3761:
2135:
6216:
5664:
is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
5569:
1454:
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra
1940:
6020:
5779:
6032:
4421:
1591:
5200:
of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by
5889:
4495:
5388:
4228:
5647:
5399:
2896:
6222:
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a
3266:
5207:, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a
4589:
is the intersection of all (two-sided) maximal ideals (in contrast, in general, a
Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
6234:
Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.
2130:{\displaystyle {\begin{aligned}\varphi (r\cdot x)&=r\cdot \varphi (x)\\\varphi (x+y)&=\varphi (x)+\varphi (y)\\\varphi (xy)&=\varphi (x)\varphi (y)\\\varphi (1)&=1\end{aligned}}}
1945:
1279:
5660:. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A
253:
2197:
or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
545:
4142:
3189:
1517:
5905:
4320:
3313:
3133:
3064:
1688:
498:
461:
4169:
3341:
3092:
3023:
207:
17:
6211:{\displaystyle \rho (x)\rho (y)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+\sigma (x)\otimes \tau (y)+\sigma (y)\otimes \tau (x)+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)}
2932:
2388:
form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
5685:
4267:
4519:
1529:
4329:
3742:
is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see
4287:
4107:
3153:
4692:
to be the
Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of
5790:
4434:
6299:
5286:
1132:
4082:
is
Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field
3639:
is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.
2563:, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.
650:
6550:
4615:
3738:. The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual
3402:. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of
5564:{\displaystyle \rho (kx)=\sigma (kx)\otimes \tau (kx)=k\sigma (x)\otimes k\tau (x)=k^{2}(\sigma (x)\otimes \tau (x))=k^{2}\rho (x)}
4180:
6562:
5605:
2829:
1358:(here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by
6661:
6541:
3196:
3513:
107:
5584:. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism
2657:(or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
1207:
790:
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called
6468:
4935:
1125:
6490:
5203:
3792:
1865:
643:
595:
3676:
may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra.
2720:
217:
6480:
5899:, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
4974:) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using
6512:
2675:
512:
4114:
1447:; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the
794:
for clarification. In some areas of mathematics this assumption is not made, and we will call such structures
6015:{\displaystyle \rho (xy)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)}
2182:
1197:
in such a way that the two additions (the ring addition and the module addition) are the same operation, and
1118:
3158:
798:
associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
4575:
3349:
2574:
1477:
636:
503:
6322:
is a full matrix ring in interesting cases and it is more conventional to let matrices act from the right.
4292:
3772:
6693:
2820:
987:
353:
3277:
3097:
3028:
1649:
5674:
4585:
is
Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of
3543:
1869:
1444:
113:
474:
437:
128:
4147:
3322:
3073:
3004:
1460:
5774:{\displaystyle x\mapsto \rho (x)=\sigma (x)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)}
4888:
having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism
3316:
2242:
1423:
588:
391:
341:
31:
2712:
form a real or complex associative algebra; here the functions are added and multiplied pointwise.
190:
3555:
2553:
2335:-algebra under matrix addition and multiplication. This coincides with the previous example when
1305:
1078:
400:
134:
93:
4239:
2901:
5017:
4570:
is a semisimple algebra, then it is a finite product of matrix algebras (over various division
4049:
2813:
2608:. The construction is the starting point for the application to the study of (discrete) groups.
2145:
1850:
557:
408:
359:
140:
1778:. In the commutative case, one can consider the category whose objects are ring homomorphisms
1586:{\displaystyle m\circ ({\operatorname {id} }\otimes m)=m\circ (m\otimes \operatorname {id} ).}
6637:
5023:
4813:-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g.,
4504:
2794:
1459:. For example, the associativity can be expressed as follows. By the universal property of a
1198:
769:
714:
4416:{\displaystyle \Gamma =\operatorname {Gal} (k_{s}/k)=\varprojlim \operatorname {Gal} (k'/k)}
6671:
6589:
6285:
4530:
3363:
2791:
2577:
of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
2324:
2194:
1887:
1628:
1324:
1191:
1065:
1057:
1029:
1024:
1015:
972:
914:
746:
734:
702:
281:
155:
6522:
2694:
form an associative algebra (using composition of operators as multiplication); this is a
8:
6280:
6252:
4979:
4796:
4709:
3421:
3067:
2747:
1901:
1874:
1448:
1427:
1083:
924:
824:
816:
807:
710:
679:
563:
371:
322:
267:
147:
75:
43:
6593:
6262:
Another example is the vector space of continuous periodic functions, together with the
30:
This article is about an algebraic structure. For other uses of the term "algebra", see
6613:
6579:
6554:
4975:
4272:
4092:
3138:
2823:
is an associative algebra together with a grading and a differential. For example, the
2560:
1184:
889:
880:
838:
726:
687:
576:
62:
4086:. Now, a reduced Artinian local ring is a field and thus the following are equivalent
6688:
6657:
6617:
6605:
6570:
Tjin, T. (October 10, 1992). "An introduction to quantized Lie groups and algebras".
6537:
6508:
6486:
6290:
6237:
One example of a non-unital associative algebra is given by the set of all functions
4898:
identifying the scalar multiples of the multiplicative identity. If the bilinear map
4693:
4013:
3929:
3870:
3520:-algebras in a manner similar to the free product of groups. The free product is the
2805:
2787:
2620:
2469:
2271:
1907:
1602:
1436:
1316:
694:
617:
414:
179:
120:
6649:
6597:
6518:
6275:
5679:
One can try to be more clever in defining a tensor product. Consider, for example,
5653:
4732:
3499:
2773:
2465:
2167:
1838:
1606:
1407:
1155:
909:
784:
675:
623:
609:
423:
365:
328:
101:
87:
934:
6667:
6645:
6531:
4424:
3626:
2956:
is a commutative associative algebra over a field together with a structure of a
2953:
2824:
2760:
2678:
2624:
2616:
2392:
1001:
995:
982:
962:
953:
919:
856:
385:
335:
173:
5884:{\displaystyle \rho (x)(v\otimes w)=(\sigma (x)v)\otimes w+v\otimes (\tau (x)w)}
5197:
2809:
2716:
2695:
2654:
2453:
2374:
1846:
1301:
1043:
795:
429:
6653:
6601:
6461:
4490:{\displaystyle A\mapsto X_{A}=\{k{\text{-algebra homomorphisms }}A\to k_{s}\}}
6682:
6626:
6609:
4067:
3625:-algebra to its underlying ring (forgetting the module structure). See also:
3460:
3357:
2798:
2357:
2231:
1854:
1775:
929:
894:
851:
761:
570:
466:
81:
1300:
in the algebra. (This definition implies that the algebra, being a ring, is
801:
Every ring is an associative algebra over its center and over the integers.
6294:
5661:
5208:
3636:
3618:
2754:
2668:
2632:
2481:
2430:
1893:
1103:
1034:
868:
738:
717:(the multiplication by the image of the ring homomorphism of an element of
602:
377:
273:
5383:{\displaystyle \rho (x)(v\otimes w)=(\sigma (x)(v))\otimes (\tau (x)(w)).}
3352:. The dual of such an algebra turns out to be an associative algebra (see
1745:
If a ring is commutative then it equals its center, so that a commutative
6263:
6223:
5212:
4994:
4697:
3356:) and is, philosophically speaking, the (quantized) coordinate ring of a
2957:
2521:
consists of infinitesimal elements; i.e., the multiplication is given as
2378:
2340:
2159:
1093:
1088:
977:
967:
941:
780:
664:
582:
293:
167:
49:
6536:, Colloquium Publications, vol. 37, American Mathematical Society,
3760:
3691:
is an associative algebra, but it also comes with the co-multiplication
6584:
4497:
is an anti-equivalence of the category of finite-dimensional separable
2601:
2385:
843:
347:
5008:. This is vaguely related to the notion of coalgebra discussed above.
6500:
5657:
5656:
if it satisfies certain axioms. The resulting structure is called a
4987:
4857:
3521:
3395:
1098:
904:
861:
829:
307:
212:
5157:
are two representations, then there is a (canonical) representation
4913:
3850:
2777:
2702:
2193:
The most basic example is a ring itself; it is an algebra over its
899:
301:
287:
2600:-algebra with the convolution as multiplication. It is called the
2144:-algebras together with algebra homomorphisms between them form a
1864:
How to weaken the commutativity assumption is a subject matter of
729:; the addition and scalar multiplication operations together give
6644:, Graduate Texts in Mathematics, vol. 66, Berlin, New York:
5005:
3391:
2781:
2473:
2222:
is determined by the fact that it must send 1 to the identity in
1418:
The definition is equivalent to saying that a unital associative
1392:
185:
69:
1413:
833:
5393:
However, such a map would not be linear, since one would have
5280:
according to how it acts on the product vector space, so that
3672:
need not have a structure of an associative algebra. However,
3135:
has a structure of an associative algebra with multiplication
2960:
so that the Lie bracket {,} satisfies the
Leibniz rule; i.e.,
6376:
5049:
to the endomorphism algebra of some vector space (or module)
4983:
6424:
4223:{\displaystyle A\otimes {\overline {k}}={\overline {k}}^{n}}
721:). The addition and multiplication operations together give
6436:
5784:
so that the action on the tensor product space is given by
1443:-modules). By definition, a ring is a monoid object in the
6412:
6400:
5642:{\displaystyle \rho =(\sigma \otimes \tau )\circ \Delta .}
3683:
to be the ring of continuous functions on a compact group
2891:{\textstyle \Omega (M)=\bigoplus _{p=0}^{n}\Omega ^{p}(M)}
783:, or, equivalently, an associative algebra that is also a
779:
is an associative algebra for which the multiplication is
5253:. One might try to form a tensor product representation
4501:-algebras to the category of finite sets with continuous
6364:
4718:
be a
Noetherian integral domain with field of fractions
3480:-algebra. It follows that any ring homomorphic image of
2230:-algebras are equivalent concepts, in the same way that
3660:
is in particular a module, we can take the dual module
1728:, the same formula in turn defines a ring homomorphism
6169:
6092:
5973:
5956:
5744:
5727:
3261:{\displaystyle f*g=fg-{\frac {1}{2}}\{f,g\}u+\cdots ,}
2904:
2832:
2708:, the continuous real- or complex-valued functions on
1796:-algebras, and whose morphisms are ring homomorphisms
6035:
5908:
5793:
5688:
5608:
5402:
5289:
4507:
4437:
4332:
4295:
4275:
4242:
4183:
4150:
4117:
4095:
3325:
3280:
3199:
3161:
3141:
3100:
3076:
3031:
3007:
2588:
is a commutative ring, the set of all functions from
2464:-algebra. The same is true for quotients such as the
1943:
1749:-algebra can be defined simply as a commutative ring
1652:
1532:
1480:
1335:
is such a homomorphism, the scalar multiplication is
1210:
515:
477:
440:
220:
193:
6388:
5599:, and defining the tensor product representation as
2441:
is an algebra of "polynomials" with coefficients in
6350:
can be used to construct a section of a surjection.
4541:is a (full) matrix algebra over a division algebra
3353:
2445:and noncommuting indeterminates taken from the set
1274:{\displaystyle r\cdot (xy)=(r\cdot x)y=x(r\cdot y)}
6210:
6014:
5883:
5773:
5641:
5563:
5382:
4843:is an order that is maximal among all the orders.
4513:
4489:
4415:
4314:
4281:
4261:
4222:
4163:
4136:
4101:
3793:Non-associative_algebra Β§ Associated_algebras
3652:be an associative algebra over a commutative ring
3335:
3307:
3260:
3183:
3147:
3127:
3086:
3058:
3017:
2926:
2890:
2492:-module (forgetting the multiplicative structure).
2377:form a 2-dimensional commutative algebra over the
2129:
1682:
1585:
1511:
1273:
539:
492:
455:
247:
201:
5196:. However, there is no natural way of defining a
5061:preserves the multiplicative operation (that is,
4533:is a (full) matrix ring over a division ring, if
3643:
3301:
3291:
3121:
3111:
3052:
3042:
6680:
5218:
4938:), then we can view an associative algebra over
3506:-algebra with the obvious scalar multiplication.
2404:-algebra. In fact, this is the free commutative
5668:
4836:but not an order (since it is not an algebra).
1523:The associativity then refers to the identity:
1753:together with a commutative ring homomorphism
6300:Deligne's conjecture on Hochschild cohomology
4058:be a finite-dimensional algebra over a field
4043:
2766:
1414:As a monoid object in the category of modules
1126:
644:
6478:
5112:) (that is, to the identity endomorphism of
4950:endowed with two morphisms (one of the form
4629:is at most one, then the natural surjection
4484:
4457:
3243:
3231:
2214:-algebra. The unique ring homomorphism from
1376:
745:. In this article we will also use the term
248:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
27:Ring that is also a vector space or a module
18:Enveloping algebra of an associative algebra
6563:Quantum Groups: an entrΓ©e to modern algebra
5223:Consider, for example, two representations
3546:for more details. Given a commutative ring
1841:of the category of commutative rings under
713:with an addition, a multiplication, and a
6636:
6418:
6406:
6331:To see the equivalence, note a section of
1596:
1133:
1119:
651:
637:
6583:
6572:International Journal of Modern Physics A
5057:being an algebra homomorphism means that
4596:states: for a finite-dimensional algebra
1774:appearing in the above is often called a
1406:is an associative algebra that is also a
540:{\displaystyle \mathbb {Z} (p^{\infty })}
517:
480:
443:
241:
228:
195:
6551:A Synopsis of Linear Associative Algebra
6529:
4936:universal property of the tensor product
4912:is reinterpreted as a linear map (i.e.,
4137:{\displaystyle A\otimes {\overline {k}}}
4040:, measures the failure of separability.
2567:
2327:with coefficients in a commutative ring
1881:
2946:
2797:are associative algebras considered in
1467:-bilinear map) corresponds to a unique
14:
6681:
6229:
4703:
4524:
3879:be an algebra over a commutative ring
3184:{\displaystyle f,g\in {\mathfrak {a}}}
2166:-algebras can be characterized as the
6566:, an overview of index-free notation.
6474:from the original on October 9, 2022.
6459:
6442:
6026:But, in general, this does not equal
3751:
3354:Β§ Dual of an associative algebra
2552:. The notion is sometimes called the
1512:{\displaystyle m:A\otimes _{R}A\to A}
1311:Equivalently, an associative algebra
1304:, since rings are supposed to have a
752:to mean an associative algebra over
6642:Introduction to affine group schemes
6624:
6569:
6499:
6430:
6394:
6382:
6370:
4993:There is also an abstract notion of
4315:{\displaystyle A\to {\overline {k}}}
3864:
3755:
3744:
2460:-module is naturally an associative
1853:of this category to the category of
1601:An associative algebra amounts to a
108:Free product of associative algebras
4846:
4073:
3328:
3283:
3176:
3103:
3079:
3034:
3010:
2942:, is a differential graded algebra.
2808:and its subalgebras, including the
24:
5652:Such a homomorphism Ξ is called a
5633:
5011:
4986:; this defines the structure of a
4574:-algebras), the fact known as the
4508:
4333:
3494:The direct product of a family of
3308:{\displaystyle {\mathfrak {a}}\!]}
3128:{\displaystyle {\mathfrak {a}}\!]}
3059:{\displaystyle {\mathfrak {a}}\!]}
2906:
2870:
2833:
1683:{\displaystyle r\cdot x=\eta (r)x}
529:
25:
6705:
6318:Editorial note: as it turns out,
5204:tensor product of representations
4604:, if the projective dimension of
3571:can be given the structure of an
2480:-module to its tensor algebra is
2364:form an associative algebra over
1866:noncommutative algebraic geometry
596:Noncommutative algebraic geometry
6505:Further Algebra and Applications
3759:
3498:-algebras is the ring-theoretic
3369:
1932:associative algebra homomorphism
1849:functor Spec then determines an
1742:whose image lies in the center.
756:. A standard first example of a
493:{\displaystyle \mathbb {Q} _{p}}
456:{\displaystyle \mathbb {Z} _{p}}
4978:by reversing all arrows in the
4427:of finite Galois extensions of
4164:{\displaystyle {\overline {k}}}
3542:-algebra in a natural way. See
3336:{\displaystyle {\mathfrak {a}}}
3087:{\displaystyle {\mathfrak {a}}}
3018:{\displaystyle {\mathfrak {a}}}
1609:. Indeed, starting with a ring
6627:"notes on quasi-free algebras"
6325:
6312:
6293:, a sort of an algebra over a
6205:
6199:
6193:
6187:
6161:
6155:
6146:
6140:
6131:
6125:
6116:
6110:
6084:
6078:
6072:
6066:
6057:
6051:
6045:
6039:
6009:
6003:
5997:
5991:
5948:
5942:
5936:
5930:
5921:
5912:
5895:This map is clearly linear in
5878:
5872:
5866:
5860:
5842:
5836:
5830:
5824:
5818:
5806:
5803:
5797:
5768:
5762:
5719:
5713:
5704:
5698:
5692:
5627:
5615:
5558:
5552:
5533:
5530:
5524:
5515:
5509:
5503:
5487:
5481:
5469:
5463:
5451:
5442:
5433:
5424:
5415:
5406:
5374:
5371:
5365:
5362:
5356:
5350:
5344:
5341:
5335:
5332:
5326:
5320:
5314:
5302:
5299:
5293:
5176:of the tensor product algebra
4471:
4441:
4410:
4391:
4366:
4345:
4299:
3857:is exactly a left module over
3644:Dual of an associative algebra
3621:to the functor which sends an
3302:
3298:
3292:
3288:
3122:
3118:
3112:
3108:
3053:
3049:
3043:
3039:
2921:
2915:
2885:
2879:
2842:
2836:
2472:. Categorically speaking, the
2110:
2104:
2094:
2088:
2082:
2076:
2063:
2054:
2044:
2038:
2029:
2023:
2010:
1998:
1988:
1982:
1963:
1951:
1674:
1668:
1577:
1565:
1553:
1539:
1503:
1377:Β§ From ring homomorphisms
1268:
1256:
1244:
1232:
1226:
1217:
534:
521:
13:
1:
6452:
5219:Motivation for a Hopf algebra
4851:
4171:is some algebraic closure of
3797:Given an associative algebra
3420:-algebra. Any ring-theoretic
2484:to the functor that sends an
2183:category of commutative rings
1383:Every ring is an associative
1145:
6357:
5669:Motivation for a Lie algebra
4874:endowed with a bilinear map
4862:An associative algebra over
4594:Wedderburn principal theorem
4465:-algebra homomorphisms
4307:
4209:
4195:
4156:
4129:
3350:quantized enveloping algebra
3025:, consider the vector space
2596:with finite support form an
2575:universal enveloping algebra
2503:, the direct sum of modules
2360:with entries from the field
202:{\displaystyle \mathbb {Z} }
7:
6557:Historical Math Monographs.
6269:
5030:is an algebra homomorphism
2927:{\textstyle \Omega ^{p}(M)}
2821:differential graded algebra
2661:
2488:-algebra to its underlying
2188:
792:unital associative algebras
354:Unique factorization domain
10:
6710:
6507:(2nd ed.). Springer.
5675:Lie algebra representation
5672:
5015:
4982:that describe the algebra
4855:
4722:(for example, they can be
4707:
4688:is an isomorphism. Taking
4537:is a simple algebra, then
4262:{\displaystyle \dim _{k}A}
4047:
4044:Finite-dimensional algebra
3932:if the multiplication map
3868:
3790:
3598:. The functor which sends
3544:tensor product of algebras
3534:The tensor product of two
2767:Geometry and combinatorics
2721:filtered probability space
2349:In particular, the square
2259:)-algebra in the same way.
2200:
1885:
1870:derived algebraic geometry
1463:, the multiplication (the
1445:category of abelian groups
1315:is a ring together with a
114:Tensor product of algebras
29:
6654:10.1007/978-1-4612-6217-6
6602:10.1142/S0217751X92002805
6549:James Byrnie Shaw (1907)
6530:Jacobson, Nathan (1956),
5215:, as demonstrated below.
4028:-projective dimension of
3476:-module and, in fact, an
2934:consists of differential
2339:is a finitely-generated,
1461:tensor product of modules
6305:
6259:nears infinity is zero.
4790:be a finite-dimensional
4747:is a finitely generated
4739:in a finite-dimensional
4576:ArtinβWedderburn theorem
4012:is separable if it is a
3846:, depending on authors.
3801:over a commutative ring
3317:deformation quantization
3001:Given a Poisson algebra
2499:over a commutative ring
2238:-modules are equivalent.
1627:whose image lies in the
1613:and a ring homomorphism
1605:whose image lies in the
1391:denotes the ring of the
764:over a commutative ring
392:Formal power series ring
342:Integrally closed domain
32:Algebra (disambiguation)
6460:Artin, Michael (1999).
4600:with a nilpotent ideal
4514:{\displaystyle \Gamma }
4289:-algebra homomorphisms
4032:, sometimes called the
3924:. Then, by definition,
3887:is a right module over
2554:algebra of dual numbers
2226:. Therefore, rings and
2210:can be considered as a
1868:and, more recently, of
1597:From ring homomorphisms
1306:multiplicative identity
401:Algebraic number theory
94:Total ring of fractions
6462:"Noncommutative Rings"
6212:
6016:
5885:
5775:
5643:
5565:
5384:
5127:are two algebras, and
5018:Algebra representation
4651:contains a subalgebra
4515:
4491:
4417:
4316:
4283:
4263:
4224:
4165:
4138:
4103:
4050:Central simple algebra
3745:Β§ Representations
3337:
3309:
3262:
3185:
3149:
3129:
3088:
3060:
3019:
2928:
2892:
2868:
2814:Temperley-Lieb algebra
2795:partially ordered sets
2776:, which are useful in
2748:stochastic integration
2646:translate to those of
2513:has a structure of an
2131:
1770:The ring homomorphism
1684:
1587:
1513:
1275:
1162:could be a field). An
760:-algebra is a ring of
558:Noncommutative algebra
541:
494:
457:
409:Algebraic number field
360:Principal ideal domain
249:
203:
141:Frobenius endomorphism
6479:Bourbaki, N. (1989).
6213:
6017:
5886:
5776:
5644:
5566:
5385:
4614:as a module over the
4566:. More generally, if
4516:
4492:
4418:
4317:
4284:
4264:
4225:
4166:
4139:
4104:
3668:. A priori, the dual
3575:-algebra by defining
3538:-algebras is also an
3338:
3310:
3263:
3186:
3150:
3130:
3089:
3061:
3020:
2938:-forms on a manifold
2929:
2893:
2848:
2642:. Many structures of
2568:Representation theory
2517:-algebra by thinking
2292:-algebra by defining
2132:
1882:Algebra homomorphisms
1685:
1643:-algebra by defining
1588:
1514:
1276:
1199:scalar multiplication
770:matrix multiplication
715:scalar multiplication
542:
495:
458:
250:
204:
6286:Algebra over a field
6033:
5906:
5791:
5686:
5606:
5400:
5287:
5186:on the vector space
4980:commutative diagrams
4964:and one of the form
4531:simple Artinian ring
4505:
4435:
4330:
4293:
4273:
4240:
4181:
4148:
4115:
4093:
3472:the structure of an
3431:is automatically an
3364:Gerstenhaber algebra
3323:
3278:
3197:
3159:
3139:
3098:
3074:
3029:
3005:
2947:Mathematical physics
2902:
2830:
2408:-algebra on the set
2148:, sometimes denoted
1941:
1888:algebra homomorphism
1792:, i.e., commutative
1650:
1530:
1478:
1208:
1173:(or more simply, an
1030:Group with operators
973:Complemented lattice
808:Algebraic structures
564:Noncommutative rings
513:
475:
438:
282:Non-associative ring
218:
191:
148:Algebraic structures
6638:Waterhouse, William
6594:1992IJMPA...7.6175T
6560:Ross Street (1998)
6281:Algebraic structure
6264:convolution product
6230:Non-unital algebras
5108:to the unit of End(
4916:in the category of
4710:Order (ring theory)
4704:Lattices and orders
4525:Noncommutative case
3967:-linear map, where
3883:. Then the algebra
3524:in the category of
3378:A subalgebra of an
3068:formal power series
2746:forms a ring under
1875:Generic matrix ring
1449:category of modules
1402:commutative algebra
1084:Composition algebra
844:Quasigroup and loop
777:commutative algebra
733:the structure of a
725:the structure of a
711:algebraic structure
669:associative algebra
323:Commutative algebra
162:Associative algebra
44:Algebraic structure
6694:Algebraic geometry
6555:Cornell University
6533:Structure of Rings
6208:
6173:
6096:
6012:
5977:
5960:
5881:
5771:
5748:
5731:
5639:
5561:
5380:
5104:sends the unit of
5053:. The property of
4976:categorial duality
4759:; in other words,
4616:enveloping algebra
4511:
4487:
4413:
4380:
4312:
4279:
4259:
4220:
4161:
4144:is reduced, where
4134:
4099:
3807:enveloping algebra
3771:. You can help by
3752:Enveloping algebra
3679:For example, take
3502:. This becomes an
3333:
3305:
3258:
3181:
3145:
3125:
3084:
3056:
3015:
2924:
2888:
2788:Incidence algebras
2619:(e.g., semisimple
2561:quasi-free algebra
2470:symmetric algebras
2127:
2125:
1680:
1583:
1509:
1271:
709:. This is thus an
577:Semiprimitive ring
537:
490:
453:
261:Related structures
245:
199:
135:Inner automorphism
121:Ring homomorphisms
6663:978-0-387-90421-4
6625:Vale, R. (2009).
6578:(25): 6175β6213.
6543:978-0-8218-1037-8
6291:Sheaf of algebras
6172:
6095:
5976:
5959:
5747:
5730:
4466:
4373:
4310:
4282:{\displaystyle k}
4269:is the number of
4212:
4198:
4159:
4132:
4102:{\displaystyle A}
4014:projective module
3871:Separable algebra
3865:Separable algebra
3789:
3788:
3687:. Then, not only
3459:. This gives the
3409:Quotient algebras
3229:
3148:{\displaystyle *}
2806:partition algebra
2774:Clifford algebras
2703:topological space
2638:corresponding to
2621:complex Lie group
2400:is a commutative
2272:endomorphism ring
2140:The class of all
1908:ring homomorphism
1721:-algebra, taking
1603:ring homomorphism
1437:monoidal category
1317:ring homomorphism
1143:
1142:
768:, with the usual
695:ring homomorphism
661:
660:
618:Geometric algebra
329:Commutative rings
180:Category of rings
16:(Redirected from
6701:
6674:
6633:
6631:
6621:
6587:
6546:
6526:
6496:
6475:
6473:
6466:
6446:
6440:
6434:
6428:
6422:
6416:
6410:
6404:
6398:
6392:
6386:
6385:, Definition 3.1
6380:
6374:
6368:
6351:
6349:
6329:
6323:
6316:
6276:Abstract algebra
6250:
6217:
6215:
6214:
6209:
6180:
6179:
6174:
6170:
6103:
6102:
6097:
6093:
6021:
6019:
6018:
6013:
5984:
5983:
5978:
5974:
5967:
5966:
5961:
5957:
5890:
5888:
5887:
5882:
5780:
5778:
5777:
5772:
5755:
5754:
5749:
5745:
5738:
5737:
5732:
5728:
5654:comultiplication
5648:
5646:
5645:
5640:
5598:
5583:
5570:
5568:
5567:
5562:
5548:
5547:
5502:
5501:
5389:
5387:
5386:
5381:
5279:
5252:
5237:
5195:
5185:
5175:
5156:
5141:
5087:
5044:
4973:
4963:
4933:
4920:-vector spaces)
4911:
4897:
4887:
4847:Related concepts
4832:is a lattice in
4828:
4826:
4825:
4822:
4819:
4777:
4687:
4679:
4678:
4677:
4674:
4646:
4628:
4613:
4565:
4520:
4518:
4517:
4512:
4496:
4494:
4493:
4488:
4483:
4482:
4467:
4464:
4453:
4452:
4422:
4420:
4419:
4414:
4406:
4401:
4381:
4362:
4357:
4356:
4321:
4319:
4318:
4313:
4311:
4303:
4288:
4286:
4285:
4280:
4268:
4266:
4265:
4260:
4252:
4251:
4229:
4227:
4226:
4221:
4219:
4218:
4213:
4205:
4199:
4191:
4170:
4168:
4167:
4162:
4160:
4152:
4143:
4141:
4140:
4135:
4133:
4125:
4108:
4106:
4105:
4100:
4074:Commutative case
4027:
4021:
4008:. Equivalently,
4007:
3976:
3962:
3923:
3906:with the action
3905:
3845:
3830:
3784:
3781:
3763:
3756:
3737:
3735:
3729:
3718:
3712:
3698:
3616:
3597:
3471:
3458:
3390:which is both a
3342:
3340:
3339:
3334:
3332:
3331:
3314:
3312:
3311:
3306:
3287:
3286:
3267:
3265:
3264:
3259:
3230:
3222:
3190:
3188:
3187:
3182:
3180:
3179:
3154:
3152:
3151:
3146:
3134:
3132:
3131:
3126:
3107:
3106:
3093:
3091:
3090:
3085:
3083:
3082:
3065:
3063:
3062:
3057:
3038:
3037:
3024:
3022:
3021:
3016:
3014:
3013:
2997:
2975:
2933:
2931:
2930:
2925:
2914:
2913:
2897:
2895:
2894:
2889:
2878:
2877:
2867:
2862:
2745:
2693:
2679:linear operators
2551:
2512:
2425:
2399:
2319:
2168:coslice category
2136:
2134:
2133:
2128:
2126:
1929:
1900:-algebras is an
1851:anti-equivalence
1839:coslice category
1836:
1825:
1806:
1787:
1766:
1741:
1727:
1712:
1702:
1689:
1687:
1686:
1681:
1626:
1592:
1590:
1589:
1584:
1546:
1518:
1516:
1515:
1510:
1499:
1498:
1408:commutative ring
1404:
1403:
1387:-algebra, where
1373:
1357:
1280:
1278:
1277:
1272:
1190:that is also an
1156:commutative ring
1135:
1128:
1121:
910:Commutative ring
839:Rack and quandle
804:
803:
785:commutative ring
693:together with a
676:commutative ring
653:
646:
639:
624:Operator algebra
610:Clifford algebra
546:
544:
543:
538:
533:
532:
520:
499:
497:
496:
491:
489:
488:
483:
462:
460:
459:
454:
452:
451:
446:
424:Ring of integers
418:
415:Integers modulo
366:Euclidean domain
254:
252:
251:
246:
244:
236:
231:
208:
206:
205:
200:
198:
102:Product of rings
88:Fractional ideal
47:
39:
38:
21:
6709:
6708:
6704:
6703:
6702:
6700:
6699:
6698:
6679:
6678:
6677:
6664:
6646:Springer-Verlag
6629:
6544:
6515:
6493:
6471:
6464:
6455:
6450:
6449:
6441:
6437:
6433:, Theorem 4.7.5
6429:
6425:
6419:Waterhouse 1979
6417:
6413:
6407:Waterhouse 1979
6405:
6401:
6393:
6389:
6381:
6377:
6369:
6365:
6360:
6355:
6354:
6341:
6332:
6330:
6326:
6317:
6313:
6308:
6272:
6238:
6232:
6175:
6168:
6167:
6098:
6091:
6090:
6034:
6031:
6030:
5979:
5972:
5971:
5962:
5955:
5954:
5907:
5904:
5903:
5792:
5789:
5788:
5750:
5743:
5742:
5733:
5726:
5725:
5687:
5684:
5683:
5677:
5671:
5607:
5604:
5603:
5585:
5575:
5543:
5539:
5497:
5493:
5401:
5398:
5397:
5288:
5285:
5284:
5254:
5239:
5224:
5221:
5187:
5177:
5158:
5143:
5128:
5062:
5031:
5020:
5014:
5012:Representations
4965:
4951:
4921:
4899:
4889:
4875:
4860:
4854:
4849:
4823:
4820:
4817:
4816:
4814:
4808:
4789:
4769:
4760:
4712:
4706:
4675:
4672:
4671:
4670:
4665:
4656:
4630:
4618:
4605:
4559:
4550:
4527:
4506:
4503:
4502:
4478:
4474:
4463:
4448:
4444:
4436:
4433:
4432:
4425:profinite group
4402:
4394:
4372:
4358:
4352:
4348:
4331:
4328:
4327:
4302:
4294:
4291:
4290:
4274:
4271:
4270:
4247:
4243:
4241:
4238:
4237:
4214:
4204:
4203:
4190:
4182:
4179:
4178:
4151:
4149:
4146:
4145:
4124:
4116:
4113:
4112:
4094:
4091:
4090:
4076:
4052:
4046:
4023:
4017:
3982:
3968:
3942:
3933:
3907:
3901:
3888:
3873:
3867:
3841:
3832:
3826:
3817:
3816:is the algebra
3795:
3785:
3779:
3776:
3769:needs expansion
3754:
3731:
3725:
3720:
3708:
3694:
3692:
3646:
3627:Change of rings
3612:
3603:
3576:
3566:
3531:Tensor products
3512:One can form a
3491:Direct products
3463:
3453:
3436:
3386:is a subset of
3372:
3327:
3326:
3324:
3321:
3320:
3282:
3281:
3279:
3276:
3275:
3221:
3198:
3195:
3194:
3175:
3174:
3160:
3157:
3156:
3155:such that, for
3140:
3137:
3136:
3102:
3101:
3099:
3096:
3095:
3078:
3077:
3075:
3072:
3071:
3033:
3032:
3030:
3027:
3026:
3009:
3008:
3006:
3003:
3002:
2971:
2961:
2954:Poisson algebra
2949:
2909:
2905:
2903:
2900:
2899:
2873:
2869:
2863:
2852:
2831:
2828:
2827:
2825:de Rham algebra
2769:
2761:Azumaya algebra
2743:
2736:
2723:
2719:defined on the
2717:semimartingales
2681:
2664:
2625:coordinate ring
2617:algebraic group
2584:is a group and
2570:
2522:
2504:
2495:Given a module
2422:
2416:
2409:
2395:
2393:polynomial ring
2375:complex numbers
2293:
2283:
2203:
2191:
2162:of commutative
2124:
2123:
2113:
2098:
2097:
2066:
2048:
2047:
2013:
1992:
1991:
1966:
1944:
1942:
1939:
1938:
1928:
1921:
1911:
1890:
1884:
1827:
1812:
1807:that are under
1797:
1779:
1754:
1729:
1722:
1704:
1694:
1651:
1648:
1647:
1614:
1599:
1542:
1531:
1528:
1527:
1494:
1490:
1479:
1476:
1475:
1416:
1401:
1400:
1372:
1359:
1336:
1209:
1206:
1205:
1148:
1139:
1110:
1109:
1108:
1079:Non-associative
1061:
1050:
1049:
1039:
1019:
1008:
1007:
996:Map of lattices
992:
988:Boolean algebra
983:Heyting algebra
957:
946:
945:
939:
920:Integral domain
884:
873:
872:
866:
820:
762:square matrices
657:
628:
627:
560:
550:
549:
528:
524:
516:
514:
511:
510:
484:
479:
478:
476:
473:
472:
447:
442:
441:
439:
436:
435:
416:
386:Polynomial ring
336:Integral domain
325:
315:
314:
240:
232:
227:
219:
216:
215:
194:
192:
189:
188:
174:Involutive ring
59:
48:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6707:
6697:
6696:
6691:
6676:
6675:
6662:
6634:
6622:
6585:hep-th/9111043
6567:
6558:
6547:
6542:
6527:
6513:
6497:
6491:
6476:
6456:
6454:
6451:
6448:
6447:
6435:
6423:
6411:
6399:
6387:
6375:
6362:
6361:
6359:
6356:
6353:
6352:
6337:
6324:
6310:
6309:
6307:
6304:
6303:
6302:
6297:
6288:
6283:
6278:
6271:
6268:
6231:
6228:
6220:
6219:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6178:
6166:
6163:
6160:
6157:
6154:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
6127:
6124:
6121:
6118:
6115:
6112:
6109:
6106:
6101:
6089:
6086:
6083:
6080:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6024:
6023:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5982:
5970:
5965:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5917:
5914:
5911:
5893:
5892:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5856:
5853:
5850:
5847:
5844:
5841:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5802:
5799:
5796:
5782:
5781:
5770:
5767:
5764:
5761:
5758:
5753:
5741:
5736:
5724:
5721:
5718:
5715:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5670:
5667:
5650:
5649:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5572:
5571:
5560:
5557:
5554:
5551:
5546:
5542:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5500:
5496:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5391:
5390:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5220:
5217:
5198:tensor product
5026:of an algebra
5024:representation
5016:Main article:
5013:
5010:
4946:-vector space
4870:-vector space
4866:is given by a
4856:Main article:
4853:
4850:
4848:
4845:
4804:
4785:
4765:
4751:-submodule of
4743:-vector space
4708:Main article:
4705:
4702:
4694:Levi's theorem
4661:
4647:splits; i.e.,
4581:The fact that
4555:
4526:
4523:
4510:
4486:
4481:
4477:
4473:
4470:
4462:
4459:
4456:
4451:
4447:
4443:
4440:
4412:
4409:
4405:
4400:
4397:
4393:
4390:
4387:
4384:
4379:
4376:
4371:
4368:
4365:
4361:
4355:
4351:
4347:
4344:
4341:
4338:
4335:
4324:
4323:
4309:
4306:
4301:
4298:
4278:
4258:
4255:
4250:
4246:
4235:
4217:
4211:
4208:
4202:
4197:
4194:
4189:
4186:
4176:
4158:
4155:
4131:
4128:
4123:
4120:
4110:
4098:
4075:
4072:
4045:
4042:
3938:
3897:
3869:Main article:
3866:
3863:
3837:
3822:
3787:
3786:
3766:
3764:
3753:
3750:
3645:
3642:
3641:
3640:
3633:
3630:
3608:
3562:
3556:tensor product
3532:
3529:
3510:
3507:
3500:direct product
3492:
3489:
3449:
3435:-module since
3410:
3407:
3376:
3371:
3368:
3367:
3366:
3361:
3345:
3344:
3330:
3304:
3300:
3297:
3294:
3290:
3285:
3271:
3270:
3269:
3268:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3228:
3225:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3178:
3173:
3170:
3167:
3164:
3144:
3124:
3120:
3117:
3114:
3110:
3105:
3081:
3055:
3051:
3048:
3045:
3041:
3036:
3012:
2999:
2948:
2945:
2944:
2943:
2923:
2920:
2917:
2912:
2908:
2887:
2884:
2881:
2876:
2872:
2866:
2861:
2858:
2855:
2851:
2847:
2844:
2841:
2838:
2835:
2817:
2810:Brauer algebra
2802:
2792:locally finite
2785:
2768:
2765:
2764:
2763:
2757:
2751:
2738:
2732:
2713:
2699:
2696:Banach algebra
2663:
2660:
2659:
2658:
2655:quiver algebra
2651:
2609:
2578:
2569:
2566:
2565:
2564:
2557:
2493:
2454:tensor algebra
2450:
2427:
2420:
2414:
2389:
2382:
2371:
2370:
2369:
2321:
2279:
2260:
2243:characteristic
2239:
2232:abelian groups
2202:
2199:
2190:
2187:
2138:
2137:
2122:
2119:
2116:
2114:
2112:
2109:
2106:
2103:
2100:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2067:
2065:
2062:
2059:
2056:
2053:
2050:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2014:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1967:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1946:
1926:
1919:
1910:. Explicitly,
1886:Main article:
1883:
1880:
1855:affine schemes
1847:prime spectrum
1691:
1690:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1635:, we can make
1598:
1595:
1594:
1593:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1545:
1541:
1538:
1535:
1521:
1520:
1508:
1505:
1502:
1497:
1493:
1489:
1486:
1483:
1422:-algebra is a
1415:
1412:
1368:
1282:
1281:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1147:
1144:
1141:
1140:
1138:
1137:
1130:
1123:
1115:
1112:
1111:
1107:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1070:
1069:
1068:
1062:
1056:
1055:
1052:
1051:
1048:
1047:
1044:Linear algebra
1038:
1037:
1032:
1027:
1021:
1020:
1014:
1013:
1010:
1009:
1006:
1005:
1002:Lattice theory
998:
991:
990:
985:
980:
975:
970:
965:
959:
958:
952:
951:
948:
947:
938:
937:
932:
927:
922:
917:
912:
907:
902:
897:
892:
886:
885:
879:
878:
875:
874:
865:
864:
859:
854:
848:
847:
846:
841:
836:
827:
821:
815:
814:
811:
810:
659:
658:
656:
655:
648:
641:
633:
630:
629:
621:
620:
592:
591:
585:
579:
573:
561:
556:
555:
552:
551:
548:
547:
536:
531:
527:
523:
519:
500:
487:
482:
463:
450:
445:
433:-adic integers
426:
420:
411:
397:
396:
395:
394:
388:
382:
381:
380:
368:
362:
356:
350:
344:
326:
321:
320:
317:
316:
313:
312:
311:
310:
298:
297:
296:
290:
278:
277:
276:
258:
257:
256:
255:
243:
239:
235:
230:
226:
223:
209:
197:
176:
170:
164:
158:
144:
143:
137:
131:
117:
116:
110:
104:
98:
97:
96:
90:
78:
72:
60:
58:Basic concepts
57:
56:
53:
52:
26:
9:
6:
4:
3:
2:
6706:
6695:
6692:
6690:
6687:
6686:
6684:
6673:
6669:
6665:
6659:
6655:
6651:
6647:
6643:
6639:
6635:
6628:
6623:
6619:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6586:
6581:
6577:
6573:
6568:
6565:
6564:
6559:
6556:
6552:
6548:
6545:
6539:
6535:
6534:
6528:
6524:
6520:
6516:
6510:
6506:
6502:
6498:
6494:
6492:3-540-64243-9
6488:
6484:
6483:
6477:
6470:
6463:
6458:
6457:
6445:, Ch. IV, Β§ 1
6444:
6439:
6432:
6427:
6420:
6415:
6408:
6403:
6396:
6391:
6384:
6379:
6372:
6367:
6363:
6348:
6344:
6340:
6335:
6328:
6321:
6315:
6311:
6301:
6298:
6296:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6273:
6267:
6265:
6260:
6258:
6254:
6249:
6245:
6241:
6235:
6227:
6225:
6202:
6196:
6190:
6184:
6181:
6176:
6164:
6158:
6152:
6149:
6143:
6137:
6134:
6128:
6122:
6119:
6113:
6107:
6104:
6099:
6087:
6081:
6075:
6069:
6063:
6060:
6054:
6048:
6042:
6036:
6029:
6028:
6027:
6006:
6000:
5994:
5988:
5985:
5980:
5968:
5963:
5951:
5945:
5939:
5933:
5927:
5924:
5918:
5915:
5909:
5902:
5901:
5900:
5898:
5875:
5869:
5863:
5857:
5854:
5851:
5848:
5845:
5839:
5833:
5827:
5821:
5815:
5812:
5809:
5800:
5794:
5787:
5786:
5785:
5765:
5759:
5756:
5751:
5739:
5734:
5722:
5716:
5710:
5707:
5701:
5695:
5689:
5682:
5681:
5680:
5676:
5666:
5663:
5659:
5655:
5636:
5630:
5624:
5621:
5618:
5612:
5609:
5602:
5601:
5600:
5597:
5593:
5589:
5582:
5578:
5555:
5549:
5544:
5540:
5536:
5527:
5521:
5518:
5512:
5506:
5498:
5494:
5490:
5484:
5478:
5475:
5472:
5466:
5460:
5457:
5454:
5448:
5445:
5439:
5436:
5430:
5427:
5421:
5418:
5412:
5409:
5403:
5396:
5395:
5394:
5377:
5368:
5359:
5353:
5347:
5338:
5329:
5323:
5317:
5311:
5308:
5305:
5296:
5290:
5283:
5282:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5250:
5246:
5242:
5235:
5231:
5227:
5216:
5214:
5210:
5206:
5205:
5199:
5194:
5190:
5184:
5180:
5173:
5169:
5165:
5161:
5154:
5150:
5146:
5139:
5135:
5131:
5126:
5122:
5117:
5115:
5111:
5107:
5103:
5099:
5095:
5091:
5085:
5081:
5077:
5073:
5069:
5065:
5060:
5056:
5052:
5048:
5042:
5038:
5034:
5029:
5025:
5019:
5009:
5007:
5003:
4999:
4997:
4991:
4989:
4985:
4981:
4977:
4972:
4968:
4962:
4958:
4954:
4949:
4945:
4941:
4937:
4932:
4928:
4924:
4919:
4915:
4910:
4906:
4902:
4896:
4892:
4886:
4882:
4878:
4873:
4869:
4865:
4859:
4844:
4842:
4841:maximal order
4837:
4835:
4831:
4812:
4807:
4803:
4799:
4798:
4794:-algebra. An
4793:
4788:
4784:
4779:
4776:
4772:
4768:
4763:
4758:
4754:
4750:
4746:
4742:
4738:
4735:
4734:
4729:
4725:
4721:
4717:
4711:
4701:
4699:
4695:
4691:
4686:
4682:
4669:
4664:
4659:
4654:
4650:
4645:
4641:
4637:
4633:
4626:
4622:
4617:
4612:
4608:
4603:
4599:
4595:
4590:
4588:
4584:
4579:
4577:
4573:
4569:
4563:
4558:
4553:
4548:
4544:
4540:
4536:
4532:
4522:
4500:
4479:
4475:
4468:
4460:
4454:
4449:
4445:
4438:
4430:
4426:
4407:
4403:
4398:
4395:
4388:
4385:
4382:
4377:
4374:
4369:
4363:
4359:
4353:
4349:
4342:
4339:
4336:
4304:
4296:
4276:
4256:
4253:
4248:
4244:
4236:
4233:
4215:
4206:
4200:
4192:
4187:
4184:
4177:
4174:
4153:
4126:
4121:
4118:
4111:
4109:is separable.
4096:
4089:
4088:
4087:
4085:
4081:
4071:
4069:
4068:Artinian ring
4065:
4061:
4057:
4051:
4041:
4039:
4035:
4031:
4026:
4020:
4015:
4011:
4006:
4002:
3998:
3994:
3990:
3986:
3980:
3975:
3971:
3966:
3963:splits as an
3961:
3957:
3953:
3949:
3945:
3941:
3936:
3931:
3927:
3922:
3918:
3914:
3910:
3904:
3900:
3895:
3891:
3886:
3882:
3878:
3872:
3862:
3860:
3856:
3852:
3847:
3844:
3840:
3835:
3829:
3825:
3820:
3815:
3811:
3808:
3804:
3800:
3794:
3783:
3774:
3770:
3767:This section
3765:
3762:
3758:
3757:
3749:
3747:
3746:
3741:
3734:
3728:
3723:
3716:
3711:
3706:
3702:
3697:
3690:
3686:
3682:
3677:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3638:
3634:
3631:
3628:
3624:
3620:
3615:
3611:
3606:
3601:
3595:
3591:
3587:
3583:
3579:
3574:
3570:
3565:
3560:
3557:
3553:
3550:and any ring
3549:
3545:
3541:
3537:
3533:
3530:
3527:
3523:
3519:
3515:
3511:
3509:Free products
3508:
3505:
3501:
3497:
3493:
3490:
3487:
3483:
3479:
3475:
3470:
3466:
3462:
3461:quotient ring
3457:
3452:
3447:
3443:
3439:
3434:
3430:
3426:
3423:
3419:
3415:
3411:
3408:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3374:
3373:
3370:Constructions
3365:
3362:
3359:
3358:quantum group
3355:
3351:
3347:
3346:
3318:
3295:
3273:
3272:
3255:
3252:
3249:
3246:
3240:
3237:
3234:
3226:
3223:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3193:
3192:
3171:
3168:
3165:
3162:
3142:
3115:
3069:
3046:
3000:
2995:
2991:
2987:
2983:
2979:
2974:
2969:
2965:
2959:
2955:
2951:
2950:
2941:
2937:
2918:
2910:
2882:
2874:
2864:
2859:
2856:
2853:
2849:
2845:
2839:
2826:
2822:
2818:
2815:
2811:
2807:
2803:
2800:
2799:combinatorics
2796:
2793:
2789:
2786:
2783:
2779:
2775:
2771:
2770:
2762:
2758:
2756:
2752:
2749:
2741:
2735:
2731:
2727:
2722:
2718:
2714:
2711:
2707:
2704:
2700:
2697:
2692:
2688:
2684:
2680:
2677:
2673:
2670:
2666:
2665:
2656:
2652:
2649:
2645:
2641:
2637:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2607:
2603:
2602:group algebra
2599:
2595:
2591:
2587:
2583:
2579:
2576:
2572:
2571:
2562:
2558:
2555:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2520:
2516:
2511:
2507:
2502:
2498:
2494:
2491:
2487:
2483:
2479:
2476:that maps an
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2448:
2444:
2440:
2436:
2434:
2428:
2423:
2413:
2407:
2403:
2398:
2394:
2390:
2387:
2383:
2380:
2376:
2372:
2367:
2363:
2359:
2356:
2352:
2348:
2347:
2345:
2342:
2338:
2334:
2330:
2326:
2322:
2317:
2313:
2309:
2305:
2301:
2297:
2291:
2287:
2282:
2278:, denoted End
2277:
2273:
2269:
2265:
2261:
2258:
2255:
2251:
2247:
2244:
2240:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2209:
2205:
2204:
2198:
2196:
2186:
2184:
2180:
2176:
2172:
2169:
2165:
2161:
2156:
2154:
2152:
2147:
2143:
2120:
2117:
2115:
2107:
2101:
2091:
2085:
2079:
2073:
2070:
2068:
2060:
2057:
2051:
2041:
2035:
2032:
2026:
2020:
2017:
2015:
2007:
2004:
2001:
1995:
1985:
1979:
1976:
1973:
1970:
1968:
1960:
1957:
1954:
1948:
1937:
1936:
1935:
1933:
1925:
1918:
1914:
1909:
1906:
1904:
1899:
1895:
1889:
1879:
1877:
1876:
1871:
1867:
1862:
1860:
1856:
1852:
1848:
1844:
1840:
1834:
1830:
1823:
1819:
1815:
1810:
1804:
1800:
1795:
1791:
1786:
1782:
1777:
1776:structure map
1773:
1768:
1765:
1761:
1757:
1752:
1748:
1743:
1740:
1736:
1732:
1725:
1720:
1716:
1711:
1707:
1701:
1697:
1677:
1671:
1665:
1662:
1659:
1656:
1653:
1646:
1645:
1644:
1642:
1638:
1634:
1630:
1625:
1621:
1617:
1612:
1608:
1604:
1580:
1574:
1571:
1568:
1562:
1559:
1556:
1550:
1547:
1543:
1536:
1533:
1526:
1525:
1524:
1506:
1500:
1495:
1491:
1487:
1484:
1481:
1474:
1473:
1472:
1470:
1466:
1462:
1458:
1452:
1450:
1446:
1442:
1438:
1434:
1433:
1431:
1425:
1424:monoid object
1421:
1411:
1409:
1405:
1396:
1394:
1390:
1386:
1381:
1379:
1378:
1371:
1366:
1362:
1356:
1352:
1348:
1344:
1340:
1334:
1330:
1326:
1322:
1318:
1314:
1309:
1307:
1303:
1299:
1295:
1291:
1287:
1265:
1262:
1259:
1253:
1250:
1247:
1241:
1238:
1235:
1229:
1223:
1220:
1214:
1211:
1204:
1203:
1202:
1200:
1196:
1194:
1189:
1186:
1182:
1181:
1177:
1172:
1171:
1167:
1161:
1157:
1153:
1136:
1131:
1129:
1124:
1122:
1117:
1116:
1114:
1113:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1071:
1067:
1064:
1063:
1059:
1054:
1053:
1046:
1045:
1041:
1040:
1036:
1033:
1031:
1028:
1026:
1023:
1022:
1017:
1012:
1011:
1004:
1003:
999:
997:
994:
993:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
960:
955:
950:
949:
944:
943:
936:
933:
931:
930:Division ring
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
887:
882:
877:
876:
871:
870:
863:
860:
858:
855:
853:
852:Abelian group
850:
849:
845:
842:
840:
837:
835:
831:
828:
826:
823:
822:
818:
813:
812:
809:
806:
805:
802:
799:
797:
793:
788:
786:
782:
778:
773:
771:
767:
763:
759:
755:
751:
749:
744:
740:
736:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
689:
685:
681:
677:
673:
670:
666:
654:
649:
647:
642:
640:
635:
634:
632:
631:
626:
625:
619:
615:
614:
613:
612:
611:
606:
605:
604:
599:
598:
597:
590:
586:
584:
580:
578:
574:
572:
571:Division ring
568:
567:
566:
565:
559:
554:
553:
525:
509:
507:
501:
485:
471:
470:-adic numbers
469:
464:
448:
434:
432:
427:
425:
421:
419:
412:
410:
406:
405:
404:
403:
402:
393:
389:
387:
383:
379:
375:
374:
373:
369:
367:
363:
361:
357:
355:
351:
349:
345:
343:
339:
338:
337:
333:
332:
331:
330:
324:
319:
318:
309:
305:
304:
303:
299:
295:
291:
289:
285:
284:
283:
279:
275:
271:
270:
269:
265:
264:
263:
262:
237:
233:
224:
221:
214:
213:Terminal ring
210:
187:
183:
182:
181:
177:
175:
171:
169:
165:
163:
159:
157:
153:
152:
151:
150:
149:
142:
138:
136:
132:
130:
126:
125:
124:
123:
122:
115:
111:
109:
105:
103:
99:
95:
91:
89:
85:
84:
83:
82:Quotient ring
79:
77:
73:
71:
67:
66:
65:
64:
55:
54:
51:
46:β Ring theory
45:
41:
40:
37:
33:
19:
6641:
6575:
6571:
6561:
6553:, link from
6532:
6504:
6485:. Springer.
6481:
6438:
6426:
6414:
6402:
6390:
6378:
6366:
6346:
6342:
6338:
6333:
6327:
6319:
6314:
6295:ringed space
6261:
6256:
6247:
6243:
6239:
6236:
6233:
6221:
6025:
5896:
5894:
5783:
5678:
5662:Hopf algebra
5651:
5595:
5591:
5587:
5580:
5576:
5573:
5392:
5275:
5271:
5267:
5263:
5259:
5255:
5248:
5244:
5240:
5233:
5229:
5225:
5222:
5209:Hopf algebra
5202:
5192:
5188:
5182:
5178:
5171:
5167:
5163:
5159:
5152:
5148:
5144:
5137:
5133:
5129:
5124:
5120:
5118:
5113:
5109:
5105:
5101:
5100:), and that
5097:
5093:
5089:
5083:
5079:
5075:
5071:
5067:
5063:
5058:
5054:
5050:
5046:
5040:
5036:
5032:
5027:
5021:
5001:
4995:
4992:
4970:
4966:
4960:
4956:
4952:
4947:
4943:
4939:
4930:
4926:
4922:
4917:
4908:
4904:
4900:
4894:
4890:
4884:
4880:
4876:
4871:
4867:
4863:
4861:
4840:
4838:
4833:
4829:
4810:
4805:
4801:
4795:
4791:
4786:
4782:
4780:
4774:
4770:
4766:
4761:
4756:
4752:
4748:
4744:
4740:
4736:
4731:
4727:
4723:
4719:
4715:
4713:
4698:Lie algebras
4689:
4684:
4680:
4667:
4662:
4657:
4652:
4648:
4643:
4639:
4635:
4631:
4624:
4620:
4610:
4606:
4601:
4597:
4593:
4591:
4586:
4582:
4580:
4571:
4567:
4561:
4556:
4551:
4546:
4542:
4538:
4534:
4528:
4498:
4428:
4325:
4231:
4172:
4083:
4079:
4077:
4063:
4059:
4055:
4053:
4037:
4033:
4029:
4024:
4022:; thus, the
4018:
4009:
4004:
4000:
3996:
3992:
3988:
3984:
3978:
3973:
3969:
3964:
3959:
3955:
3951:
3947:
3943:
3939:
3934:
3925:
3920:
3916:
3912:
3908:
3902:
3898:
3893:
3889:
3884:
3880:
3876:
3874:
3858:
3854:
3849:Note that a
3848:
3842:
3838:
3833:
3827:
3823:
3818:
3813:
3809:
3806:
3802:
3798:
3796:
3777:
3773:adding to it
3768:
3743:
3739:
3732:
3726:
3721:
3719:and co-unit
3714:
3709:
3704:
3700:
3695:
3688:
3684:
3680:
3678:
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3647:
3637:free algebra
3632:Free algebra
3622:
3619:left adjoint
3613:
3609:
3604:
3599:
3593:
3589:
3585:
3581:
3577:
3572:
3568:
3563:
3558:
3551:
3547:
3539:
3535:
3525:
3517:
3514:free product
3503:
3495:
3485:
3481:
3477:
3473:
3468:
3464:
3455:
3450:
3445:
3441:
3437:
3432:
3428:
3424:
3417:
3413:
3403:
3399:
3387:
3383:
3379:
3315:is called a
2993:
2989:
2985:
2981:
2977:
2972:
2967:
2963:
2939:
2935:
2755:Weyl algebra
2739:
2733:
2729:
2725:
2709:
2705:
2690:
2686:
2682:
2671:
2669:Banach space
2647:
2643:
2639:
2635:
2633:Hopf algebra
2628:
2623:), then the
2612:
2605:
2597:
2593:
2589:
2585:
2581:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2518:
2514:
2509:
2505:
2500:
2496:
2489:
2485:
2482:left adjoint
2477:
2461:
2457:
2446:
2442:
2438:
2432:
2418:
2411:
2405:
2401:
2396:
2379:real numbers
2365:
2361:
2354:
2350:
2343:
2336:
2332:
2328:
2323:Any ring of
2315:
2311:
2307:
2303:
2299:
2295:
2289:
2285:
2280:
2275:
2267:
2263:
2256:
2253:
2249:
2245:
2241:Any ring of
2235:
2227:
2223:
2219:
2215:
2211:
2207:
2192:
2178:
2174:
2170:
2163:
2157:
2150:
2149:
2141:
2139:
1931:
1923:
1916:
1912:
1902:
1897:
1896:between two
1894:homomorphism
1891:
1873:
1872:. See also:
1863:
1858:
1842:
1832:
1828:
1821:
1817:
1813:
1808:
1802:
1798:
1793:
1789:
1788:for a fixed
1784:
1780:
1771:
1769:
1763:
1759:
1755:
1750:
1746:
1744:
1738:
1734:
1730:
1723:
1718:
1714:
1709:
1705:
1699:
1695:
1692:
1640:
1636:
1632:
1623:
1619:
1615:
1610:
1600:
1522:
1471:-linear map
1468:
1464:
1456:
1453:
1440:
1429:
1428:
1419:
1417:
1399:
1397:
1388:
1384:
1382:
1375:
1374:. (See also
1369:
1364:
1360:
1354:
1350:
1346:
1342:
1338:
1332:
1328:
1320:
1312:
1310:
1297:
1293:
1289:
1285:
1283:
1192:
1187:
1179:
1175:
1174:
1169:
1165:
1164:associative
1163:
1159:
1151:
1149:
1104:Hopf algebra
1073:
1042:
1035:Vector space
1000:
940:
869:Group theory
867:
832: /
800:
791:
789:
776:
774:
765:
757:
753:
747:
742:
739:vector space
730:
722:
718:
706:
698:
690:
683:
671:
668:
662:
622:
608:
607:
603:Free algebra
601:
600:
594:
593:
562:
505:
467:
430:
399:
398:
378:Finite field
327:
274:Finite field
260:
259:
186:Initial ring
161:
146:
145:
119:
118:
61:
36:
6373:, Example 1
6224:Lie algebra
5213:Lie algebra
4755:that spans
4034:bidimension
3981:-module by
3928:is said to
3484:is also an
3375:Subalgebras
2958:Lie algebra
2715:The set of
2386:quaternions
2160:subcategory
1837:(i.e., the
1089:Lie algebra
1074:Associative
978:Total order
968:Semilattice
942:Ring theory
781:commutative
665:mathematics
583:Simple ring
294:Jordan ring
168:Graded ring
50:Ring theory
6683:Categories
6523:1006.00001
6514:1852336676
6501:Cohn, P.M.
6453:References
6443:Artin 1999
5673:See also:
4998:-coalgebra
4852:Coalgebras
4655:such that
4521:-actions.
4048:See also:
3791:See also:
3780:March 2023
3528:-algebras.
2701:Given any
2676:continuous
2667:Given any
1857:over Spec
1201:satisfies
1146:Definition
796:non-unital
589:Commutator
348:GCD domain
6618:119087306
6610:0217-751X
6482:Algebra I
6431:Cohn 2003
6395:Cohn 2003
6383:Vale 2009
6371:Tjin 1992
6358:Citations
6197:τ
6185:τ
6182:⊗
6153:τ
6150:⊗
6138:σ
6123:τ
6120:⊗
6108:σ
6088:⊗
6076:σ
6064:σ
6049:ρ
6037:ρ
6001:τ
5989:τ
5986:⊗
5952:⊗
5940:σ
5928:σ
5910:ρ
5864:τ
5858:⊗
5846:⊗
5828:σ
5813:⊗
5795:ρ
5760:τ
5757:⊗
5723:⊗
5711:σ
5696:ρ
5693:↦
5658:bialgebra
5634:Δ
5631:∘
5625:τ
5622:⊗
5619:σ
5610:ρ
5586:Ξ :
5550:ρ
5522:τ
5519:⊗
5507:σ
5479:τ
5473:⊗
5461:σ
5440:τ
5437:⊗
5422:σ
5404:ρ
5354:τ
5348:⊗
5324:σ
5309:⊗
5291:ρ
4988:coalgebra
4858:Coalgebra
4509:Γ
4472:→
4442:↦
4389:
4383:
4378:←
4343:
4334:Γ
4308:¯
4300:→
4254:
4230:for some
4210:¯
4196:¯
4188:⊗
4157:¯
4130:¯
4122:⊗
3930:separable
3892: :=
3522:coproduct
3488:-algebra.
3396:submodule
3382:-algebra
3253:⋯
3219:−
3204:∗
3172:∈
3143:∗
2907:Ω
2871:Ω
2850:⨁
2834:Ω
2437:on a set
2346:-module.
2331:forms an
2262:Given an
2206:Any ring
2102:φ
2086:φ
2074:φ
2052:φ
2036:φ
2021:φ
1996:φ
1980:φ
1977:⋅
1958:⋅
1949:φ
1666:η
1657:⋅
1572:⊗
1563:∘
1548:⊗
1537:∘
1504:→
1492:⊗
1263:⋅
1239:⋅
1215:⋅
1178:-algebra
1168:-algebra
1099:Bialgebra
905:Near-ring
862:Lie group
830:Semigroup
701:into the
678:(often a
530:∞
308:Semifield
6689:Algebras
6640:(1979),
6503:(2003).
6469:Archived
6270:See also
6242: :
5258: :
5243: :
5228: :
5147: :
5132: :
5088:for all
5035: :
5000:, where
4934:(by the
4914:morphism
4666: :
4634: :
4549:; i.e.,
4529:Since a
4399:′
3950: :
3851:bimodule
3748:below).
3656:. Since
2898:, where
2812:and the
2778:geometry
2685: :
2662:Analysis
2466:exterior
2435:-algebra
2358:matrices
2325:matrices
2288:) is an
2266:-module
2189:Examples
2146:category
1915: :
1845:.) The
1811:; i.e.,
1758: :
1733: :
1693:for all
1618: :
1393:integers
1380:below).
1284:for all
935:Lie ring
900:Semiring
750:-algebra
302:Semiring
288:Lie ring
70:Subrings
6672:0547117
6590:Bibcode
6421:, Β§ 6.3
6409:, Β§ 6.2
6397:, Β§ 4.7
5006:functor
4827:
4815:
4733:lattice
4431:. Then
4062:. Then
3561: β
3392:subring
3094:. If
2782:physics
2631:is the
2474:functor
2417:, ...,
2201:Algebra
2181:is the
1905:-linear
1323:to the
1195:-module
1183:) is a
1066:Algebra
1058:Algebra
963:Lattice
954:Lattice
674:over a
504:PrΓΌfer
106:β’
6670:
6660:
6616:
6608:
6540:
6521:
6511:
6489:
6251:whose
5247:β End(
5232:β End(
5166:β End(
5151:β End(
5136:β End(
5039:β End(
4984:axioms
4809:is an
4423:, the
4066:is an
3977:is an
3805:, the
3567:
3416:be an
3394:and a
2674:, the
2615:is an
2456:of an
2391:Every
2270:, the
2248:is a (
2195:center
2177:where
1930:is an
1717:is an
1629:center
1607:center
1325:center
1302:unital
1094:Graded
1025:Module
1016:Module
915:Domain
834:Monoid
735:module
703:center
156:Module
129:Kernel
6630:(PDF)
6614:S2CID
6580:arXiv
6472:(PDF)
6465:(PDF)
6306:Notes
6253:limit
5211:or a
5045:from
5004:is a
4942:as a
4797:order
4730:). A
4545:over
4016:over
3991:) β
(
3853:over
3588:) = (
3422:ideal
3274:then
3070:over
2431:free
2179:CRing
2175:CRing
1713:. If
1435:(the
1331:. If
1319:from
1154:be a
1060:-like
1018:-like
956:-like
925:Field
883:-like
857:Magma
825:Group
819:-like
817:Group
741:over
697:from
686:is a
680:field
667:, an
508:-ring
372:Field
268:Field
76:Ideal
63:Rings
6658:ISBN
6606:ISSN
6538:ISBN
6509:ISBN
6487:ISBN
5574:for
5270:) β
5238:and
5142:and
5123:and
5092:and
5070:) =
4781:Let
4714:Let
4696:for
4592:The
4326:Let
4054:Let
3999:) =
3919:) =
3875:Let
3730:) =
3707:) =
3648:Let
3554:the
3412:Let
2984:} +
2970:} =
2804:The
2780:and
2772:The
2753:The
2744:, P)
2724:(Ξ©,
2573:The
2539:) =
2468:and
2452:The
2429:The
2384:The
2373:The
2353:-by-
2341:free
2306:) =
2234:and
2158:The
2153:-Alg
1703:and
1432:-Mod
1345:) β¦
1292:and
1185:ring
1158:(so
1150:Let
890:Ring
881:Ring
727:ring
688:ring
6650:doi
6598:doi
6519:Zbl
6255:as
5119:If
5116:).
5096:in
4800:in
4554:= M
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