
1519:y
1516:x
1514:(
1500:(
1498:z
1492:(
1490:y
1484:(
1482:x
1475:y
1473:)
1467:x
1465:)
1459:z
1457:)
1453:(
1445:.
1429:.
1413:.
1411:)
1407:(
1405:x
1401:z
1399:)
1395:(
1387:.
1384:e
1380:x
1370:x
1361:e
1345:x
1339:x
1336:x
1332:x
1326:x
1322:x
1317:K
1313:A
1309:z
1305:y
1301:x
1270:e
1263:t
1256:v
920:Z
898:Z
887:R
883:R
877:R
872:R
862:A
848:K
844:A
838:-
836:K
832:A
817:x
809:x
801:e
767:(
765:b
763:(
761:a
757:d
751:(
749:a
721:A
717:A
713:A
707:-
705:K
701:K
693:K
686:A
650:e
643:t
636:v
533:)
524:p
520:(
516:Z
504:p
484:p
479:Q
466:p
447:p
442:Z
429:p
415:n
240:Z
236:1
232:/
227:Z
223:=
220:0
194:Z
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.