36:
1168:
are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about
1802:
2940:
3197:
2822:
1419:
1355:
1480:
306:
598:
551:
514:
260:
1517:
1044:
3661:
on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
703:
1172:
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "
2839:
4420:
4314:
160:
57:
1037:
4403:
4395:
696:
648:
79:
3549:
of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules that are not simple (e.g.
3113:
2778:
3207:
of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphism of
1923:
are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
4472:
4443:
1030:
3256:. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
2598:
689:
1188:) the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique
4438:
899:
406:
50:
44:
1362:
1298:
4107:
3672:
1426:
166:
1912:
is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
270:
4285:
3658:
3601:
3475:
3375:
3357:
2943:
2480:
2215:
1526:. Often the symbol Β· is omitted, but in this article we use it and reserve juxtaposition for multiplication in
1180:. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a
61:
4091:
2594:
641:
565:
444:
394:
2953:, together with the two binary operations + (the module spanned by the union of the arguments) and β©, forms a
1920:
1806:
1590:
990:
453:
146:
4477:
3675:
on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
2464:
2267:
2201:
1988:
1893:
1193:
1181:
1177:
610:
461:
412:
193:
4275:
4270:
3546:
3542:
3435:
1108:
1097:
4245:
are a module (in this generalized sense only). This allows a further generalization of the concept of
527:
490:
4084:
3885:
2018:
1905:
1145:
1074:
977:
969:
941:
884:
826:
334:
4206:
has only a single point, then this is a module category in the old sense over the commutative ring O
1585:
omit condition 4 in the definition above; they would call the structures defined above "unital left
4175:
3619:
3345:
3314:
3286:
3075:
2193:
1713:
1161:
1149:
1120:
1116:
1112:
1070:
995:
985:
836:
736:
728:
719:
616:
424:
375:
320:
214:
200:
128:
96:
243:
4199:
4054:
3859:
3493:
3403:
1901:
1810:
1487:
1225:
1205:
1157:
1128:
1124:
1078:
801:
792:
750:
629:
187:
115:
4349:
1908:
set, since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a
3531:
is a direct sum (finite or not) of simple modules. Historically these modules are also called
1160:, so the module concept represents a significant generalization. In commutative algebra, both
4416:
4399:
4391:
4310:
3897:
3654:
3622:
is a module over a ring such that 0 is the only element annihilated by a regular element (non
3528:
3448:
3057:
2613:
2460:
2448:
2436:
2251:
1939:
1916:
670:
467:
232:
173:
2463:, every projective module is isomorphic to the module of sections of some vector bundle; the
4433:
4370:
4185:
4135:
4099:
3461:
3333:
3260:
1684:
1153:
1101:
821:
676:
662:
476:
418:
381:
181:
154:
140:
4408:
4280:
3668:
3560:
2959:
2954:
2396:
2392:
1897:
1748:
1744:
1201:
913:
907:
894:
874:
865:
831:
768:
438:
388:
226:
1096:
Like a vector space, a module is an additive abelian group, and scalar multiplication is
556:
4331:
1152:
and acts on the vectors by scalar multiplication, subject to certain axioms such as the
3781:
3550:
3479:
3452:
2533:
1935:
1582:
955:
482:
4466:
3682:
3507:
3417:
2526:
2444:
1979:-module {0} consisting only of its identity element. Modules of this type are called
1821:
1608:
is an abelian group together with both a left scalar multiplication Β· by elements of
1240:
1196:
condition, unlike vector spaces, which always have a (possibly infinite) basis whose
1165:
1086:
841:
806:
763:
623:
519:
134:
4390:, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992,
4246:
4161:
3623:
3204:
2432:
2416:
1909:
1721:
1173:
1066:
1015:
946:
780:
655:
430:
326:
1208:
vector spaces, or certain well-behaved infinite-dimensional vector spaces such as
4224:
3709:
3497:
3471:
3427:
3241:
2404:
1980:
1197:
1189:
1185:
1058:
1005:
1000:
889:
879:
853:
846:
635:
346:
220:
102:
3784:
is a module in which all pairs of nonzero submodules have nonzero intersection.
2935:{\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}
1968:-module, where the scalar multiplication is just ring multiplication. The case
4265:
4256:, one can consider near-ring modules, a nonabelian generalization of modules.
4253:
3807:
3609:
3216:
2495:
2129:
1803:
structure theorem for finitely generated modules over a principal ideal domain
1790:
1100:
over the operations of addition between elements of the ring or module and is
755:
400:
4223:. Modules over rings are abelian groups, but modules over semirings are only
17:
4016:
3223:
2248:
1010:
816:
773:
741:
360:
265:
3434:
is a module that has a basis, or equivalently, one that is isomorphic to a
4230:. Most applications of modules are still possible. In particular, for any
4231:
4220:
2689:
1605:
1209:
811:
354:
340:
4450:
4050:
1904:
3, one cannot find even one element that satisfies the definition of a
1825:
1090:
238:
122:
2536:
and the same addition operation, but the opposite multiplication: if
4227:
4118:-modules are contravariant additive functors. This suggests that, if
2760:
745:
1089:, since the abelian groups are exactly the modules over the ring of
1820:-module agrees with the notion of an abelian group. That is, every
4415:. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964,
3513:
is a module that is not {0} and whose only submodules are {0} and
3442:. These are the modules that behave very much like vector spaces.
4454:
1919:(including negative ones) form a module over the integers. Only
4249:
incorporating the semirings from theoretical computer science.
4122:
is any preadditive category, a covariant additive functor from
29:
4145:, which is the natural generalization of the module category
3455:
of free modules and share many of their desirable properties.
3192:{\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}
3332:-modules together with their module homomorphisms forms an
4160:
rings can be generalized in a different direction: take a
3317:
familiar from groups and vector spaces are also valid for
2817:{\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}
1769:
by a group homomorphism that commutes with the action of
4241:
form a semiring over which the tuples of elements from
1200:
is then unique. (These last two assertions require the
2842:
2781:
3946:; an alternative and equivalent way of defining left
3858:
in the case of a right module), and is necessarily a
3116:
1490:
1429:
1365:
1301:
568:
530:
493:
273:
246:
4130:
should be considered a generalized left module over
4049:. Every abelian group is a faithful module over the
3281:
consisting of all elements that are sent to zero by
1953:
if we use the component-wise operations. Hence when
27:
Generalization of vector spaces from fields to rings
1612:and a right scalar multiplication β by elements of
3406:of those elements with coefficients from the ring
3191:
2934:
2816:
2475:)-modules and the category of vector bundles over
2283:, then with addition and scalar multiplication in
1593:, all rings and modules are assumed to be unital.
1511:
1474:
1413:
1349:
592:
545:
508:
300:
254:
4305:Dummit, David S. & Foote, Richard M. (2004).
3685:is a module with a decomposition as a direct sum
3545:is a non-zero module that cannot be written as a
1991:(e.g. any commutative ring or field) the number
1589:-modules". In this article, consistent with the
3001:, then the following two submodules are equal:
1900:do not. (For example, in the group of integers
2351:-module case is analogous. In particular, if
1563:is defined similarly in terms of an operation
1414:{\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x}
1350:{\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y}
1192:) if the underlying ring does not satisfy the
1115:. They are also one of the central notions of
1624:-module, satisfying the additional condition
1475:{\displaystyle (rs)\cdot x=r\cdot (s\cdot x)}
1038:
697:
8:
2788:
2782:
301:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
4309:. Hoboken, NJ: John Wiley & Sons, Inc.
2597:are (associative algebra) modules over its
1805:to this example shows the existence of the
1156:. In a module, the scalars need only be a
4019:. In terms of modules, this means that if
2942:, which is important in the definition of
1228:, and 1 is its multiplicative identity. A
1045:
1031:
715:
704:
690:
91:
4357:Abstract Algebra: The Basic Graduate Year
3870:. The set of all group endomorphisms of
3464:are defined dually to projective modules.
3115:
2915:
2896:
2883:
2873:
2863:
2852:
2841:
2799:
2794:
2780:
2510:-module, and analogously right ideals in
1489:
1428:
1364:
1300:
593:{\displaystyle \mathbb {Z} (p^{\infty })}
581:
570:
569:
567:
537:
533:
532:
529:
500:
496:
495:
492:
294:
293:
285:
281:
280:
272:
248:
247:
245:
80:Learn how and when to remove this message
2949:The set of submodules of a given module
1107:Modules are very closely related to the
43:This article includes a list of general
4297:
4198:, and play an important role in modern
3517:. Simple modules are sometimes called
2443:. More generally, the sections of any
1581:Authors who do not require rings to be
718:
94:
3884:) and forms a ring under addition and
3378:if there exist finitely many elements
2355:is commutative then the collection of
2218:. The special case is that the module
4219:One can also consider modules over a
2587:can be considered a left module over
1761:-module with an additional action of
1065:is a generalization of the notion of
7:
1995:is then the rank of the free module.
161:Free product of associative algebras
2070:-entry (and zeros elsewhere), then
1204:in general, but not in the case of
4094:. With this understanding, a left
3958:together with a representation of
3420:if it is generated by one element.
1616:, making it simultaneously a left
582:
49:it lacks sufficient corresponding
25:
3896:to its action actually defines a
3794:Relation to representation theory
649:Noncommutative algebraic geometry
3566:is one where the action of each
2771:, then the submodule spanned by
1892:. Such a module need not have a
546:{\displaystyle \mathbb {Q} _{p}}
509:{\displaystyle \mathbb {Z} _{p}}
34:
4388:Rings and Categories of Modules
4386:F.W. Anderson and K.R. Fuller:
4332:"ALGEBRA II: RINGS AND MODULES"
3962:over it. Such a representation
3950:-modules is to say that a left
3671:is a module that satisfies the
3657:is a module that satisfies the
1695:-modules and are simply called
1691:-modules are the same as right
1085:also generalizes the notion of
3186:
3180:
3165:
3159:
3144:
3120:
2226:as a module over itself, then
1469:
1457:
1439:
1430:
1378:
1366:
1320:
1308:
1184:, and even for those that do (
1144:In a vector space, the set of
1104:with the ring multiplication.
587:
574:
1:
3888:, and sending a ring element
3486:-modules preserves exactness.
2525:is a ring, we can define the
1824:is a module over the ring of
1169:left ideals or left modules.
4098:-module is just a covariant
3954:-module is an abelian group
3798:A representation of a group
3626:) of the ring, equivalently
2828:runs over the submodules of
2672:Submodules and homomorphisms
2599:universal enveloping algebra
2583:, and any right module over
255:{\displaystyle \mathbb {Z} }
4439:Encyclopedia of Mathematics
3994:A representation is called
3398:such that every element of
1512:{\displaystyle 1\cdot x=x.}
1135:Introduction and definition
407:Unique factorization domain
4494:
4188:). These form a category O
3673:descending chain condition
3355:
2944:tensor products of modules
2595:Modules over a Lie algebra
1522:The operation Β· is called
167:Tensor product of algebras
3917:Such a ring homomorphism
3836:is defined to be the map
3659:ascending chain condition
3358:Glossary of module theory
3263:of a module homomorphism
2575:can then be seen to be a
2415:). The set of all smooth
2266:is the collection of all
1945:is both a left and right
1123:, and are used widely in
4134:. These functors form a
4055:ring of integers modulo
445:Formal power series ring
395:Integrally closed domain
4330:Mcgerty, Kevin (2016).
3998:if and only if the map
3942:over the abelian group
2708:-submodule) if for any
2704:(or more explicitly an
2376:-module (and in fact a
2192:)-module. In fact, the
2164:. Conversely, given an
1732:-modules are identical.
1591:glossary of ring theory
454:Algebraic number theory
147:Total ring of fractions
64:more precise citations.
3600:). Equivalently, the
3496:if it embeds into its
3438:of copies of the ring
3328:, the set of all left
3244:, and the two modules
3193:
2936:
2868:
2818:
2357:R-module homomorphisms
1989:invariant basis number
1513:
1476:
1415:
1351:
1194:invariant basis number
1178:principal ideal domain
611:Noncommutative algebra
594:
547:
510:
462:Algebraic number field
413:Principal ideal domain
302:
256:
194:Frobenius endomorphism
4350:"Module Fundamentals"
4276:Module (model theory)
4271:Algebra (ring theory)
3979:may also be called a
3862:of the abelian group
3806:is a module over the
3543:indecomposable module
3416:A module is called a
3297:consisting of values
3194:
2937:
2848:
2819:
2532:, which has the same
2287:defined pointwise by
2058:matrix with 1 in the
1831:in a unique way. For
1524:scalar multiplication
1514:
1477:
1416:
1352:
1109:representation theory
595:
548:
511:
303:
257:
4473:Algebraic structures
4237:, the matrices over
4085:preadditive category
3581:is nontrivial (i.e.
3533:completely reducible
3315:isomorphism theorems
3293:is the submodule of
3277:is the submodule of
3226:module homomorphism
3114:
2840:
2779:
2641:-module by defining
2423:forms a module over
2347:-module. The right
1906:linearly independent
1777:. In other words, a
1724:(vector spaces over
1620:-module and a right
1488:
1427:
1363:
1299:
942:Group with operators
885:Complemented lattice
720:Algebraic structures
617:Noncommutative rings
566:
528:
491:
335:Non-associative ring
271:
244:
201:Algebraic structures
4174:) and consider the
3620:torsion-free module
3492:A module is called
3470:A module is called
3346:category of modules
3305:) for all elements
3240:is called a module
2963:: Given submodules
2957:that satisfies the
1975:yields the trivial
1896:βgroups containing
1121:homological algebra
1117:commutative algebra
996:Composition algebra
756:Quasigroup and loop
376:Commutative algebra
215:Associative algebra
97:Algebraic structure
4413:Structure of rings
4200:algebraic geometry
3860:group endomorphism
3824:-module, then the
3449:Projective modules
3404:linear combination
3376:finitely generated
3363:Finitely generated
3189:
2932:
2814:
2810:
2663:itself is such an
2437:differential forms
2372:(see below) is an
1539:to emphasize that
1509:
1472:
1411:
1347:
1268:such that for all
1206:finite-dimensional
1176:" ring, such as a
1129:algebraic topology
1125:algebraic geometry
630:Semiprimitive ring
590:
543:
506:
314:Related structures
298:
252:
188:Inner automorphism
174:Ring homomorphisms
4421:978-0-8218-1037-8
4369:Jacobson (1964),
4316:978-0-471-43334-7
4112:of abelian groups
4083:corresponds to a
4023:is an element of
3898:ring homomorphism
3655:Noetherian module
3529:semisimple module
3462:Injective modules
3056:-modules, then a
2795:
2775:is defined to be
2659:. In particular,
2614:ring homomorphism
2612:are rings with a
2449:projective module
2431:), and so do the
2128:breaks up as the
2008:) is the ring of
1940:cartesian product
1917:decimal fractions
1816:The concept of a
1250:and an operation
1216:Formal definition
1081:. The concept of
1077:is replaced by a
1055:
1054:
714:
713:
671:Geometric algebra
382:Commutative rings
233:Category of rings
90:
89:
82:
16:(Redirected from
4485:
4447:
4374:
4367:
4361:
4360:
4354:
4345:
4339:
4338:
4336:
4327:
4321:
4320:
4307:Abstract Algebra
4302:
4186:sheaf of modules
4136:functor category
4100:additive functor
4048:
4033:
4014:
3978:
3933:
3869:
3846:that sends each
3845:
3765:
3733:
3719:
3707:
3693:
3646:
3639:
3632:
3591:
3572:
3352:Types of modules
3334:abelian category
3276:
3239:
3198:
3196:
3195:
3190:
3076:homomorphism of
3072:
3040:
3000:
2941:
2939:
2938:
2933:
2931:
2927:
2920:
2919:
2901:
2900:
2888:
2887:
2878:
2877:
2867:
2862:
2836:, or explicitly
2823:
2821:
2820:
2815:
2809:
2743:
2733:
2658:
2628:
2559:
2545:
2490:is any ring and
2397:smooth functions
2371:
2338:
2318:
2282:
2163:
2123:
2069:
2057:
2017:
1974:
1959:
1930:is any ring and
1898:torsion elements
1891:
1871:
1859:
1837:
1811:Jordan canonical
1789:combined with a
1739:is a field, and
1651:
1577:
1530:. One may write
1518:
1516:
1515:
1510:
1481:
1479:
1478:
1473:
1420:
1418:
1417:
1412:
1356:
1354:
1353:
1348:
1267:
1249:
1154:distributive law
1047:
1040:
1033:
822:Commutative ring
751:Rack and quandle
716:
706:
699:
692:
677:Operator algebra
663:Clifford algebra
599:
597:
596:
591:
586:
585:
573:
552:
550:
549:
544:
542:
541:
536:
515:
513:
512:
507:
505:
504:
499:
477:Ring of integers
471:
468:Integers modulo
419:Euclidean domain
307:
305:
304:
299:
297:
289:
284:
261:
259:
258:
253:
251:
155:Product of rings
141:Fractional ideal
100:
92:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
4493:
4492:
4488:
4487:
4486:
4484:
4483:
4482:
4463:
4462:
4432:
4429:
4409:Nathan Jacobson
4383:
4378:
4377:
4368:
4364:
4352:
4347:
4346:
4342:
4334:
4329:
4328:
4324:
4317:
4304:
4303:
4299:
4294:
4281:Module spectrum
4262:
4211:
4193:
4183:
4173:
4077:
4075:Generalizations
4043:
4028:
4008:
3999:
3972:
3963:
3927:
3918:
3909:
3879:
3863:
3837:
3796:
3791:
3789:Further notions
3764:
3751:
3743:
3735:
3732:
3724:
3717:
3712:
3706:
3698:
3691:
3686:
3669:Artinian module
3641:
3634:
3627:
3582:
3567:
3561:faithful module
3551:uniform modules
3478:of it with any
3453:direct summands
3393:
3384:
3360:
3354:
3264:
3227:
3211:-modules is an
3203:This, like any
3112:
3111:
3060:
3038:
3027:
3020:
3009:
3002:
2999:
2992:
2986:
2980:
2973:
2911:
2892:
2879:
2869:
2847:
2843:
2838:
2837:
2777:
2776:
2748:-module) is in
2735:
2725:
2674:
2642:
2616:
2551:
2537:
2393:smooth manifold
2359:
2320:
2288:
2270:
2235:
2209:
2187:
2181:
2174:
2159:
2147:
2137:
2119:
2107:
2095:
2087:
2086:-module, since
2078:
2059:
2049:
2047:
2034:
2013: Γ
2009:
2003:
1969:
1954:
1873:
1865:
1839:
1832:
1801:. Applying the
1745:polynomial ring
1705:
1625:
1564:
1562:
1535:
1486:
1485:
1425:
1424:
1361:
1360:
1297:
1296:
1251:
1243:
1239:consists of an
1218:
1202:axiom of choice
1142:
1137:
1051:
1022:
1021:
1020:
991:Non-associative
973:
962:
961:
951:
931:
920:
919:
908:Map of lattices
904:
900:Boolean algebra
895:Heyting algebra
869:
858:
857:
851:
832:Integral domain
796:
785:
784:
778:
732:
710:
681:
680:
613:
603:
602:
577:
564:
563:
531:
526:
525:
494:
489:
488:
469:
439:Polynomial ring
389:Integral domain
378:
368:
367:
269:
268:
242:
241:
227:Involutive ring
112:
101:
95:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
4491:
4489:
4481:
4480:
4475:
4465:
4464:
4461:
4460:
4448:
4428:
4427:External links
4425:
4424:
4423:
4406:
4382:
4379:
4376:
4375:
4362:
4340:
4322:
4315:
4296:
4295:
4293:
4290:
4289:
4288:
4283:
4278:
4273:
4268:
4261:
4258:
4207:
4189:
4184:-modules (see
4179:
4169:
4090:with a single
4076:
4073:
4004:
3968:
3936:representation
3923:
3905:
3875:
3874:is denoted End
3828:of an element
3795:
3792:
3790:
3787:
3786:
3785:
3782:uniform module
3778:
3775:
3756:
3747:
3739:
3728:
3720:
3702:
3694:
3679:
3676:
3665:
3662:
3651:
3648:
3616:
3613:
3557:
3554:
3539:
3538:Indecomposable
3536:
3525:
3522:
3504:
3501:
3498:algebraic dual
3490:
3487:
3480:exact sequence
3476:tensor product
3474:if taking the
3468:
3465:
3459:
3456:
3446:
3443:
3424:
3421:
3414:
3411:
3389:
3382:
3364:
3353:
3350:
3201:
3200:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3036:
3025:
3018:
3007:
2997:
2990:
2978:
2971:
2930:
2926:
2923:
2918:
2914:
2910:
2907:
2904:
2899:
2895:
2891:
2886:
2882:
2876:
2872:
2866:
2861:
2858:
2855:
2851:
2846:
2813:
2808:
2805:
2802:
2798:
2793:
2790:
2787:
2784:
2724:, the product
2673:
2670:
2669:
2668:
2602:
2592:
2534:underlying set
2519:
2484:
2461:Swan's theorem
2385:
2241:
2231:
2214:)-modules are
2205:
2183:
2179:
2172:
2155:
2145:
2115:
2103:
2091:
2074:
2043:
2039:)-module, and
2030:
1999:
1996:
1936:natural number
1924:
1913:
1814:
1785:-vector space
1733:
1704:
1701:
1558:
1531:
1520:
1519:
1508:
1505:
1502:
1499:
1496:
1493:
1483:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1422:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1358:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1217:
1214:
1166:quotient rings
1141:
1138:
1136:
1133:
1053:
1052:
1050:
1049:
1042:
1035:
1027:
1024:
1023:
1019:
1018:
1013:
1008:
1003:
998:
993:
988:
982:
981:
980:
974:
968:
967:
964:
963:
960:
959:
956:Linear algebra
950:
949:
944:
939:
933:
932:
926:
925:
922:
921:
918:
917:
914:Lattice theory
910:
903:
902:
897:
892:
887:
882:
877:
871:
870:
864:
863:
860:
859:
850:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
798:
797:
791:
790:
787:
786:
777:
776:
771:
766:
760:
759:
758:
753:
748:
739:
733:
727:
726:
723:
722:
712:
711:
709:
708:
701:
694:
686:
683:
682:
674:
673:
645:
644:
638:
632:
626:
614:
609:
608:
605:
604:
601:
600:
589:
584:
580:
576:
572:
553:
540:
535:
516:
503:
498:
486:-adic integers
479:
473:
464:
450:
449:
448:
447:
441:
435:
434:
433:
421:
415:
409:
403:
397:
379:
374:
373:
370:
369:
366:
365:
364:
363:
351:
350:
349:
343:
331:
330:
329:
311:
310:
309:
308:
296:
292:
288:
283:
279:
276:
262:
250:
229:
223:
217:
211:
197:
196:
190:
184:
170:
169:
163:
157:
151:
150:
149:
143:
131:
125:
113:
111:Basic concepts
110:
109:
106:
105:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4490:
4479:
4478:Module theory
4476:
4474:
4471:
4470:
4468:
4459:
4457:
4452:
4449:
4445:
4441:
4440:
4435:
4431:
4430:
4426:
4422:
4418:
4414:
4410:
4407:
4405:
4404:3-540-97845-3
4401:
4397:
4396:0-387-97845-3
4393:
4389:
4385:
4384:
4380:
4372:
4366:
4363:
4358:
4351:
4348:Ash, Robert.
4344:
4341:
4333:
4326:
4323:
4318:
4312:
4308:
4301:
4298:
4291:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4263:
4259:
4257:
4255:
4250:
4248:
4244:
4240:
4236:
4233:
4229:
4226:
4222:
4217:
4215:
4210:
4205:
4201:
4197:
4192:
4187:
4182:
4177:
4172:
4167:
4163:
4159:
4156:Modules over
4154:
4152:
4148:
4144:
4140:
4137:
4133:
4129:
4125:
4121:
4117:
4113:
4111:
4105:
4101:
4097:
4093:
4089:
4086:
4082:
4074:
4072:
4070:
4067:
4063:
4059:
4058:
4053:or over some
4052:
4046:
4041:
4037:
4031:
4026:
4022:
4018:
4012:
4007:
4002:
3997:
3992:
3990:
3986:
3982:
3976:
3971:
3966:
3961:
3957:
3953:
3949:
3945:
3941:
3937:
3931:
3926:
3921:
3915:
3913:
3908:
3903:
3899:
3895:
3891:
3887:
3883:
3878:
3873:
3867:
3861:
3857:
3853:
3849:
3844:
3840:
3835:
3831:
3827:
3823:
3819:
3814:
3812:
3809:
3805:
3802:over a field
3801:
3793:
3788:
3783:
3779:
3776:
3773:
3769:
3763:
3759:
3755:
3750:
3746:
3742:
3738:
3731:
3727:
3723:
3715:
3711:
3705:
3701:
3697:
3689:
3684:
3683:graded module
3680:
3677:
3674:
3670:
3666:
3663:
3660:
3656:
3652:
3649:
3644:
3637:
3630:
3625:
3621:
3617:
3614:
3611:
3607:
3603:
3599:
3595:
3589:
3585:
3580:
3576:
3570:
3565:
3562:
3558:
3555:
3552:
3548:
3544:
3540:
3537:
3534:
3530:
3526:
3523:
3520:
3516:
3512:
3509:
3508:simple module
3505:
3502:
3499:
3495:
3491:
3488:
3485:
3481:
3477:
3473:
3469:
3466:
3463:
3460:
3457:
3454:
3450:
3447:
3444:
3441:
3437:
3433:
3431:
3425:
3422:
3419:
3418:cyclic module
3415:
3412:
3409:
3405:
3401:
3397:
3392:
3388:
3381:
3377:
3373:
3369:
3365:
3362:
3361:
3359:
3351:
3349:
3347:
3343:
3339:
3336:, denoted by
3335:
3331:
3327:
3324:Given a ring
3322:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3275:
3271:
3267:
3262:
3257:
3255:
3251:
3247:
3243:
3238:
3234:
3230:
3225:
3220:
3218:
3214:
3210:
3206:
3183:
3177:
3174:
3171:
3168:
3162:
3156:
3153:
3150:
3147:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3117:
3110:
3109:
3108:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3081:
3079:
3071:
3067:
3063:
3059:
3055:
3051:
3047:
3042:
3035:
3031:
3024:
3017:
3013:
3006:
2996:
2989:
2984:
2977:
2970:
2966:
2962:
2961:
2956:
2952:
2947:
2945:
2928:
2924:
2921:
2916:
2912:
2908:
2905:
2902:
2897:
2893:
2889:
2884:
2880:
2874:
2870:
2864:
2859:
2856:
2853:
2849:
2844:
2835:
2832:that contain
2831:
2827:
2811:
2806:
2803:
2800:
2796:
2791:
2785:
2774:
2770:
2766:
2762:
2758:
2753:
2751:
2747:
2742:
2738:
2732:
2728:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2671:
2666:
2662:
2657:
2653:
2649:
2645:
2640:
2636:
2632:
2629:, then every
2627:
2623:
2619:
2615:
2611:
2607:
2603:
2600:
2596:
2593:
2590:
2586:
2582:
2578:
2574:
2570:
2567:
2563:
2558:
2554:
2549:
2544:
2540:
2535:
2531:
2528:
2527:opposite ring
2524:
2520:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2445:vector bundle
2442:
2438:
2434:
2433:tensor fields
2430:
2426:
2422:
2418:
2417:vector fields
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2383:
2379:
2375:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2336:
2332:
2328:
2324:
2316:
2312:
2308:
2304:
2300:
2296:
2292:
2286:
2281:
2277:
2273:
2269:
2265:
2262:-module, and
2261:
2257:
2253:
2250:
2246:
2242:
2239:
2234:
2229:
2225:
2221:
2217:
2213:
2208:
2203:
2199:
2197:
2191:
2186:
2178:
2171:
2167:
2162:
2158:
2154:
2150:
2144:
2140:
2135:
2131:
2127:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2090:
2085:
2081:
2077:
2073:
2067:
2063:
2056:
2052:
2046:
2042:
2038:
2033:
2028:
2024:
2020:
2016:
2012:
2007:
2002:
1997:
1994:
1990:
1986:
1982:
1978:
1972:
1967:
1963:
1957:
1952:
1949:-module over
1948:
1944:
1941:
1937:
1933:
1929:
1925:
1922:
1918:
1914:
1911:
1907:
1903:
1899:
1895:
1889:
1885:
1881:
1877:
1869:
1863:
1858:
1854:
1850:
1846:
1842:
1835:
1830:
1827:
1823:
1822:abelian group
1819:
1815:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1781:-module is a
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1753:
1751:
1746:
1743:a univariate
1742:
1738:
1734:
1731:
1727:
1723:
1722:vector spaces
1719:
1715:
1711:
1707:
1706:
1702:
1700:
1698:
1694:
1690:
1686:
1682:
1677:
1675:
1671:
1667:
1663:
1659:
1655:
1649:
1645:
1641:
1637:
1633:
1629:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1594:
1592:
1588:
1584:
1579:
1576:
1572:
1568:
1561:
1557:
1554:
1552:
1546:
1542:
1538:
1534:
1529:
1525:
1506:
1503:
1500:
1497:
1494:
1491:
1484:
1466:
1463:
1460:
1454:
1451:
1448:
1445:
1442:
1436:
1433:
1423:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1375:
1372:
1369:
1359:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1317:
1314:
1311:
1305:
1302:
1295:
1294:
1293:
1291:
1287:
1283:
1279:
1275:
1271:
1266:
1262:
1258:
1254:
1247:
1242:
1241:abelian group
1238:
1235:
1233:
1227:
1223:
1220:Suppose that
1215:
1213:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1170:
1167:
1163:
1159:
1155:
1151:
1147:
1139:
1134:
1132:
1130:
1126:
1122:
1118:
1114:
1110:
1105:
1103:
1099:
1094:
1092:
1088:
1087:abelian group
1084:
1080:
1076:
1072:
1069:in which the
1068:
1064:
1060:
1048:
1043:
1041:
1036:
1034:
1029:
1028:
1026:
1025:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
983:
979:
976:
975:
971:
966:
965:
958:
957:
953:
952:
948:
945:
943:
940:
938:
935:
934:
929:
924:
923:
916:
915:
911:
909:
906:
905:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
872:
867:
862:
861:
856:
855:
848:
845:
843:
842:Division ring
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
799:
794:
789:
788:
783:
782:
775:
772:
770:
767:
765:
764:Abelian group
762:
761:
757:
754:
752:
749:
747:
743:
740:
738:
735:
734:
730:
725:
724:
721:
717:
707:
702:
700:
695:
693:
688:
687:
685:
684:
679:
678:
672:
668:
667:
666:
665:
664:
659:
658:
657:
652:
651:
650:
643:
639:
637:
633:
631:
627:
625:
624:Division ring
621:
620:
619:
618:
612:
607:
606:
578:
562:
560:
554:
538:
524:
523:-adic numbers
522:
517:
501:
487:
485:
480:
478:
474:
472:
465:
463:
459:
458:
457:
456:
455:
446:
442:
440:
436:
432:
428:
427:
426:
422:
420:
416:
414:
410:
408:
404:
402:
398:
396:
392:
391:
390:
386:
385:
384:
383:
377:
372:
371:
362:
358:
357:
356:
352:
348:
344:
342:
338:
337:
336:
332:
328:
324:
323:
322:
318:
317:
316:
315:
290:
286:
277:
274:
267:
266:Terminal ring
263:
240:
236:
235:
234:
230:
228:
224:
222:
218:
216:
212:
210:
206:
205:
204:
203:
202:
195:
191:
189:
185:
183:
179:
178:
177:
176:
175:
168:
164:
162:
158:
156:
152:
148:
144:
142:
138:
137:
136:
135:Quotient ring
132:
130:
126:
124:
120:
119:
118:
117:
108:
107:
104:
99:β Ring theory
98:
93:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
18:Module theory
4455:
4437:
4412:
4387:
4365:
4356:
4343:
4325:
4306:
4300:
4251:
4247:vector space
4242:
4238:
4234:
4218:
4213:
4208:
4203:
4195:
4190:
4180:
4170:
4165:
4162:ringed space
4157:
4155:
4150:
4146:
4142:
4138:
4131:
4127:
4123:
4119:
4115:
4114:, and right
4109:
4103:
4095:
4087:
4080:
4078:
4068:
4065:
4061:
4056:
4044:
4039:
4035:
4029:
4024:
4020:
4010:
4005:
4000:
3995:
3993:
3988:
3984:
3980:
3974:
3969:
3964:
3959:
3955:
3951:
3947:
3943:
3939:
3935:
3934:is called a
3929:
3924:
3919:
3916:
3911:
3906:
3901:
3893:
3889:
3881:
3876:
3871:
3865:
3855:
3851:
3847:
3842:
3838:
3833:
3829:
3825:
3821:
3817:
3815:
3810:
3803:
3799:
3797:
3771:
3767:
3761:
3757:
3753:
3748:
3744:
3740:
3736:
3729:
3725:
3721:
3713:
3703:
3699:
3695:
3687:
3642:
3635:
3628:
3624:zero-divisor
3615:Torsion-free
3605:
3597:
3593:
3587:
3583:
3578:
3574:
3568:
3563:
3532:
3518:
3514:
3510:
3483:
3439:
3429:
3407:
3399:
3395:
3390:
3386:
3379:
3371:
3367:
3341:
3337:
3329:
3325:
3323:
3318:
3310:
3306:
3302:
3298:
3294:
3290:
3282:
3278:
3273:
3269:
3265:
3258:
3253:
3249:
3245:
3236:
3232:
3228:
3221:
3212:
3208:
3205:homomorphism
3202:
3104:
3100:
3096:
3092:
3088:
3084:
3077:
3074:
3069:
3065:
3061:
3053:
3049:
3045:
3043:
3033:
3029:
3022:
3015:
3011:
3004:
2994:
2987:
2982:
2975:
2968:
2964:
2958:
2950:
2948:
2833:
2829:
2825:
2772:
2768:
2764:
2756:
2754:
2749:
2745:
2744:for a right
2740:
2736:
2730:
2726:
2721:
2717:
2713:
2709:
2705:
2701:
2697:
2693:
2685:
2684:-module and
2681:
2677:
2675:
2664:
2660:
2655:
2651:
2647:
2643:
2638:
2634:
2630:
2625:
2621:
2617:
2609:
2605:
2588:
2584:
2580:
2579:module over
2576:
2572:
2568:
2565:
2561:
2556:
2552:
2547:
2542:
2538:
2529:
2522:
2515:
2511:
2507:
2503:
2499:
2491:
2487:
2476:
2472:
2468:
2456:
2452:
2440:
2428:
2424:
2420:
2412:
2408:
2407:form a ring
2405:real numbers
2400:
2388:
2381:
2377:
2373:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2334:
2330:
2326:
2322:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2284:
2279:
2275:
2271:
2263:
2259:
2255:
2244:
2237:
2232:
2227:
2223:
2219:
2211:
2206:
2195:
2194:category of
2189:
2184:
2176:
2169:
2165:
2160:
2156:
2152:
2148:
2142:
2138:
2133:
2125:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2083:
2079:
2075:
2071:
2065:
2061:
2054:
2050:
2044:
2040:
2036:
2031:
2026:
2022:
2021:over a ring
2014:
2010:
2005:
2000:
1992:
1984:
1976:
1970:
1965:
1961:
1955:
1950:
1946:
1942:
1931:
1927:
1910:finite field
1887:
1883:
1879:
1875:
1867:
1861:
1856:
1852:
1848:
1844:
1840:
1833:
1828:
1817:
1798:
1794:
1786:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1749:
1740:
1736:
1729:
1725:
1717:
1709:
1696:
1692:
1688:
1687:, then left
1680:
1678:
1673:
1669:
1665:
1661:
1657:
1653:
1647:
1643:
1639:
1635:
1631:
1627:
1621:
1617:
1613:
1609:
1601:
1597:
1595:
1586:
1580:
1574:
1570:
1566:
1559:
1555:
1550:
1548:
1547:-module. A
1544:
1540:
1536:
1532:
1527:
1523:
1521:
1289:
1285:
1281:
1277:
1273:
1269:
1264:
1260:
1256:
1252:
1245:
1236:
1231:
1229:
1221:
1219:
1186:free modules
1174:well-behaved
1171:
1143:
1106:
1098:distributive
1095:
1082:
1067:vector space
1062:
1056:
1016:Hopf algebra
954:
947:Vector space
936:
927:
912:
852:
781:Group theory
779:
744: /
675:
661:
660:
656:Free algebra
654:
653:
647:
646:
615:
558:
520:
483:
452:
451:
431:Finite field
380:
327:Finite field
313:
312:
239:Initial ring
208:
199:
198:
172:
171:
114:
76:
67:
48:
4286:Annihilator
4225:commutative
4158:commutative
3981:ring action
3886:composition
3710:graded ring
3602:annihilator
3519:irreducible
3494:torsionless
3489:Torsionless
3252:are called
3242:isomorphism
3083:if for any
2960:modular law
2419:defined on
2395:, then the
1938:, then the
1864:summands),
1685:commutative
1198:cardinality
1059:mathematics
1001:Lie algebra
986:Associative
890:Total order
880:Semilattice
854:Ring theory
636:Simple ring
347:Jordan ring
221:Graded ring
103:Ring theory
62:introducing
4467:Categories
4381:References
4266:Group ring
4254:near-rings
4027:such that
3820:is a left
3808:group ring
3734:such that
3650:Noetherian
3610:zero ideal
3547:direct sum
3524:Semisimple
3445:Projective
3436:direct sum
3356:See also:
3321:-modules.
3285:, and the
3254:isomorphic
3217:linear map
2985:such that
2680:is a left
2514:are right
2506:is a left
2496:left ideal
2481:equivalent
2459:), and by
2343:is a left
2258:is a left
2216:equivalent
2136:-modules,
2130:direct sum
1921:singletons
1791:linear map
1699:-modules.
1543:is a left
1292:, we have
1140:Motivation
1102:compatible
642:Commutator
401:GCD domain
45:references
4444:EMS Press
4108:category
4017:injective
3592:for some
3458:Injective
3224:bijective
3175:⋅
3154:⋅
3139:⋅
3127:⋅
3052:are left
2922:∈
2903:∈
2890:∣
2850:∑
2804:⊇
2797:⋂
2789:⟩
2783:⟨
2702:submodule
2518:-modules.
2378:submodule
2268:functions
2240:)-module.
1747:, then a
1565:Β· :
1495:⋅
1464:⋅
1455:⋅
1443:⋅
1406:⋅
1394:⋅
1382:⋅
1342:⋅
1330:⋅
1306:⋅
1011:Bialgebra
817:Near-ring
774:Lie group
742:Semigroup
583:∞
361:Semifield
4434:"Module"
4373:, Def. 1
4260:See also
4232:semiring
4221:semiring
4051:integers
4034:for all
3996:faithful
3766:for all
3664:Artinian
3633:implies
3556:Faithful
3370:-module
3268: :
3231: :
3080:-modules
3064: :
2767:-module
2716:and any
2696:. Then
2690:subgroup
2676:Suppose
2667:-module.
2633:-module
2620: :
2571:-module
2465:category
2435:and the
2363: :
2274: :
2249:nonempty
2222:is just
2202:category
2200:and the
2198:-modules
2168:-module
2151:β ... β
2019:matrices
1855:+ ... +
1826:integers
1807:rational
1703:Examples
1652:for all
1606:bimodule
1255: :
1210:L spaces
1091:integers
847:Lie ring
812:Semiring
355:Semiring
341:Lie ring
123:Subrings
70:May 2015
4453:at the
4446:, 2001
4228:monoids
4176:sheaves
4106:to the
4079:A ring
4042:, then
3777:Uniform
3708:over a
3608:is the
3432:-module
3385:, ...,
2955:lattice
2759:is any
2564:. Any
2550:, then
2502:, then
2494:is any
2447:form a
2403:to the
2230:is an M
2182:is an M
2175:, then
2048:is the
2029:is an M
1983:and if
1752:-module
1716:, then
1553:-module
1234:-module
1146:scalars
1075:scalars
978:Algebra
970:Algebra
875:Lattice
866:Lattice
557:PrΓΌfer
159:β’
58:improve
4451:module
4419:
4402:
4394:
4313:
4092:object
3904:to End
3826:action
3678:Graded
3503:Simple
3413:Cyclic
3313:. The
3261:kernel
2824:where
2763:of an
2761:subset
2637:is an
2082:is an
1964:is an
1902:modulo
1872:, and
1838:, let
1836:> 0
1813:forms.
1728:) and
1668:, and
1583:unital
1549:right
1162:ideals
1113:groups
1083:module
1063:module
1006:Graded
937:Module
928:Module
827:Domain
746:Monoid
209:Module
182:Kernel
47:, but
4353:(PDF)
4335:(PDF)
4292:Notes
4252:Over
4202:. If
4102:from
4003:β End
3967:β End
3922:β End
3900:from
3428:free
3402:is a
3344:(see
3287:image
3073:is a
2700:is a
2688:is a
2577:right
2451:over
2399:from
2391:is a
2247:is a
2124:. So
1894:basis
1793:from
1757:is a
1714:field
1712:is a
1230:left
1224:is a
1182:basis
1150:field
1148:is a
1071:field
972:-like
930:-like
868:-like
837:Field
795:-like
769:Magma
737:Group
731:-like
729:Group
561:-ring
425:Field
321:Field
129:Ideal
116:Rings
4417:ISBN
4400:ISBN
4392:ISBN
4371:p. 4
4311:ISBN
4178:of O
3868:, +)
3854:(or
3770:and
3472:flat
3467:Flat
3451:are
3423:Free
3259:The
3248:and
3095:and
3048:and
3014:) β©
2734:(or
2608:and
2566:left
2479:are
2329:) =
2319:and
2309:) +
2301:) =
2204:of M
1998:If M
1987:has
1981:free
1915:The
1882:= β(
1878:) β
1866:0 β
1809:and
1634:) β
1596:An (
1280:and
1248:, +)
1226:ring
1190:rank
1164:and
1158:ring
1127:and
1119:and
1079:ring
1061:, a
802:Ring
793:Ring
4458:Lab
4216:).
4196:Mod
4168:, O
4151:Mod
4143:Mod
4126:to
4047:= 0
4038:in
4032:= 0
4015:is
3987:on
3983:of
3938:of
3914:).
3892:of
3850:to
3832:in
3816:If
3667:An
3645:= 0
3640:or
3638:= 0
3631:= 0
3604:of
3596:in
3590:β 0
3577:on
3573:in
3571:β 0
3541:An
3482:of
3394:in
3374:is
3366:An
3348:).
3342:Mod
3309:of
3289:of
3103:in
3091:in
3058:map
3044:If
3028:+ (
2981:of
2946:.
2755:If
2720:in
2712:in
2692:of
2604:If
2560:in
2546:in
2521:If
2498:in
2486:If
2467:of
2439:on
2387:If
2380:of
2252:set
2243:If
2132:of
1973:= 0
1958:= 1
1926:If
1870:= 0
1797:to
1773:on
1765:on
1735:If
1708:If
1683:is
1679:If
1672:in
1664:in
1656:in
1642:β
(
1288:in
1276:in
1212:.)
1111:of
1073:of
1057:In
807:Rng
4469::
4442:,
4436:,
4411:.
4398:,
4355:.
4153:.
4128:Ab
4110:Ab
4071:.
4060:,
4030:rx
3991:.
3856:xr
3852:rx
3841:β
3813:.
3780:A
3752:β
3716:=
3690:=
3681:A
3653:A
3629:rm
3618:A
3586:β
3559:A
3553:).
3527:A
3506:A
3426:A
3272:β
3235:β
3222:A
3219:.
3107:,
3099:,
3087:,
3068:β
3041:.
3032:β©
3021:=
3010:+
2993:β
2974:,
2967:,
2752:.
2739:β
2729:β
2646:=
2644:rm
2624:β
2555:=
2553:ba
2541:=
2539:ab
2384:).
2367:β
2339:,
2331:rf
2325:)(
2323:rf
2297:)(
2293:+
2278:β
2254:,
2141:=
2111:β
2109:rm
2099:=
2089:re
2064:,
2053:Γ
2025:,
1960:,
1934:a
1886:β
1874:(β
1851:+
1847:=
1843:β
1676:.
1660:,
1646:β
1638:=
1630:Β·
1604:)-
1578:.
1573:β
1569:Γ
1284:,
1272:,
1263:β
1259:Γ
1131:.
1093:.
669:β’
640:β’
634:β’
628:β’
622:β’
555:β’
518:β’
481:β’
475:β’
466:β’
460:β’
443:β’
437:β’
429:β’
423:β’
417:β’
411:β’
405:β’
399:β’
393:β’
387:β’
359:β’
353:β’
345:β’
339:β’
333:β’
325:β’
319:β’
264:β’
237:β’
231:β’
225:β’
219:β’
213:β’
207:β’
192:β’
186:β’
180:β’
165:β’
153:β’
145:β’
139:β’
133:β’
127:β’
121:β’
4456:n
4359:.
4337:.
4319:.
4243:S
4239:S
4235:S
4214:X
4212:(
4209:X
4204:X
4194:-
4191:X
4181:X
4171:X
4166:X
4164:(
4149:-
4147:R
4141:-
4139:C
4132:C
4124:C
4120:C
4116:R
4104:R
4096:R
4088:R
4081:R
4069:Z
4066:n
4064:/
4062:Z
4057:n
4045:r
4040:M
4036:x
4025:R
4021:r
4013:)
4011:M
4009:(
4006:Z
4001:R
3989:M
3985:R
3977:)
3975:M
3973:(
3970:Z
3965:R
3960:R
3956:M
3952:R
3948:R
3944:M
3940:R
3932:)
3930:M
3928:(
3925:Z
3920:R
3912:M
3910:(
3907:Z
3902:R
3894:R
3890:r
3882:M
3880:(
3877:Z
3872:M
3866:M
3864:(
3848:x
3843:M
3839:M
3834:R
3830:r
3822:R
3818:M
3811:k
3804:k
3800:G
3774:.
3772:y
3768:x
3762:y
3760:+
3758:x
3754:M
3749:y
3745:M
3741:x
3737:R
3730:x
3726:R
3722:x
3718:β¨
3714:R
3704:x
3700:M
3696:x
3692:β¨
3688:M
3647:.
3643:m
3636:r
3612:.
3606:M
3598:M
3594:x
3588:x
3584:r
3579:M
3575:R
3569:r
3564:M
3535:.
3521:.
3515:S
3511:S
3500:.
3484:R
3440:R
3430:R
3410:.
3408:R
3400:M
3396:M
3391:n
3387:x
3383:1
3380:x
3372:M
3368:R
3340:-
3338:R
3330:R
3326:R
3319:R
3311:M
3307:m
3303:m
3301:(
3299:f
3295:N
3291:f
3283:f
3279:M
3274:N
3270:M
3266:f
3250:N
3246:M
3237:N
3233:M
3229:f
3215:-
3213:R
3209:R
3199:.
3187:)
3184:n
3181:(
3178:f
3172:s
3169:+
3166:)
3163:m
3160:(
3157:f
3151:r
3148:=
3145:)
3142:n
3136:s
3133:+
3130:m
3124:r
3121:(
3118:f
3105:R
3101:s
3097:r
3093:M
3089:n
3085:m
3078:R
3070:N
3066:M
3062:f
3054:R
3050:N
3046:M
3039:)
3037:2
3034:N
3030:U
3026:1
3023:N
3019:2
3016:N
3012:U
3008:1
3005:N
3003:(
2998:2
2995:N
2991:1
2988:N
2983:M
2979:2
2976:N
2972:1
2969:N
2965:U
2951:M
2929:}
2925:X
2917:i
2913:x
2909:,
2906:R
2898:i
2894:r
2885:i
2881:x
2875:i
2871:r
2865:k
2860:1
2857:=
2854:i
2845:{
2834:X
2830:M
2826:N
2812:N
2807:X
2801:N
2792:=
2786:X
2773:X
2769:M
2765:R
2757:X
2750:N
2746:R
2741:r
2737:n
2731:n
2727:r
2722:R
2718:r
2714:N
2710:n
2706:R
2698:N
2694:M
2686:N
2682:R
2678:M
2665:R
2661:S
2656:m
2654:)
2652:r
2650:(
2648:Ο
2639:R
2635:M
2631:S
2626:S
2622:R
2618:Ο
2610:S
2606:R
2601:.
2591:.
2589:R
2585:R
2581:R
2573:M
2569:R
2562:R
2557:c
2548:R
2543:c
2530:R
2523:R
2516:R
2512:R
2508:R
2504:I
2500:R
2492:I
2488:R
2483:.
2477:X
2473:X
2471:(
2469:C
2457:X
2455:(
2453:C
2441:X
2429:X
2427:(
2425:C
2421:X
2413:X
2411:(
2409:C
2401:X
2389:X
2382:N
2374:R
2369:N
2365:M
2361:h
2353:R
2349:R
2345:R
2341:M
2337:)
2335:s
2333:(
2327:s
2321:(
2317:)
2315:s
2313:(
2311:g
2307:s
2305:(
2303:f
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2224:R
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2185:n
2180:0
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2011:n
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2004:(
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1993:n
1985:R
1977:R
1971:n
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