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product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. Often, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are
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In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic, even for abelian groups, vector spaces, or modules. As an example, consider the direct sum and direct product of (countably) infinitely many copies of the integers. An element in the direct
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A distinction is made between internal and external direct sums, though the two are isomorphic. If the summands are defined first, and then the direct sum is defined in terms of the summands, we have an external direct sum. For example, if we define the real numbers
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the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.
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if it is not a complemented subspace. For example, every vector subspace of a
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of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
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4552:{\displaystyle {\begin{alignedat}{4}\ \;&&M\times N&&\;\to \;&X\\&&(m,n)&&\;\mapsto \;&m+n\\\end{alignedat}}}
2846:
1490:. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as
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are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is
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154:. It is defined differently, but analogously, for different kinds of structures. As an example, the direct sum of two abelian groups
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2212:{\displaystyle \left(a_{1},b_{1}\right)\cdot \left(a_{2},b_{2}\right)=\left(a_{1}\circ a_{2},b_{1}\bullet b_{2}\right).}
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Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If
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4364:{\displaystyle \mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}\mathbf {A} &0\\0&\mathbf {B} \end{bmatrix}}.}
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2837:. So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.
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3605:{\displaystyle g\mapsto {\begin{pmatrix}\rho _{V}(g)&0\\0&\rho _{W}(g)\end{pmatrix}}.}
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4098:. In the category of rings, the coproduct is given by a construction similar to the
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398:; in other words addition is defined coordinate-wise. For example, the direct sum
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Operation in abstract algebra composing objects into "more complicated" objects
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804:. This is false, however, for some algebraic objects, like nonabelian groups.
1675:, then the direct sum is said to be internal. In this case, each element of
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The direct sum of finitely many abelian groups, vector spaces, or modules is
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We can also form direct sums with any finite number of summands, for example
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This definition generalizes to direct sums of finitely many abelian groups.
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Representation theory of finite groups § Direct sum of representations
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the phrase "direct sum" is used, while if the group operation is written
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The most familiar examples of this construction occur when considering
5875:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
1695:
is expressible uniquely as an algebraic combination of an element of
1293:{\displaystyle (x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})}
4090:, and should not be written as a direct sum. (The coproduct in the
5429:{\textstyle \alpha _{j}\colon \,A_{j}\to \bigoplus _{i\in I}A_{i}}
583:
5839:. Pallas Proefschriften. Amsterdam University Press. p. 26.
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Another equivalent way of defining the direct sum is as follows:
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is the natural map obtained by coordinate-wise action as above.
2398:{\textstyle \left(a_{i}\right)_{i\in I}\in \prod _{i\in I}A_{i}}
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If, on the other hand, we first define some algebraic structure
5348:{\textstyle \pi _{j}\colon \,\bigoplus _{i\in I}A_{i}\to A_{j}}
487:. A similar process can be used to form the direct sum of two
31:
2773:(defined identically to the direct sum of abelian groups) is
644:{\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}
3367:{\displaystyle \alpha :GL(V)\times GL(W)\to GL(V\oplus W)}
1917:
is a prototypical example of a direct sum. Given two such
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is the field, then the direct sum of the representations
3287:{\displaystyle \alpha \circ (\rho _{V}\times \rho _{W}),}
1884:{\displaystyle \mathbb {Z} _{6}=\{0,2,4\}\oplus \{0,3\}}
5464:(with the same additional structure) and homomorphisms
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is a vector subspace of a real or complex vector space
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is not a ring homomorphism since it fails to send 1 to
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1735:. For an example of an internal direct sum, consider
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5567:{\textstyle g\colon \,\bigoplus _{i\in I}A_{i}\to B}
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of the direct product that consists of the elements
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on direct sum of rings vs. direct product of rings.
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topological direct sum of the topological subgroups
2727:
Direct sums versus coproducts in category of groups
62:. Unsourced material may be challenged and removed.
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4847:then there always exists another vector subspace
4291:if both are square matrices (and to an analogous
3858:does not receive natural ring homomorphisms from
3092:{\displaystyle g\cdot (v,w)=(g\cdot v,g\cdot w).}
2698:agree and the direct sum is either of them, cf.
1820:. This is expressible as an internal direct sum
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5000:(which happens if and only if the addition map
4641:This is true if and only if when considered as
4236:{\displaystyle \mathbf {A} \oplus \mathbf {B} }
3404:are finite dimensional, then, given a basis of
1585:{\displaystyle \mathbb {R} \oplus \mathbb {R} }
421:{\displaystyle \mathbb {R} \oplus \mathbb {R} }
1764:(the integers modulo six), whose elements are
8:
2976:), the direct sum of the representations is
2766:{\displaystyle S_{3}\oplus \mathbb {Z} _{2}}
2668:. The construction may also be extended to
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2579:{\displaystyle \left(A_{i}\right)_{i\in I}}
2450:{\displaystyle \left(a_{i}\right)_{i\in I}}
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3777:Some authors will speak of the direct sum
304:. To add ordered pairs, we define the sum
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711:of the same kind. The direct sum is also
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122:Learn how and when to remove this message
4561:isomorphism of topological vector spaces
2317:{\displaystyle \bigoplus _{i\in I}A_{i}}
1300:, which is the same as vector addition.
890:{\displaystyle \bigoplus _{i\in I}A_{i}}
746:{\displaystyle A\oplus B\cong B\oplus A}
5684:
4648:(so scalar multiplication is ignored),
4374:Direct sum of topological vector spaces
1592:the direct sum is said to be external.
1468:{\textstyle A=\bigoplus _{i\in I}A_{i}}
5503:{\displaystyle g_{j}\colon A_{j}\to B}
3180:the vector space of the direct sum is
5837:Categorical Quantum Models and Logics
5737:The Theory of Groups: an Introduction
5444:. Given another algebraic structure
5270:{\textstyle \bigoplus _{i\in I}A_{i}}
5109:if there exists some vector subspace
2538:The direct sum of an infinite family
2017:. That is, the underlying set is the
1635:as a direct sum of two substructures
1056:{\textstyle \bigoplus _{i\in I}A_{i}}
7:
5822:
60:adding citations to reliable sources
3832:, but this should be avoided since
2853:direct sum of group representations
2841:Direct sum of group representations
1109:, but is strictly smaller when the
897:is defined to be the set of tuples
5619:{\displaystyle g\alpha _{j}=g_{j}}
2629:{\textstyle \prod _{i\in I}A_{i}.}
2222:For an arbitrary family of groups
25:
5518:, there is a unique homomorphism
5169:is the topological direct sum of
4165:Matrix addition § Direct sum
3511:{\displaystyle \rho _{V\oplus W}}
3485:are matrix-valued. In this case,
3232:{\displaystyle \rho _{V\oplus W}}
1535:Internal and external direct sums
1343:, their direct sum is written as
1102:{\textstyle \prod _{i\in I}A_{i}}
523:{\displaystyle A\oplus B\oplus C}
5658:Direct sum of topological groups
4384:Direct sum of topological groups
4346:
4327:
4311:
4303:
4277:
4255:
4229:
4221:
4199:
4177:
4144:{\displaystyle (R_{i})_{i\in I}}
3803:of two rings when they mean the
3740:is equal to their direct sum as
2826:{\displaystyle \mathbb {Z} _{2}}
1757:{\displaystyle \mathbb {Z} _{6}}
1426:, the direct sum may be written
936:{\displaystyle (a_{i})_{i\in I}}
847:{\displaystyle (A_{i})_{i\in I}}
480:{\displaystyle \mathbb {R} ^{2}}
220:consisting of the ordered pairs
36:
5739:, p. 177, Allyn and Bacon, 1965
3028:given component-wise, that is,
1813:{\displaystyle \{0,1,2,3,4,5\}}
47:needs additional citations for
5630:. Thus the direct sum is the
5558:
5494:
5397:
5332:
5025:{\displaystyle M\times N\to X}
5016:
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3923:{\displaystyle R\to R\times S}
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1971:
1959:
1939:
1927:
1287:
1235:
1229:
1203:
1197:
1171:
1156:axes. In this direct sum, the
976:{\displaystyle a_{i}\in A_{i}}
918:
904:
829:
815:
638:
626:
608:
596:
385:
361:
341:
329:
323:
311:
239:
227:
1:
5873:Graduate Texts in Mathematics
4092:category of commutative rings
3173:{\displaystyle (W,\rho _{W})}
3134:{\displaystyle (V,\rho _{V})}
2867:to it. Specifically, given a
1980:{\displaystyle (B,\bullet ),}
753:for any algebraic structures
651:for any algebraic structures
5220:is complemented. But every
5212:A vector subspace is called
4284:{\displaystyle \mathbf {B} }
4262:{\displaystyle \mathbf {A} }
4206:{\displaystyle \mathbf {B} }
4184:{\displaystyle \mathbf {A} }
1900:Direct sum of abelian groups
1555:{\displaystyle \mathbb {R} }
443:{\displaystyle \mathbb {R} }
18:Direct sum of abelian groups
4714:If this is the case and if
4169:For any arbitrary matrices
2664:, which are modules over a
2586:of non-trivial groups is a
2488:is the identity element of
2068:is defined component-wise:
347:{\displaystyle (a,b)+(c,d)}
5931:
5653:Direct sum of permutations
4377:
4162:
4086:is not a coproduct in the
3770:
3102:Given two representations
2844:
2777:a coproduct of the groups
2683:
2642:
2515:for all but finitely many
2061:{\displaystyle \,\cdot \,}
1945:{\displaystyle (A,\circ )}
1903:
1144:-plane, a two-dimensional
1016:for all but finitely many
4079:{\displaystyle R\times S}
3898:: in particular, the map
3851:{\displaystyle R\times S}
3825:{\displaystyle R\times S}
3796:{\displaystyle R\oplus S}
3478:{\displaystyle \rho _{W}}
3451:{\displaystyle \rho _{V}}
3199:{\displaystyle V\oplus W}
2995:{\displaystyle V\oplus W}
2953:(or, more generally, two
2039:{\displaystyle A\times B}
2006:{\displaystyle A\oplus B}
1362:{\displaystyle A\oplus B}
391:{\displaystyle (a+c,b+d)}
213:{\displaystyle A\oplus B}
194:is another abelian group
5698:, p.60, Springer, 1974,
5574:, called the sum of the
5034:vector space isomorphism
4892:algebraic complement of
4425:of two vector subspaces
4390:topological vector space
2731:However, the direct sum
2680:Direct sum in categories
2046:and the group operation
1906:Direct product of groups
715:up to isomorphism, i.e.
5673:Feferman-Vaught theorem
4616:topological complements
4096:tensor product of rings
4033:{\displaystyle 0\neq 1}
2409:, where by definition,
2271:{\displaystyle i\in I,}
1009:{\displaystyle a_{i}=0}
5835:Heunen, Chris (2009).
5620:
5568:
5504:
5458:
5430:
5349:
5277:comes equipped with a
5271:
5206:
5183:
5163:
5143:
5123:
5102:
5073:
5053:
5026:
4994:
4974:
4950:
4929:
4906:
4884:
4861:
4841:
4821:
4799:
4772:
4752:
4728:
4708:
4685:
4662:
4635:
4608:
4588:
4553:
4459:
4439:
4422:topological direct sum
4409:
4365:
4285:
4263:
4237:
4207:
4185:
4159:Direct sum of matrices
4145:
4080:
4054:
4034:
4008:
3976:
3944:
3924:
3892:
3872:
3852:
3826:
3797:
3757:
3734:
3714:
3694:
3674:
3648:
3628:
3614:Moreover, if we treat
3606:
3512:
3479:
3452:
3425:
3398:
3368:
3288:
3233:
3200:
3174:
3135:
3093:
3022:
3021:{\displaystyle g\in G}
2996:
2968:
2947:
2927:
2907:
2884:
2827:
2798:
2767:
2630:
2580:
2532:
2509:
2482:
2451:
2399:
2318:
2272:
2243:
2213:
2062:
2040:
2007:
1981:
1946:
1885:
1814:
1758:
1729:
1709:
1689:
1669:
1649:
1629:
1609:
1586:
1556:
1524:
1504:
1469:
1420:
1419:{\displaystyle i\in I}
1394:
1363:
1337:
1317:
1294:
1126:
1103:
1057:
1010:
977:
937:
891:
848:
798:canonically isomorphic
787:
767:
747:
705:
685:
665:
645:
573:
553:
524:
481:
444:
422:
392:
348:
298:
297:{\displaystyle b\in B}
272:
271:{\displaystyle a\in A}
246:
214:
188:
168:
5621:
5569:
5505:
5459:
5431:
5350:
5272:
5207:
5184:
5164:
5144:
5124:
5103:
5086:complemented subspace
5074:
5054:
5027:
4995:
4975:
4951:
4930:
4907:
4885:
4862:
4842:
4822:
4800:
4773:
4753:
4729:
4709:
4686:
4663:
4636:
4609:
4589:
4554:
4460:
4440:
4410:
4380:Complemented subspace
4366:
4286:
4264:
4245:block diagonal matrix
4238:
4208:
4186:
4146:
4081:
4055:
4035:
4009:
4007:{\displaystyle (1,1)}
3977:
3975:{\displaystyle (r,0)}
3945:
3925:
3893:
3873:
3853:
3827:
3798:
3758:
3735:
3715:
3695:
3675:
3649:
3629:
3607:
3513:
3480:
3453:
3426:
3424:{\displaystyle V,\,W}
3399:
3397:{\displaystyle V,\,W}
3369:
3289:
3234:
3206:and the homomorphism
3201:
3175:
3136:
3094:
3023:
2997:
2969:
2948:
2928:
2908:
2885:
2828:
2799:
2797:{\displaystyle S_{3}}
2768:
2651:direct sum of modules
2645:Direct sum of modules
2639:Direct sum of modules
2631:
2590:of the product group
2581:
2533:
2510:
2508:{\displaystyle A_{i}}
2483:
2481:{\displaystyle a_{i}}
2452:
2400:
2319:
2273:
2244:
2242:{\displaystyle A_{i}}
2214:
2063:
2041:
2013:is the same as their
2008:
1982:
1947:
1886:
1815:
1759:
1730:
1710:
1690:
1670:
1650:
1630:
1610:
1587:
1557:
1525:
1505:
1470:
1421:
1395:
1393:{\displaystyle A_{i}}
1364:
1338:
1318:
1303:Given two structures
1295:
1127:
1104:
1058:
1011:
978:
938:
892:
849:
800:to the corresponding
788:
768:
748:
706:
686:
666:
646:
574:
554:
525:
482:
452:real coordinate space
445:
423:
393:
349:
299:
273:
247:
245:{\displaystyle (a,b)}
215:
189:
169:
5692:Thomas W. Hungerford
5648:Direct sum of groups
5587:
5522:
5468:
5448:
5370:
5289:
5238:
5193:
5173:
5153:
5133:
5113:
5092:
5063:
5043:
5004:
4984:
4964:
4958:algebraic direct sum
4940:
4916:
4896:
4871:
4851:
4831:
4811:
4786:
4762:
4742:
4718:
4695:
4675:
4652:
4622:
4598:
4578:
4469:
4465:if the addition map
4449:
4429:
4396:
4299:
4273:
4251:
4217:
4195:
4173:
4109:
4064:
4044:
4018:
3986:
3954:
3934:
3902:
3882:
3862:
3836:
3810:
3781:
3744:
3724:
3704:
3684:
3661:
3654:as modules over the
3638:
3618:
3522:
3489:
3462:
3435:
3408:
3381:
3298:
3243:
3210:
3184:
3145:
3106:
3032:
3006:
2980:
2958:
2937:
2917:
2897:
2874:
2808:
2781:
2735:
2594:
2542:
2519:
2492:
2465:
2413:
2332:
2285:
2253:
2226:
2072:
2050:
2024:
1991:
1956:
1924:
1824:
1768:
1739:
1719:
1699:
1679:
1659:
1639:
1619:
1599:
1566:
1544:
1514:
1494:
1430:
1404:
1377:
1347:
1327:
1307:
1168:
1116:
1070:
1063:is contained in the
1024:
987:
947:
901:
858:
812:
777:
757:
719:
695:
675:
655:
593:
563:
552:{\displaystyle A,B,}
534:
502:
462:
432:
402:
358:
308:
282:
256:
224:
198:
178:
158:
56:improve this article
5634:in the appropriate
4563:(meaning that this
3767:Direct sum of rings
3002:with the action of
2657:into a new module.
1895:Types of direct sum
5811:Math StackExchange
5735:Joseph J. Rotman,
5663:Restricted product
5616:
5564:
5547:
5500:
5454:
5426:
5415:
5345:
5321:
5267:
5256:
5205:{\displaystyle N.}
5202:
5179:
5159:
5139:
5119:
5098:
5069:
5049:
5039:A vector subspace
5022:
4990:
4970:
4946:
4928:{\displaystyle X,}
4925:
4902:
4883:{\displaystyle X,}
4880:
4857:
4837:
4817:
4798:{\displaystyle X.}
4795:
4768:
4748:
4724:
4707:{\displaystyle N.}
4704:
4681:
4658:
4646:topological groups
4634:{\displaystyle X.}
4631:
4604:
4584:
4549:
4547:
4455:
4435:
4419:, is said to be a
4408:{\displaystyle X,}
4405:
4361:
4352:
4281:
4259:
4243:is defined as the
4233:
4203:
4181:
4141:
4076:
4050:
4030:
4004:
3972:
3940:
3920:
3888:
3868:
3848:
3822:
3793:
3756:{\displaystyle kG}
3753:
3730:
3710:
3690:
3673:{\displaystyle kG}
3670:
3644:
3624:
3602:
3593:
3508:
3475:
3448:
3421:
3394:
3364:
3284:
3229:
3196:
3170:
3131:
3089:
3018:
2992:
2964:
2943:
2923:
2903:
2880:
2859:of the underlying
2835:category of groups
2823:
2794:
2763:
2626:
2612:
2576:
2531:{\displaystyle i.}
2528:
2505:
2478:
2447:
2395:
2384:
2314:
2303:
2268:
2239:
2209:
2058:
2036:
2003:
1977:
1942:
1881:
1810:
1754:
1725:
1715:and an element of
1705:
1685:
1665:
1645:
1625:
1605:
1582:
1552:
1520:
1500:
1465:
1454:
1416:
1390:
1359:
1333:
1313:
1290:
1122:
1099:
1088:
1053:
1042:
1020:. The direct sum
1006:
973:
933:
887:
876:
844:
793:of the same kind.
783:
763:
743:
701:
681:
661:
641:
569:
549:
520:
477:
440:
418:
388:
344:
294:
268:
242:
210:
184:
164:
5882:978-0-387-95385-4
5532:
5457:{\displaystyle B}
5400:
5306:
5241:
5182:{\displaystyle M}
5162:{\displaystyle X}
5142:{\displaystyle X}
5122:{\displaystyle N}
5101:{\displaystyle X}
5079:is said to be a (
5072:{\displaystyle X}
5052:{\displaystyle M}
4993:{\displaystyle N}
4973:{\displaystyle M}
4949:{\displaystyle X}
4905:{\displaystyle M}
4860:{\displaystyle N}
4840:{\displaystyle X}
4820:{\displaystyle M}
4771:{\displaystyle N}
4751:{\displaystyle M}
4727:{\displaystyle X}
4684:{\displaystyle M}
4661:{\displaystyle X}
4607:{\displaystyle N}
4587:{\displaystyle M}
4574:), in which case
4478:
4458:{\displaystyle N}
4438:{\displaystyle M}
4213:, the direct sum
4088:category of rings
4053:{\displaystyle S}
3943:{\displaystyle r}
3891:{\displaystyle S}
3871:{\displaystyle R}
3733:{\displaystyle W}
3713:{\displaystyle V}
3693:{\displaystyle k}
3647:{\displaystyle W}
3627:{\displaystyle V}
2967:{\displaystyle G}
2946:{\displaystyle G}
2926:{\displaystyle W}
2906:{\displaystyle V}
2883:{\displaystyle G}
2705:General case: In
2692:additive category
2597:
2405:that have finite
2369:
2288:
2019:Cartesian product
1987:their direct sum
1728:{\displaystyle W}
1708:{\displaystyle V}
1688:{\displaystyle S}
1668:{\displaystyle W}
1648:{\displaystyle V}
1628:{\displaystyle S}
1608:{\displaystyle S}
1523:{\displaystyle *}
1503:{\displaystyle +}
1439:
1336:{\displaystyle B}
1316:{\displaystyle A}
1125:{\displaystyle I}
1073:
1027:
861:
854:, the direct sum
786:{\displaystyle B}
766:{\displaystyle A}
704:{\displaystyle C}
684:{\displaystyle B}
664:{\displaystyle A}
572:{\displaystyle C}
187:{\displaystyle B}
167:{\displaystyle A}
132:
131:
124:
106:
16:(Redirected from
5922:
5915:Abstract algebra
5901:
5851:
5850:
5832:
5826:
5820:
5814:
5808:
5802:
5801:
5799:
5798:
5792:
5786:. Archived from
5785:
5777:
5771:
5770:
5768:
5767:
5761:
5755:. Archived from
5754:
5746:
5740:
5733:
5727:
5715:
5706:
5689:
5625:
5623:
5622:
5617:
5615:
5614:
5602:
5601:
5573:
5571:
5570:
5565:
5557:
5556:
5546:
5509:
5507:
5506:
5501:
5493:
5492:
5480:
5479:
5463:
5461:
5460:
5455:
5435:
5433:
5432:
5427:
5425:
5424:
5414:
5396:
5395:
5382:
5381:
5354:
5352:
5351:
5346:
5344:
5343:
5331:
5330:
5320:
5301:
5300:
5276:
5274:
5273:
5268:
5266:
5265:
5255:
5211:
5209:
5208:
5203:
5188:
5186:
5185:
5180:
5168:
5166:
5165:
5160:
5148:
5146:
5145:
5140:
5128:
5126:
5125:
5120:
5107:
5105:
5104:
5099:
5078:
5076:
5075:
5070:
5058:
5056:
5055:
5050:
5031:
5029:
5028:
5023:
4999:
4997:
4996:
4991:
4979:
4977:
4976:
4971:
4955:
4953:
4952:
4947:
4934:
4932:
4931:
4926:
4911:
4909:
4908:
4903:
4889:
4887:
4886:
4881:
4866:
4864:
4863:
4858:
4846:
4844:
4843:
4838:
4826:
4824:
4823:
4818:
4804:
4802:
4801:
4796:
4778:are necessarily
4777:
4775:
4774:
4769:
4757:
4755:
4754:
4749:
4733:
4731:
4730:
4725:
4713:
4711:
4710:
4705:
4690:
4688:
4687:
4682:
4667:
4665:
4664:
4659:
4640:
4638:
4637:
4632:
4613:
4611:
4610:
4605:
4593:
4591:
4590:
4585:
4558:
4556:
4555:
4550:
4548:
4527:
4509:
4508:
4493:
4481:
4476:
4464:
4462:
4461:
4456:
4444:
4442:
4441:
4436:
4414:
4412:
4411:
4406:
4370:
4368:
4367:
4362:
4357:
4356:
4349:
4330:
4314:
4306:
4290:
4288:
4287:
4282:
4280:
4268:
4266:
4265:
4260:
4258:
4242:
4240:
4239:
4234:
4232:
4224:
4212:
4210:
4209:
4204:
4202:
4190:
4188:
4187:
4182:
4180:
4150:
4148:
4147:
4142:
4140:
4139:
4124:
4123:
4085:
4083:
4082:
4077:
4059:
4057:
4056:
4051:
4039:
4037:
4036:
4031:
4013:
4011:
4010:
4005:
3981:
3979:
3978:
3973:
3949:
3947:
3946:
3941:
3929:
3927:
3926:
3921:
3897:
3895:
3894:
3889:
3877:
3875:
3874:
3869:
3857:
3855:
3854:
3849:
3831:
3829:
3828:
3823:
3802:
3800:
3799:
3794:
3773:Product of rings
3762:
3760:
3759:
3754:
3739:
3737:
3736:
3731:
3719:
3717:
3716:
3711:
3699:
3697:
3696:
3691:
3679:
3677:
3676:
3671:
3653:
3651:
3650:
3645:
3633:
3631:
3630:
3625:
3611:
3609:
3608:
3603:
3598:
3597:
3581:
3580:
3548:
3547:
3517:
3515:
3514:
3509:
3507:
3506:
3484:
3482:
3481:
3476:
3474:
3473:
3457:
3455:
3454:
3449:
3447:
3446:
3430:
3428:
3427:
3422:
3403:
3401:
3400:
3395:
3377:Furthermore, if
3373:
3371:
3370:
3365:
3293:
3291:
3290:
3285:
3277:
3276:
3264:
3263:
3238:
3236:
3235:
3230:
3228:
3227:
3205:
3203:
3202:
3197:
3179:
3177:
3176:
3171:
3166:
3165:
3140:
3138:
3137:
3132:
3127:
3126:
3098:
3096:
3095:
3090:
3027:
3025:
3024:
3019:
3001:
2999:
2998:
2993:
2973:
2971:
2970:
2965:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2912:
2910:
2909:
2904:
2889:
2887:
2886:
2881:
2855:generalizes the
2832:
2830:
2829:
2824:
2822:
2821:
2816:
2803:
2801:
2800:
2795:
2793:
2792:
2772:
2770:
2769:
2764:
2762:
2761:
2756:
2747:
2746:
2714:
2713:
2635:
2633:
2632:
2627:
2622:
2621:
2611:
2585:
2583:
2582:
2577:
2575:
2574:
2563:
2559:
2558:
2537:
2535:
2534:
2529:
2514:
2512:
2511:
2506:
2504:
2503:
2487:
2485:
2484:
2479:
2477:
2476:
2457:is said to have
2456:
2454:
2453:
2448:
2446:
2445:
2434:
2430:
2429:
2404:
2402:
2401:
2396:
2394:
2393:
2383:
2365:
2364:
2353:
2349:
2348:
2323:
2321:
2320:
2315:
2313:
2312:
2302:
2277:
2275:
2274:
2269:
2248:
2246:
2245:
2240:
2238:
2237:
2218:
2216:
2215:
2210:
2205:
2201:
2200:
2199:
2187:
2186:
2174:
2173:
2161:
2160:
2143:
2139:
2138:
2137:
2125:
2124:
2107:
2103:
2102:
2101:
2089:
2088:
2067:
2065:
2064:
2059:
2045:
2043:
2042:
2037:
2012:
2010:
2009:
2004:
1986:
1984:
1983:
1978:
1951:
1949:
1948:
1943:
1890:
1888:
1887:
1882:
1838:
1837:
1832:
1819:
1817:
1816:
1811:
1763:
1761:
1760:
1755:
1753:
1752:
1747:
1734:
1732:
1731:
1726:
1714:
1712:
1711:
1706:
1694:
1692:
1691:
1686:
1674:
1672:
1671:
1666:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1591:
1589:
1588:
1583:
1581:
1573:
1562:and then define
1561:
1559:
1558:
1553:
1551:
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1474:
1472:
1471:
1466:
1464:
1463:
1453:
1425:
1423:
1422:
1417:
1399:
1397:
1396:
1391:
1389:
1388:
1368:
1366:
1365:
1360:
1342:
1340:
1339:
1334:
1322:
1320:
1319:
1314:
1299:
1297:
1296:
1291:
1286:
1285:
1273:
1272:
1260:
1259:
1247:
1246:
1228:
1227:
1215:
1214:
1196:
1195:
1183:
1182:
1131:
1129:
1128:
1123:
1108:
1106:
1105:
1100:
1098:
1097:
1087:
1062:
1060:
1059:
1054:
1052:
1051:
1041:
1015:
1013:
1012:
1007:
999:
998:
982:
980:
979:
974:
972:
971:
959:
958:
942:
940:
939:
934:
932:
931:
916:
915:
896:
894:
893:
888:
886:
885:
875:
853:
851:
850:
845:
843:
842:
827:
826:
792:
790:
789:
784:
772:
770:
769:
764:
752:
750:
749:
744:
710:
708:
707:
702:
690:
688:
687:
682:
670:
668:
667:
662:
650:
648:
647:
642:
578:
576:
575:
570:
558:
556:
555:
550:
529:
527:
526:
521:
486:
484:
483:
478:
476:
475:
470:
449:
447:
446:
441:
439:
427:
425:
424:
419:
417:
409:
397:
395:
394:
389:
353:
351:
350:
345:
303:
301:
300:
295:
277:
275:
274:
269:
251:
249:
248:
243:
219:
217:
216:
211:
193:
191:
190:
185:
173:
171:
170:
165:
148:abstract algebra
127:
120:
116:
113:
107:
105:
64:
40:
32:
21:
5930:
5929:
5925:
5924:
5923:
5921:
5920:
5919:
5905:
5904:
5883:
5863:
5860:
5855:
5854:
5847:
5834:
5833:
5829:
5821:
5817:
5809:
5805:
5796:
5794:
5790:
5783:
5779:
5778:
5774:
5765:
5763:
5759:
5752:
5748:
5747:
5743:
5734:
5730:
5716:
5709:
5690:
5686:
5681:
5644:
5606:
5593:
5585:
5584:
5582:
5548:
5520:
5519:
5484:
5471:
5466:
5465:
5446:
5445:
5416:
5387:
5373:
5368:
5367:
5335:
5322:
5292:
5287:
5286:
5257:
5236:
5235:
5234:The direct sum
5230:
5191:
5190:
5171:
5170:
5151:
5150:
5131:
5130:
5111:
5110:
5090:
5089:
5061:
5060:
5041:
5040:
5002:
5001:
4982:
4981:
4962:
4961:
4938:
4937:
4914:
4913:
4894:
4893:
4869:
4868:
4849:
4848:
4829:
4828:
4809:
4808:
4784:
4783:
4760:
4759:
4740:
4739:
4716:
4715:
4693:
4692:
4673:
4672:
4650:
4649:
4620:
4619:
4614:are said to be
4596:
4595:
4576:
4575:
4546:
4545:
4534:
4526:
4506:
4505:
4500:
4492:
4480:
4467:
4466:
4447:
4446:
4427:
4426:
4394:
4393:
4386:
4378:Main articles:
4376:
4351:
4350:
4343:
4337:
4336:
4331:
4319:
4297:
4296:
4271:
4270:
4249:
4248:
4215:
4214:
4193:
4192:
4171:
4170:
4167:
4161:
4125:
4115:
4107:
4106:
4062:
4061:
4042:
4041:
4016:
4015:
4014:(assuming that
3984:
3983:
3952:
3951:
3932:
3931:
3900:
3899:
3880:
3879:
3860:
3859:
3834:
3833:
3808:
3807:
3779:
3778:
3775:
3769:
3742:
3741:
3722:
3721:
3702:
3701:
3682:
3681:
3659:
3658:
3636:
3635:
3616:
3615:
3592:
3591:
3572:
3570:
3564:
3563:
3558:
3539:
3532:
3520:
3519:
3492:
3487:
3486:
3465:
3460:
3459:
3438:
3433:
3432:
3406:
3405:
3379:
3378:
3296:
3295:
3268:
3255:
3241:
3240:
3213:
3208:
3207:
3182:
3181:
3157:
3143:
3142:
3118:
3104:
3103:
3030:
3029:
3004:
3003:
2978:
2977:
2956:
2955:
2935:
2934:
2915:
2914:
2895:
2894:
2892:representations
2872:
2871:
2849:
2843:
2811:
2806:
2805:
2784:
2779:
2778:
2751:
2738:
2733:
2732:
2729:
2711:
2710:
2707:category theory
2688:
2682:
2647:
2641:
2613:
2592:
2591:
2588:proper subgroup
2550:
2546:
2545:
2540:
2539:
2517:
2516:
2495:
2490:
2489:
2468:
2463:
2462:
2421:
2417:
2416:
2411:
2410:
2385:
2340:
2336:
2335:
2330:
2329:
2304:
2283:
2282:
2251:
2250:
2229:
2224:
2223:
2191:
2178:
2165:
2152:
2151:
2147:
2129:
2116:
2115:
2111:
2093:
2080:
2079:
2075:
2070:
2069:
2048:
2047:
2022:
2021:
1989:
1988:
1954:
1953:
1922:
1921:
1908:
1902:
1897:
1827:
1822:
1821:
1766:
1765:
1742:
1737:
1736:
1717:
1716:
1697:
1696:
1677:
1676:
1657:
1656:
1637:
1636:
1617:
1616:
1615:and then write
1597:
1596:
1564:
1563:
1542:
1541:
1537:
1512:
1511:
1492:
1491:
1480:
1455:
1428:
1427:
1402:
1401:
1400:, indexed with
1380:
1375:
1374:
1345:
1344:
1325:
1324:
1305:
1304:
1277:
1264:
1251:
1238:
1219:
1206:
1187:
1174:
1166:
1165:
1138:
1114:
1113:
1089:
1068:
1067:
1043:
1022:
1021:
990:
985:
984:
963:
950:
945:
944:
917:
907:
899:
898:
877:
856:
855:
828:
818:
810:
809:
775:
774:
755:
754:
717:
716:
693:
692:
673:
672:
653:
652:
591:
590:
561:
560:
532:
531:
500:
499:
465:
460:
459:
456:Cartesian plane
430:
429:
400:
399:
356:
355:
306:
305:
280:
279:
254:
253:
222:
221:
196:
195:
176:
175:
156:
155:
128:
117:
111:
108:
65:
63:
53:
41:
30:
23:
22:
15:
12:
11:
5:
5928:
5926:
5918:
5917:
5907:
5906:
5903:
5902:
5881:
5859:
5856:
5853:
5852:
5846:978-9085550242
5845:
5827:
5825:, section I.11
5815:
5803:
5772:
5741:
5728:
5707:
5683:
5682:
5680:
5677:
5676:
5675:
5670:
5665:
5660:
5655:
5650:
5643:
5640:
5613:
5609:
5605:
5600:
5596:
5592:
5578:
5563:
5560:
5555:
5551:
5545:
5542:
5539:
5535:
5530:
5527:
5499:
5496:
5491:
5487:
5483:
5478:
5474:
5453:
5423:
5419:
5413:
5410:
5407:
5403:
5399:
5394:
5390:
5385:
5380:
5376:
5342:
5338:
5334:
5329:
5325:
5319:
5316:
5313:
5309:
5304:
5299:
5295:
5264:
5260:
5254:
5251:
5248:
5244:
5229:
5226:
5215:
5214:uncomplemented
5201:
5198:
5178:
5158:
5138:
5118:
5108:
5097:
5082:
5068:
5048:
5021:
5018:
5015:
5012:
5009:
4989:
4969:
4959:
4945:
4935:
4924:
4921:
4901:
4879:
4876:
4856:
4836:
4816:
4794:
4791:
4767:
4747:
4723:
4703:
4700:
4680:
4657:
4630:
4627:
4617:
4603:
4583:
4544:
4541:
4538:
4535:
4532:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4494:
4491:
4488:
4485:
4482:
4475:
4474:
4454:
4434:
4424:
4404:
4401:
4375:
4372:
4360:
4355:
4348:
4344:
4342:
4339:
4338:
4335:
4332:
4329:
4325:
4324:
4322:
4317:
4313:
4309:
4305:
4279:
4257:
4231:
4227:
4223:
4201:
4179:
4160:
4157:
4138:
4135:
4132:
4128:
4122:
4118:
4114:
4075:
4072:
4069:
4049:
4029:
4026:
4023:
4003:
4000:
3997:
3994:
3991:
3971:
3968:
3965:
3962:
3959:
3939:
3919:
3916:
3913:
3910:
3907:
3887:
3867:
3847:
3844:
3841:
3821:
3818:
3815:
3805:direct product
3792:
3789:
3786:
3771:Main article:
3768:
3765:
3752:
3749:
3729:
3709:
3689:
3669:
3666:
3643:
3623:
3601:
3596:
3590:
3587:
3584:
3579:
3575:
3571:
3569:
3566:
3565:
3562:
3559:
3557:
3554:
3551:
3546:
3542:
3538:
3537:
3535:
3530:
3527:
3505:
3502:
3499:
3495:
3472:
3468:
3445:
3441:
3420:
3416:
3413:
3393:
3389:
3386:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3283:
3280:
3275:
3271:
3267:
3262:
3258:
3254:
3251:
3248:
3226:
3223:
3220:
3216:
3195:
3192:
3189:
3169:
3164:
3160:
3156:
3153:
3150:
3130:
3125:
3121:
3117:
3114:
3111:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3017:
3014:
3011:
2991:
2988:
2985:
2963:
2942:
2922:
2902:
2879:
2842:
2839:
2820:
2815:
2791:
2787:
2776:
2760:
2755:
2750:
2745:
2741:
2728:
2725:
2684:Main article:
2681:
2678:
2674:Hilbert spaces
2643:Main article:
2640:
2637:
2625:
2620:
2616:
2610:
2607:
2604:
2600:
2573:
2570:
2567:
2562:
2557:
2553:
2549:
2527:
2524:
2502:
2498:
2475:
2471:
2460:
2459:finite support
2444:
2441:
2438:
2433:
2428:
2424:
2420:
2392:
2388:
2382:
2379:
2376:
2372:
2368:
2363:
2360:
2357:
2352:
2347:
2343:
2339:
2311:
2307:
2301:
2298:
2295:
2291:
2281:
2267:
2264:
2261:
2258:
2236:
2232:
2208:
2204:
2198:
2194:
2190:
2185:
2181:
2177:
2172:
2168:
2164:
2159:
2155:
2150:
2146:
2142:
2136:
2132:
2128:
2123:
2119:
2114:
2110:
2106:
2100:
2096:
2092:
2087:
2083:
2078:
2056:
2035:
2032:
2029:
2015:direct product
2002:
1999:
1996:
1976:
1973:
1970:
1967:
1964:
1961:
1941:
1938:
1935:
1932:
1929:
1914:abelian groups
1912:direct sum of
1904:Main article:
1901:
1898:
1896:
1893:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1836:
1831:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1751:
1746:
1724:
1704:
1684:
1664:
1644:
1624:
1604:
1580:
1576:
1572:
1550:
1536:
1533:
1519:
1499:
1484:direct summand
1478:
1462:
1458:
1452:
1449:
1446:
1442:
1438:
1435:
1415:
1412:
1409:
1387:
1383:
1373:of structures
1371:indexed family
1358:
1355:
1352:
1332:
1312:
1289:
1284:
1280:
1276:
1271:
1267:
1263:
1258:
1254:
1250:
1245:
1241:
1237:
1234:
1231:
1226:
1222:
1218:
1213:
1209:
1205:
1202:
1199:
1194:
1190:
1186:
1181:
1177:
1173:
1137:
1134:
1121:
1096:
1092:
1086:
1083:
1080:
1076:
1065:direct product
1050:
1046:
1040:
1037:
1034:
1030:
1005:
1002:
997:
993:
970:
966:
962:
957:
953:
930:
927:
924:
920:
914:
910:
906:
884:
880:
874:
871:
868:
864:
841:
838:
835:
831:
825:
821:
817:
802:direct product
782:
762:
742:
739:
736:
733:
730:
727:
724:
700:
680:
660:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
568:
548:
545:
542:
539:
519:
516:
513:
510:
507:
474:
469:
438:
416:
412:
408:
387:
384:
381:
378:
375:
372:
369:
366:
363:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
293:
290:
287:
267:
264:
261:
241:
238:
235:
232:
229:
209:
206:
203:
183:
163:
150:, a branch of
130:
129:
44:
42:
35:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5927:
5916:
5913:
5912:
5910:
5900:
5896:
5892:
5888:
5884:
5878:
5874:
5870:
5866:
5862:
5861:
5857:
5848:
5842:
5838:
5831:
5828:
5824:
5819:
5816:
5812:
5807:
5804:
5793:on 2006-09-17
5789:
5782:
5776:
5773:
5762:on 2013-05-22
5758:
5751:
5745:
5742:
5738:
5732:
5729:
5726:
5724:
5719:
5714:
5712:
5708:
5705:
5701:
5697:
5693:
5688:
5685:
5678:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5645:
5641:
5639:
5637:
5633:
5629:
5611:
5607:
5603:
5598:
5594:
5590:
5581:
5577:
5561:
5553:
5549:
5543:
5540:
5537:
5533:
5528:
5525:
5517:
5513:
5497:
5489:
5485:
5481:
5476:
5472:
5451:
5443:
5439:
5421:
5417:
5411:
5408:
5405:
5401:
5392:
5388:
5383:
5378:
5374:
5366:
5362:
5358:
5340:
5336:
5327:
5323:
5317:
5314:
5311:
5307:
5302:
5297:
5293:
5285:
5282:
5281:
5262:
5258:
5252:
5249:
5246:
5242:
5232:
5228:Homomorphisms
5227:
5225:
5223:
5219:
5218:Hilbert space
5213:
5199:
5196:
5176:
5156:
5136:
5116:
5095:
5087:
5084:
5081:topologically
5080:
5066:
5046:
5037:
5035:
5019:
5013:
5010:
5007:
4987:
4967:
4957:
4943:
4922:
4919:
4899:
4891:
4877:
4874:
4854:
4834:
4814:
4805:
4792:
4789:
4782:subspaces of
4781:
4765:
4745:
4737:
4721:
4701:
4698:
4678:
4671:
4655:
4647:
4644:
4628:
4625:
4615:
4601:
4581:
4573:
4572:homeomorphism
4570:
4566:
4562:
4542:
4539:
4536:
4520:
4517:
4514:
4502:
4489:
4486:
4483:
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3158:
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3115:
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3100:
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3009:
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2718:
2708:
2703:
2701:
2697:
2693:
2687:
2679:
2677:
2675:
2671:
2670:Banach spaces
2667:
2663:
2662:vector spaces
2658:
2656:
2652:
2646:
2638:
2636:
2623:
2618:
2614:
2608:
2605:
2602:
2598:
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2500:
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2256:
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2196:
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2183:
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2175:
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2166:
2162:
2157:
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2148:
2144:
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2134:
2130:
2126:
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2117:
2112:
2108:
2104:
2098:
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2090:
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2076:
2054:
2033:
2030:
2027:
2020:
2016:
2000:
1997:
1994:
1974:
1968:
1965:
1962:
1936:
1933:
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1920:
1916:
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1899:
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968:
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928:
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922:
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585:
582:
566:
546:
543:
540:
537:
517:
514:
511:
508:
505:
496:
494:
490:
489:vector spaces
472:
457:
453:
410:
382:
379:
376:
373:
370:
367:
364:
338:
335:
332:
326:
320:
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314:
291:
288:
285:
265:
262:
259:
236:
233:
230:
207:
204:
201:
181:
161:
153:
149:
145:
141:
137:
126:
123:
115:
112:December 2013
104:
101:
97:
94:
90:
87:
83:
80:
76:
73: –
72:
68:
67:Find sources:
61:
57:
51:
50:
45:This article
43:
39:
34:
33:
27:
19:
5868:
5836:
5830:
5818:
5806:
5795:. Retrieved
5788:the original
5775:
5764:. Retrieved
5757:the original
5744:
5736:
5731:
5722:
5695:
5687:
5627:
5583:, such that
5579:
5575:
5515:
5511:
5441:
5437:
5365:coprojection
5364:
5360:
5356:
5284:homomorphism
5278:
5233:
5231:
5222:Banach space
5038:
4806:
4417:Banach space
4387:
4293:block matrix
4168:
4104:
4102:of groups.)
4100:free product
3776:
3613:
3518:is given as
3376:
3239:is given by
3101:
2865:group action
2852:
2850:
2730:
2704:
2689:
2659:
2650:
2648:
2221:
1911:
1909:
1594:
1538:
1487:
1483:
1482:is called a
1476:
1302:
1161:
1157:
1153:
1149:
1146:vector space
1141:
1139:
1017:
806:
795:
497:
135:
133:
118:
109:
99:
92:
85:
78:
71:"Direct sum"
66:
54:Please help
49:verification
46:
26:
5865:Lang, Serge
5668:Whitney sum
4295:, if not).
2863:, adding a
2249:indexed by
1369:. Given an
713:commutative
589:. That is,
587:isomorphism
581:associative
530:, provided
152:mathematics
5899:0984.00001
5858:References
5797:2014-01-14
5781:"Appendix"
5766:2014-01-14
5718:Direct Sum
5704:0387905189
5510:for every
5280:projection
5149:such that
4936:such that
4890:called an
4565:linear map
4415:such as a
4163:See also:
3656:group ring
2857:direct sum
2845:See also:
2712:direct sum
2696:coproducts
2280:direct sum
983:such that
144:structures
136:direct sum
82:newspapers
5823:Lang 2002
5632:coproduct
5595:α
5559:→
5541:∈
5534:⨁
5529::
5495:→
5482::
5436:for each
5409:∈
5402:⨁
5398:→
5384::
5375:α
5355:for each
5333:→
5315:∈
5308:⨁
5303::
5294:π
5250:∈
5243:⨁
5017:→
5011:×
4736:Hausdorff
4569:bijective
4531:↦
4497:→
4487:×
4308:⊕
4226:⊕
4134:∈
4071:×
4060:). Thus
4025:≠
3915:×
3909:→
3843:×
3817:×
3788:⊕
3763:modules.
3574:ρ
3541:ρ
3529:↦
3501:⊕
3494:ρ
3467:ρ
3440:ρ
3356:⊕
3341:→
3323:×
3302:α
3270:ρ
3266:×
3257:ρ
3250:∘
3247:α
3222:⊕
3215:ρ
3191:⊕
3159:ρ
3120:ρ
3078:⋅
3066:⋅
3039:⋅
3013:∈
2987:⊕
2749:⊕
2717:coproduct
2700:biproduct
2686:Coproduct
2606:∈
2599:∏
2569:∈
2440:∈
2378:∈
2371:∏
2367:∈
2359:∈
2297:∈
2290:⨁
2260:∈
2189:∙
2163:∘
2109:⋅
2055:⋅
2031:×
1998:⊕
1969:∙
1937:∘
1864:⊕
1575:⊕
1518:∗
1448:∈
1441:⨁
1411:∈
1354:⊕
1111:index set
1082:∈
1075:∏
1036:∈
1029:⨁
961:∈
926:∈
870:∈
863:⨁
837:∈
738:⊕
732:≅
726:⊕
633:⊕
624:⊕
618:≅
612:⊕
603:⊕
515:⊕
509:⊕
454:, is the
411:⊕
289:∈
263:∈
205:⊕
140:operation
5909:Category
5867:(2002),
5750:""p.45""
5642:See also
5636:category
5626:for all
4643:additive
3930:sending
3680:, where
2974:-modules
2890:and two
2721:category
2326:subgroup
1475:. Each
1136:Examples
428:, where
142:between
5891:1878556
5869:Algebra
5720:at the
5696:Algebra
4956:is the
4668:is the
4094:is the
2861:modules
2833:in the
2719:in the
2655:modules
2407:support
2324:is the
493:modules
491:or two
96:scholar
5897:
5889:
5879:
5843:
5702:
5363:and a
4780:closed
4559:is an
4477:
4392:(TVS)
3294:where
2278:their
1919:groups
691:, and
354:to be
252:where
138:is an
98:
91:
84:
77:
69:
5791:(PDF)
5784:(PDF)
5760:(PDF)
5753:(PDF)
5679:Notes
5032:is a
4738:then
4567:is a
2869:group
2666:field
943:with
584:up to
103:JSTOR
89:books
5877:ISBN
5841:ISBN
5700:ISBN
5189:and
4980:and
4758:and
4691:and
4594:and
4445:and
4382:and
4269:and
4191:and
3878:and
3720:and
3634:and
3458:and
3141:and
2913:and
2851:The
2804:and
2709:the
2672:and
2649:The
1952:and
1910:The
1655:and
1323:and
1160:and
1152:and
1140:The
773:and
559:and
278:and
174:and
134:The
75:news
5895:Zbl
5725:Lab
5514:in
5440:in
5359:in
5129:of
5088:of
5059:of
4960:of
4912:in
4867:of
4807:If
4734:is
4618:in
4247:of
4153:rng
4040:in
3950:to
2933:of
2775:not
2690:An
2461:if
1486:of
450:is
146:in
58:by
5911::
5893:,
5887:MR
5885:,
5871:,
5710:^
5694:,
5638:.
5083:)
4388:A
3431:,
2702:.
2676:.
1891:.
1142:xy
671:,
495:.
458:,
5849:.
5800:.
5769:.
5723:n
5628:j
5612:j
5608:g
5604:=
5599:j
5591:g
5580:j
5576:g
5562:B
5554:i
5550:A
5544:I
5538:i
5526:g
5516:I
5512:j
5498:B
5490:j
5486:A
5477:j
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5406:i
5393:j
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5379:j
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5337:A
5328:i
5324:A
5318:I
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5298:j
5263:i
5259:A
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5247:i
5200:.
5197:N
5177:M
5157:X
5137:X
5117:N
5096:X
5067:X
5047:M
5020:X
5014:N
5008:M
4988:N
4968:M
4944:X
4923:,
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4878:,
4875:X
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4815:M
4793:.
4790:X
4766:N
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4722:X
4702:.
4699:N
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4656:X
4629:.
4626:X
4602:N
4582:M
4543:n
4540:+
4537:m
4524:)
4521:n
4518:,
4515:m
4512:(
4503:X
4490:N
4484:M
4453:N
4433:M
4403:,
4400:X
4359:.
4354:]
4347:B
4341:0
4334:0
4328:A
4321:[
4316:=
4312:B
4304:A
4278:B
4256:A
4230:B
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4200:B
4178:A
4137:I
4131:i
4127:)
4121:i
4117:R
4113:(
4074:S
4068:R
4048:S
4028:1
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3999:1
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3993:1
3990:(
3970:)
3967:0
3964:,
3961:r
3958:(
3938:r
3918:S
3912:R
3906:R
3886:S
3866:R
3846:S
3840:R
3820:S
3814:R
3791:S
3785:R
3751:G
3748:k
3728:W
3708:V
3688:k
3668:G
3665:k
3642:W
3622:V
3600:.
3595:)
3589:)
3586:g
3583:(
3578:W
3568:0
3561:0
3556:)
3553:g
3550:(
3545:V
3534:(
3526:g
3504:W
3498:V
3471:W
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3419:W
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3385:V
3362:)
3359:W
3353:V
3350:(
3347:L
3344:G
3338:)
3335:W
3332:(
3329:L
3326:G
3320:)
3317:V
3314:(
3311:L
3308:G
3305::
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3225:W
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3163:W
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3152:W
3149:(
3129:)
3124:V
3116:,
3113:V
3110:(
3087:.
3084:)
3081:w
3075:g
3072:,
3069:v
3063:g
3060:(
3057:=
3054:)
3051:w
3048:,
3045:v
3042:(
3036:g
3016:G
3010:g
2990:W
2984:V
2962:G
2941:G
2921:W
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2878:G
2819:2
2814:Z
2790:3
2786:S
2759:2
2754:Z
2744:3
2740:S
2624:.
2619:i
2615:A
2609:I
2603:i
2572:I
2566:i
2561:)
2556:i
2552:A
2548:(
2526:.
2523:i
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2474:i
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2443:I
2437:i
2432:)
2427:i
2423:a
2419:(
2391:i
2387:A
2381:I
2375:i
2362:I
2356:i
2351:)
2346:i
2342:a
2338:(
2310:i
2306:A
2300:I
2294:i
2266:,
2263:I
2257:i
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2231:A
2207:.
2203:)
2197:2
2193:b
2184:1
2180:b
2176:,
2171:2
2167:a
2158:1
2154:a
2149:(
2145:=
2141:)
2135:2
2131:b
2127:,
2122:2
2118:a
2113:(
2105:)
2099:1
2095:b
2091:,
2086:1
2082:a
2077:(
2034:B
2028:A
2001:B
1995:A
1975:,
1972:)
1966:,
1963:B
1960:(
1940:)
1934:,
1931:A
1928:(
1879:}
1876:3
1873:,
1870:0
1867:{
1861:}
1858:4
1855:,
1852:2
1849:,
1846:0
1843:{
1840:=
1835:6
1830:Z
1808:}
1805:5
1802:,
1799:4
1796:,
1793:3
1790:,
1787:2
1784:,
1781:1
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1775:0
1772:{
1750:6
1745:Z
1723:W
1703:V
1683:S
1663:W
1643:V
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1603:S
1579:R
1571:R
1549:R
1498:+
1488:A
1479:i
1477:A
1461:i
1457:A
1451:I
1445:i
1437:=
1434:A
1414:I
1408:i
1386:i
1382:A
1357:B
1351:A
1331:B
1311:A
1288:)
1283:2
1279:y
1275:+
1270:1
1266:y
1262:,
1257:2
1253:x
1249:+
1244:1
1240:x
1236:(
1233:=
1230:)
1225:2
1221:y
1217:,
1212:2
1208:x
1204:(
1201:+
1198:)
1193:1
1189:y
1185:,
1180:1
1176:x
1172:(
1162:y
1158:x
1154:y
1150:x
1120:I
1095:i
1091:A
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1045:A
1039:I
1033:i
1018:i
1004:0
1001:=
996:i
992:a
969:i
965:A
956:i
952:a
929:I
923:i
919:)
913:i
909:a
905:(
883:i
879:A
873:I
867:i
840:I
834:i
830:)
824:i
820:A
816:(
781:B
761:A
741:A
735:B
729:B
723:A
699:C
679:B
659:A
639:)
636:C
630:B
627:(
621:A
615:C
609:)
606:B
600:A
597:(
567:C
547:,
544:B
541:,
538:A
518:C
512:B
506:A
473:2
468:R
437:R
415:R
407:R
386:)
383:d
380:+
377:b
374:,
371:c
368:+
365:a
362:(
342:)
339:d
336:,
333:c
330:(
327:+
324:)
321:b
318:,
315:a
312:(
292:B
286:b
266:A
260:a
240:)
237:b
234:,
231:a
228:(
208:B
202:A
182:B
162:A
125:)
119:(
114:)
110:(
100:·
93:·
86:·
79:·
52:.
20:)
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