340:
133:
modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include
2774:
is injective if and only if it is divisible (the case of vector spaces is an example of this theorem, as every field is a principal ideal domain and every vector space is divisible). Over a general integral domain, we still have one implication: every injective module over an integral domain is
993:, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its
1836:
3187:
Over a
Noetherian ring, every injective module is the direct sum of (uniquely determined) indecomposable injective modules. Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in
2949:
One can use injective hulls to define a minimal injective resolution (see below). If each term of the injective resolution is the injective hull of the cokernel of the previous map, then the injective resolution has minimal length.
2843:. So in particular, every abelian group is a subgroup of an injective one. It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left
2146:
2475:
1766:
693:, §3I). Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients
2311:
2230:
2033:
1399:
1459:
1925:
1670:
3579:
1223:
1337:
2526:
1102:
2565:
1529:
1262:
1173:
2408:
2636:
638:
1134:
2371:
2254:
2173:
2100:
1973:
1949:
1868:
1055:
665:
2349:, every injective module is a direct sum of indecomposable injective modules and every indecomposable injective module is the injective hull of the residue field at a prime
1771:
800:
1724:
3470:
1564:
1702:
109:
are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the
2656:
2605:
2585:
2347:
2076:
2056:
1888:
1584:
1479:
1031:
896:
that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the
4018:
In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all. For instance, a
1586:. This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle. A simple non-example is the ring
4392:
2670:
of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective (
3840:, Th. 15.1). An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.
2700:
Bass-Papp
Theorem states that every infinite direct sum of right (left) injective modules is injective if and only if the ring is right (left)
4352:
3766:-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.
2716:
In Baer's original paper, he proved a useful result, usually known as Baer's
Criterion, for checking whether a module is injective: a left
3968:
The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term
2108:
2674:, p. 61). Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite
2416:
1729:
4004:, and injective modules is more clear, as it simply refers to certain divisibility properties of module elements by integers.
2678:
of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is
4323:
4293:
4255:
4225:
4188:
2259:
2178:
1981:
1927:
has a relatively straightforward description of its injective modules. Using the universal enveloping algebra any injective
1342:
339:
4635:
1404:
1893:
1589:
3518:
1461:
is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over
3801:, §3B). If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every
3049:
does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. (
4509:
2797:. The dual of Baer's criterion, which would give a test for projectivity, is false in general. For instance, the
1178:
4595:
4552:
1267:
105:
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them:
2835:, which means that it is injective and any other module is contained in a suitably large product of copies of
2483:
1060:
4756:
2531:
1492:
1228:
1139:
2376:
2610:
923:
606:
3115:
Every injective submodule of an injective module is a direct summand, so it is important to understand
2832:
1831:{\displaystyle {\text{Hom}}_{\mathbb {C} }(\mathbb {C} \cdot x\oplus \mathbb {C} \cdot y,\mathbb {C} )}
1107:
675:. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.
2847:-modules has enough injectives." To prove this, one uses the peculiar properties of the abelian group
2689:, p. 152); every factor module of every injective module is injective if and only if the ring is
3493:
2667:
2352:
2235:
2154:
2081:
1954:
1930:
1849:
1036:
998:
646:
769:
3037:
admits a finite injective resolution, the minimal length among all finite injective resolutions of
889:
764:
1707:
982:, the duality is particularly well-behaved and projective modules and injective modules coincide.
431:
4761:
3864:
3774:
2771:
601:
87:
3797:
as a module over itself, but it is rarer for a ring to be injective as a module over itself, (
3443:
4470:
4019:
4013:
3382:-module. The same statement holds of course after interchanging left- and right- attributes.
3116:
2828:
2814:
2675:
941:
597:
350:
206:
177:
is injective if it satisfies one (and therefore all) of the following equivalent conditions:
130:
106:
4212:, de Gruyter Expositions in Mathematics, vol. 30, Berlin: Walter de Gruyter & Co.,
2942:
of a module is the smallest injective module containing the given one and was described in (
1534:
4740:
4712:
4684:
4656:
4626:
4583:
4540:
4493:
4458:
4423:
4375:
4333:
4303:
4265:
4235:
3829:
2963:
1675:
893:
261:
241:
36:
4383:
4250:, Monographs and Textbooks in Pure and Applied Mathematics, vol. 147, Marcel Dekker,
8:
3872:
3810:
3361:
2737:
915:
865:
401:
328:
143:
122:
71:
4090:
4590:
4435:
4411:
4244:
3418:
2641:
2590:
2570:
2332:
2061:
2041:
1873:
1569:
1464:
1016:
171:
4700:
4644:
4614:
4571:
4528:
4449:
4350:(1940), "Abelian groups that are direct summands of every containing abelian group",
4319:
4289:
4251:
4221:
4184:
3802:
3794:
3130:
2897:
979:
752:
91:
4366:
4728:
4672:
4604:
4561:
4518:
4479:
4444:
4401:
4379:
4361:
4281:
4213:
3868:
3860:
3854:
3513:
3139:
if every two nonzero submodules have nonzero intersection. For an injective module
2767:
990:
679:
151:
114:
24:
4736:
4708:
4680:
4652:
4622:
4579:
4536:
4489:
4454:
4430:
4419:
4371:
4329:
4299:
4277:
4261:
4231:
4180:
4174:
3969:
3963:
3825:
3818:
3806:
3311:
2999:
2701:
2690:
2682:
2327:
682:
586:
582:
551:
361:
155:
147:
126:
51:
2896:-module that exhibits an interesting duality, not between injective modules and
829:
4500:
4315:
4205:
4023:
4001:
3192:). The indecomposable injective modules are the injective hulls of the modules
3135:
2967:
2939:
2933:
2661:
994:
706:
118:
63:
4732:
4676:
4285:
856:= 0, 1, 2, …. Multiplication by scalars is as expected, and multiplication by
4750:
4704:
4648:
4618:
4575:
4532:
4347:
3833:
3814:
3307:
3175:
2759:
2679:
986:
373:
369:
307:
is an arbitrary module homomorphism, then there exists a module homomorphism
28:
4125:
4048:
3742:
is injective as a module over itself. While it is easy to convert injective
4609:
4566:
4523:
4465:
3904:
3882:
881:
505:
494:
412:
139:
4217:
4547:
4108:
3997:
3790:
3485:
3345:
3252:
is isomorphic as finite-dimensional vector space over the quotient field
3003:
2901:
2791:
997:. For finite-dimensional algebras over fields, these injective hulls are
686:
641:
443:
353:
20:
3069:→ 0 indicates that the arrow in the center is an isomorphism, and hence
4484:
4415:
4139:
3127:
2790:) that for a commutative Noetherian ring, it suffices to consider only
897:
748:
668:
135:
4504:
3061:) = 0. In this situation, the exactness of the sequence 0 →
2766:). More generally, an abelian group is injective if and only if it is
3314:. In general, this is difficult, but a number of results are known, (
2916:-module is flat if and only if its character module is injective. If
2321:
848:. This module has a basis consisting of "inverse monomials", that is
840:(the ring of inverse polynomials). The latter is easily described as
292:
59:
4406:
3341:
2998:
are injective modules. Injective resolutions can be used to define
1768:, which is 2-dimensional. The residue field has the injective hull
489:(with addition) form an injective abelian group (i.e. an injective
4159:
A module isomorphic to an injective module is of course injective.
2924:-module is injective if and only if its character module is flat.
3303:
2141:{\displaystyle {\mathfrak {g}}\hookrightarrow U({\mathfrak {g}})}
3156:
is nonzero and is the injective hull of every nonzero submodule
2819:
Maybe the most important injective module is the abelian group
2855:
to construct an injective cogenerator in the category of left
2805:
satisfies the dual of Baer's criterion but is not projective.
4400:(3), American Mathematical Society, Vol. 97, No. 3: 457–473,
2470:{\displaystyle I\cong \bigoplus _{i}E(R/{\mathfrak {p}}_{i})}
1761:{\displaystyle \mathbb {C} \cdot x\oplus \mathbb {C} \cdot y}
4719:
Vámos, P. (1983), "Ideals and modules testing injectivity",
2662:
Submodules, quotients, products, and sums, Bass-Papp
Theorem
4663:
Smith, P. F. (1981), "Injective modules and prime ideals",
4276:, Graduate Texts in Mathematics No. 189, Berlin, New York:
952:-modules. Therefore, the finitely generated injective left
4314:, Pure and Applied Mathematics, vol. 85, Boston, MA:
125:, and turn out to be minimal injective extensions. Over a
3777:
preserves injectives, but a counterexample was given in (
4633:
Papp, Zoltán (1959), "On algebraically closed modules",
4390:
Chase, Stephen U. (1960), "Direct products of modules",
3895:). The following general definition is used: an object
3080:
is the minimal integer (if there is such, otherwise ∞)
4126:"Structure of injective modules over Noetherian rings"
3843:
2306:{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}
2225:{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}
2028:{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}
3867:
more general than module categories, for instance in
3521:
3446:
2644:
2613:
2593:
2573:
2534:
2486:
2419:
2379:
2355:
2335:
2262:
2238:
2181:
2157:
2111:
2084:
2064:
2044:
1984:
1957:
1933:
1896:
1876:
1852:
1774:
1732:
1710:
1678:
1592:
1572:
1537:
1495:
1467:
1407:
1394:{\displaystyle {\mathfrak {p}}=(x_{1},\ldots ,x_{n})}
1345:
1270:
1231:
1181:
1142:
1110:
1063:
1039:
1019:
772:
649:
609:
129:, every injective module is uniquely a direct sum of
3302:
It is important to be able to consider modules over
3013:
of a finite injective resolution is the first index
864:·1 = 0. The endomorphism ring is simply the ring of
98:) and are discussed in some detail in the textbook (
66:
of that module; also, given a submodule of a module
4550:(1958), "Injective modules over Noetherian rings",
4243:
3754:-modules, this process does not convert injective
3573:
3484:bimodule, by left and right multiplication. Being
3464:
2650:
2630:
2599:
2579:
2559:
2520:
2469:
2402:
2365:
2341:
2322:Structure theorem for commutative Noetherian rings
2305:
2248:
2224:
2167:
2140:
2094:
2070:
2050:
2027:
1967:
1943:
1919:
1882:
1862:
1830:
1760:
1718:
1696:
1664:
1578:
1558:
1523:
1473:
1454:{\displaystyle E=\oplus _{i}{\text{Hom}}(R_{i},k)}
1453:
1393:
1331:
1256:
1217:
1167:
1128:
1096:
1049:
1025:
956:-modules are precisely the modules of the form Hom
794:
659:
632:
42:that shares certain desirable properties with the
4593:(1964), "On ring properties of injective hulls",
4393:Transactions of the American Mathematical Society
3041:is called its injective dimension and denoted id(
482:in the above definition is typically not unique.
4748:
2778:Baer's criterion has been refined in many ways (
3440:In the opposite direction, a ring homomorphism
1920:{\displaystyle {\mathcal {M}}({\mathfrak {g}})}
1665:{\displaystyle R=\mathbb {C} /(x^{2},xy,y^{2})}
671:is also prime and corresponds to the injective
4468:; Schopf, A. (1953), "Über injektive Moduln",
4173:Anderson, Frank Wylie; Fuller, Kent R (1992),
4172:
3574:{\displaystyle f_{*}M=\mathrm {Hom} _{S}(R,M)}
3500:-module. Specializing the above statement for
1264:. In particular, for the standard graded ring
78:can be extended to a homomorphism from all of
4464:
4433:(1981), "Localization of injective modules",
4353:Bulletin of the American Mathematical Society
3608:, the change of rings is also very clear. An
2943:
2724:is injective if and only if any homomorphism
1890:of characteristic 0, the category of modules
1104:as the injective hull. The injective hull of
907:is not zero, the following example may help.
397:Trivially, the zero module {0} is injective.
3620:-module precisely when it is annihilated by
3126:Every indecomposable injective module has a
1841:
1008:
678:A particularly rich theory is available for
4203:
4022:is a module in which a homomorphism from a
3773:, p. 103) has an erroneous proof that
2905:
1218:{\displaystyle (0:_{E}{\mathfrak {p}}^{k})}
813:Two examples are the injective hull of the
573:-module, and indeed the smallest injective
62:of some other module, then it is already a
747:. In other words, it suffices to consider
384:-modules are defined in complete analogy.
110:
4691:Utumi, Yuzo (1956), "On quotient rings",
4608:
4565:
4522:
4483:
4448:
4405:
4365:
4241:
3168:is the injective hull of a uniform module
3076:Equivalently, the injective dimension of
2808:
2779:
2686:
1821:
1807:
1793:
1783:
1748:
1734:
1712:
1600:
1332:{\displaystyle R_{\bullet }=k_{\bullet }}
974:is a finitely generated projective right
4049:"Lemma 47.7.5 (08Z6)—The Stacks project"
3828:, right self-injective ring is called a
3229:given by the annihilators of the ideals
3220:has an increasing filtration by modules
3053:, §5C) As an example, consider a module
2953:
2770:. More generally still: a module over a
2754:Using this criterion, one can show that
2521:{\displaystyle E(R/{\mathfrak {p}}_{i})}
1531:is injective over itself if and only if
903:are injective. If the characteristic of
94:. Injective modules were introduced in (
4589:
3784:
2782:, p. 119), including a result of (
2528:are the injective hulls of the modules
1225:. It is a module of the same length as
545:
327:, i.e. such that the following diagram
16:Mathematical object in abstract algebra
4749:
4546:
4499:
4312:An introduction to homological algebra
4309:
4242:Golan, Jonathan S.; Head, Tom (1991),
3770:
3189:
1097:{\displaystyle E=E(R/{\mathfrak {p}})}
948:-modules and finitely generated right
193:, then there exists another submodule
4718:
4690:
4662:
4389:
4147:
4026:can be extended to the whole module.
2787:
2783:
2587:is the injective hull of some module
2560:{\displaystyle R/{\mathfrak {p}}_{i}}
1566:is a 1-dimensional vector space over
1524:{\displaystyle (R,{\mathfrak {m}},K)}
1257:{\displaystyle R/{\mathfrak {p}}^{k}}
1168:{\displaystyle R/{\mathfrak {p}}^{k}}
150:and are generalized by the notion of
4632:
4429:
4346:
4143:
4074:
4070:
4068:
3848:
3778:
3429:-module. Similarly, every injective
2900:, but between injective modules and
2403:{\displaystyle I\in {\text{Mod}}(R)}
2256:-module is a direct summand of some
2102:-module structure from the injection
1951:-module can be constructed from the
871:
95:
4636:Publicationes Mathematicae Debrecen
4271:
4077:Introduction to Commutative Algebra
3957:
3844:Generalizations and specializations
3837:
3798:
3315:
3174:is the injective hull of a uniform
3120:
3050:
2870:, the so-called "character module"
2711:
2705:
2694:
2671:
2631:{\displaystyle {\mathfrak {p}}_{i}}
2617:
2546:
2504:
2453:
2358:
2286:
2241:
2205:
2175:-module has an injection into some
2160:
2130:
2114:
2087:
2008:
1960:
1936:
1909:
1855:
1507:
1484:
1348:
1243:
1201:
1154:
1121:
1086:
1042:
1002:
892:0, then one shows in the theory of
709:of the ring. The injective hull of
690:
652:
633:{\displaystyle R_{\mathfrak {p}}/R}
616:
99:
13:
4340:
4246:Modules and the structure of rings
4007:
3731:, one gets the familiar fact that
3662:, and is the largest submodule of
3654:} is a left submodule of the left
3546:
3543:
3540:
3297:
3110:
2927:
1899:
466:. Note that the direct complement
442:. The new extending basis vectors
185:is a submodule of some other left
14:
4773:
4106:
4065:
3996:. Here the relationships between
1129:{\displaystyle R/{\mathfrak {p}}}
832:), and the injective hull of the
478:, and likewise the extending map
392:
3813:is self-injective. Every proper
3585:-module. Thus, coinduction over
3352:module. For any injective right
2920:is left noetherian, then a left
2908:, pp. 78–80). For any ring
944:between finitely generated left
338:
4510:Canadian Journal of Mathematics
4367:10.1090/S0002-9904-1940-07306-9
4176:Rings and Categories of Modules
3421:, then every vector space over
3208:. Moreover, the injective hull
2762:(i.e. an injective module over
2366:{\displaystyle {\mathfrak {p}}}
2249:{\displaystyle {\mathfrak {g}}}
2168:{\displaystyle {\mathfrak {g}}}
2095:{\displaystyle {\mathfrak {g}}}
2078:. Note this vector space has a
1968:{\displaystyle {\mathfrak {g}}}
1944:{\displaystyle {\mathfrak {g}}}
1863:{\displaystyle {\mathfrak {g}}}
1050:{\displaystyle {\mathfrak {p}}}
660:{\displaystyle {\mathfrak {p}}}
542:injective as an abelian group.
4596:Canadian Mathematical Bulletin
4553:Pacific Journal of Mathematics
4505:"On Utumi's ring of quotients"
4153:
4132:
4118:
4100:
4083:
4041:
3568:
3556:
3456:
3401:-module, then every injective
3143:the following are equivalent:
2515:
2490:
2464:
2439:
2397:
2391:
2300:
2291:
2281:
2275:
2219:
2210:
2200:
2194:
2135:
2125:
2119:
2022:
2013:
2003:
1997:
1914:
1904:
1825:
1789:
1691:
1679:
1659:
1624:
1616:
1604:
1553:
1547:
1518:
1496:
1448:
1429:
1388:
1356:
1320:
1287:
1212:
1182:
1175:can be computed as the module
1091:
1073:
795:{\displaystyle {\hat {R}}_{P}}
780:
474:is not uniquely determined by
458:is the internal direct sum of
1:
4274:Lectures on modules and rings
4029:
3183:has a local endomorphism ring
2638:are the associated primes of
860:behaves normally except that
161:
113:and represent modules in the
4693:Osaka Journal of Mathematics
4450:10.1016/0021-8693(81)90213-1
4210:Relative homological algebra
4166:
3980:is injective if and only if
3758:-resolutions into injective
2373:. That is, for an injective
1719:{\displaystyle \mathbb {C} }
438:and extend it to a basis of
368:-modules to the category of
146:. Injective modules include
23:, especially in the area of
7:
3789:Every ring with unity is a
3409:-module. In particular, if
2708:, p. 80-81, Th 3.46).
596:is also injective, and its
512:-modules. The factor group
387:
10:
4778:
4310:Rotman, Joseph J. (1979),
4011:
3992:for every nonzero integer
3961:
3852:
3836:and two-sided injective, (
3805:is self-injective, but no
3700:-module. Applying this to
3413:is an integral domain and
3310:, especially for instance
3204:a prime ideal of the ring
2931:
2833:category of abelian groups
2812:
2747:can be extended to all of
999:finitely-generated modules
721:-module is canonically an
4733:10.1080/00927878308822975
4721:Communications in Algebra
4677:10.1080/00927878108822627
4665:Communications in Algebra
4286:10.1007/978-1-4612-0525-8
2944:Eckmann & Schopf 1953
2410:, there is an isomorphism
2316:
1842:Modules over Lie algebras
1033:is a Noetherian ring and
1009:Computing injective hulls
755:of the injective hull of
527:> 1 is injective as a
4109:"Lie Algebra Cohomology"
4053:stacks.math.columbia.edu
4034:
3934:there exists a morphism
3903:is injective if for any
3746:-modules into injective
3593:-modules from injective
3465:{\displaystyle f:S\to R}
3433:-module is an injective
3405:-module is an injective
3378:) is an injective right
1672:which has maximal ideal
557:with field of fractions
550:More generally, for any
295:module homomorphism and
4272:Lam, Tsit-Yuen (1999),
3692:) is an injective left
3133:. A module is called a
2906:Enochs & Jenda 2000
1136:over the Artinian ring
74:from this submodule to
4610:10.4153/CMB-1964-039-3
4567:10.2140/pjm.1958.8.511
4524:10.4153/CJM-1963-041-4
3581:is an injective right
3575:
3508:is an injective right
3466:
2962:also has an injective
2809:Injective cogenerators
2772:principal ideal domain
2652:
2632:
2601:
2581:
2561:
2522:
2478:
2471:
2404:
2367:
2343:
2307:
2250:
2226:
2169:
2149:
2142:
2096:
2072:
2052:
2036:
2029:
1969:
1945:
1921:
1884:
1864:
1832:
1762:
1720:
1698:
1666:
1580:
1560:
1559:{\displaystyle soc(R)}
1525:
1475:
1455:
1395:
1333:
1258:
1219:
1169:
1130:
1098:
1057:is a prime ideal, set
1051:
1027:
796:
661:
634:
107:Injective cogenerators
4471:Archiv der Mathematik
4218:10.1515/9783110803662
4206:Jenda, Overtoun M. G.
4020:pure injective module
4014:pure injective module
3859:One also talks about
3678:is an injective left
3576:
3467:
3073:itself is injective.
2954:Injective resolutions
2829:injective cogenerator
2815:injective cogenerator
2780:Golan & Head 1991
2687:Golan & Head 1991
2653:
2633:
2602:
2582:
2562:
2523:
2472:
2412:
2405:
2368:
2344:
2308:
2251:
2227:
2170:
2143:
2104:
2097:
2073:
2053:
2030:
1977:
1970:
1946:
1922:
1885:
1865:
1833:
1763:
1721:
1699:
1697:{\displaystyle (x,y)}
1667:
1581:
1561:
1526:
1476:
1456:
1396:
1334:
1259:
1220:
1170:
1131:
1099:
1052:
1028:
894:group representations
797:
662:
635:
4179:, Berlin, New York:
4142:-Papp theorem, see (
4079:. pp. 624, 625.
3871:or in categories of
3830:quasi-Frobenius ring
3785:Self-injective rings
3519:
3504:, it says that when
3444:
3362:module homomorphisms
3119:injective modules, (
2642:
2611:
2591:
2571:
2532:
2484:
2417:
2377:
2353:
2333:
2260:
2236:
2232:and every injective
2179:
2155:
2109:
2082:
2062:
2042:
1982:
1955:
1931:
1894:
1874:
1850:
1772:
1730:
1708:
1676:
1590:
1570:
1535:
1493:
1489:An Artin local ring
1465:
1405:
1343:
1268:
1229:
1179:
1140:
1108:
1061:
1037:
1017:
770:
647:
607:
546:Commutative examples
422:-module. Reason: if
242:short exact sequence
123:essential extensions
4757:Homological algebra
4091:"Injective Modules"
3881:-modules over some
3832:, and is two-sided
3821:is self-injective.
3624:. The submodule ann
3600:For quotient rings
3589:produces injective
3025: = 0 for
2326:Over a commutative
916:associative algebra
866:formal power series
739:-injective hull of
730:module, and is the
577:-module containing
508:are also injective
207:internal direct sum
111:injective dimension
72:module homomorphism
54:. Specifically, if
4485:10.1007/BF01899665
4436:Journal of Algebra
4204:Enochs, Edgar E.;
3869:functor categories
3571:
3462:
3419:field of fractions
2898:projective modules
2648:
2628:
2597:
2577:
2567:. In addition, if
2557:
2518:
2467:
2435:
2400:
2363:
2339:
2303:
2246:
2222:
2165:
2138:
2092:
2068:
2048:
2025:
1965:
1941:
1917:
1880:
1860:
1846:For a Lie algebra
1828:
1758:
1716:
1704:and residue field
1694:
1662:
1576:
1556:
1521:
1471:
1451:
1391:
1329:
1254:
1215:
1165:
1126:
1094:
1047:
1023:
980:symmetric algebras
792:
657:
630:
92:projective modules
86:. This concept is
4727:(22): 2495–2505,
4325:978-0-12-599250-3
4295:978-0-387-98428-5
4257:978-0-8247-8555-0
4227:978-3-11-016633-0
4190:978-0-387-97845-1
3922:and any morphism
3861:injective objects
3849:Injective objects
3803:Frobenius algebra
3682:-module, then ann
3385:For instance, if
3150:is indecomposable
3131:endomorphism ring
2651:{\displaystyle M}
2600:{\displaystyle M}
2580:{\displaystyle I}
2426:
2389:
2342:{\displaystyle R}
2267:
2186:
2071:{\displaystyle V}
2051:{\displaystyle k}
1989:
1883:{\displaystyle k}
1779:
1579:{\displaystyle K}
1474:{\displaystyle k}
1427:
1026:{\displaystyle R}
991:commutative rings
872:Artinian examples
783:
753:endomorphism ring
600:summands are the
426:is a subspace of
152:injective objects
4769:
4743:
4715:
4687:
4659:
4629:
4612:
4586:
4569:
4543:
4526:
4496:
4487:
4461:
4452:
4431:Dade, Everett C.
4426:
4409:
4386:
4369:
4336:
4306:
4268:
4249:
4238:
4200:
4199:
4197:
4160:
4157:
4151:
4136:
4130:
4129:
4122:
4116:
4115:
4113:
4104:
4098:
4097:
4095:
4087:
4081:
4080:
4072:
4063:
4062:
4060:
4059:
4045:
3958:Divisible groups
3899:of the category
3855:injective object
3580:
3578:
3577:
3572:
3555:
3554:
3549:
3531:
3530:
3514:coinduced module
3471:
3469:
3468:
3463:
3425:is an injective
3389:is a subring of
3312:polynomial rings
3094:
3093:
3000:derived functors
2758:is an injective
2712:Baer's criterion
2657:
2655:
2654:
2649:
2637:
2635:
2634:
2629:
2627:
2626:
2621:
2620:
2606:
2604:
2603:
2598:
2586:
2584:
2583:
2578:
2566:
2564:
2563:
2558:
2556:
2555:
2550:
2549:
2542:
2527:
2525:
2524:
2519:
2514:
2513:
2508:
2507:
2500:
2476:
2474:
2473:
2468:
2463:
2462:
2457:
2456:
2449:
2434:
2409:
2407:
2406:
2401:
2390:
2387:
2372:
2370:
2369:
2364:
2362:
2361:
2348:
2346:
2345:
2340:
2312:
2310:
2309:
2304:
2290:
2289:
2274:
2273:
2268:
2265:
2255:
2253:
2252:
2247:
2245:
2244:
2231:
2229:
2228:
2223:
2209:
2208:
2193:
2192:
2187:
2184:
2174:
2172:
2171:
2166:
2164:
2163:
2147:
2145:
2144:
2139:
2134:
2133:
2118:
2117:
2101:
2099:
2098:
2093:
2091:
2090:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2049:
2034:
2032:
2031:
2026:
2012:
2011:
1996:
1995:
1990:
1987:
1974:
1972:
1971:
1966:
1964:
1963:
1950:
1948:
1947:
1942:
1940:
1939:
1926:
1924:
1923:
1918:
1913:
1912:
1903:
1902:
1889:
1887:
1886:
1881:
1869:
1867:
1866:
1861:
1859:
1858:
1837:
1835:
1834:
1829:
1824:
1810:
1796:
1788:
1787:
1786:
1780:
1777:
1767:
1765:
1764:
1759:
1751:
1737:
1725:
1723:
1722:
1717:
1715:
1703:
1701:
1700:
1695:
1671:
1669:
1668:
1663:
1658:
1657:
1636:
1635:
1623:
1603:
1585:
1583:
1582:
1577:
1565:
1563:
1562:
1557:
1530:
1528:
1527:
1522:
1511:
1510:
1485:Self-injectivity
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1441:
1440:
1428:
1425:
1423:
1422:
1400:
1398:
1397:
1392:
1387:
1386:
1368:
1367:
1352:
1351:
1338:
1336:
1335:
1330:
1328:
1327:
1318:
1317:
1299:
1298:
1280:
1279:
1263:
1261:
1260:
1255:
1253:
1252:
1247:
1246:
1239:
1224:
1222:
1221:
1216:
1211:
1210:
1205:
1204:
1197:
1196:
1174:
1172:
1171:
1166:
1164:
1163:
1158:
1157:
1150:
1135:
1133:
1132:
1127:
1125:
1124:
1118:
1103:
1101:
1100:
1095:
1090:
1089:
1083:
1056:
1054:
1053:
1048:
1046:
1045:
1032:
1030:
1029:
1024:
801:
799:
798:
793:
791:
790:
785:
784:
776:
705:varies over the
683:noetherian rings
666:
664:
663:
658:
656:
655:
640:for the nonzero
639:
637:
636:
631:
626:
621:
620:
619:
569:is an injective
430:, we can find a
418:is an injective
380:Injective right
342:
148:divisible groups
115:derived category
52:rational numbers
33:injective module
25:abstract algebra
4777:
4776:
4772:
4771:
4770:
4768:
4767:
4766:
4747:
4746:
4501:Lambek, Joachim
4407:10.2307/1993382
4360:(10): 800–807,
4343:
4341:Primary sources
4326:
4296:
4278:Springer-Verlag
4258:
4228:
4195:
4193:
4191:
4181:Springer-Verlag
4169:
4164:
4163:
4158:
4154:
4137:
4133:
4124:
4123:
4119:
4111:
4105:
4101:
4093:
4089:
4088:
4084:
4073:
4066:
4057:
4055:
4047:
4046:
4042:
4037:
4032:
4016:
4010:
4008:Pure injectives
4002:pure submodules
3970:divisible group
3966:
3964:divisible group
3960:
3894:
3880:
3857:
3851:
3846:
3819:Dedekind domain
3807:integral domain
3793:and hence is a
3787:
3687:
3629:
3550:
3539:
3538:
3526:
3522:
3520:
3517:
3516:
3445:
3442:
3441:
3369:
3318:, p. 62).
3300:
3298:Change of rings
3277:
3251:
3242:
3228:
3113:
3111:Indecomposables
3092:
3087:
3086:
3085:
3021:is nonzero and
2956:
2936:
2930:
2928:Injective hulls
2879:
2817:
2811:
2714:
2664:
2643:
2640:
2639:
2622:
2616:
2615:
2614:
2612:
2609:
2608:
2592:
2589:
2588:
2572:
2569:
2568:
2551:
2545:
2544:
2543:
2538:
2533:
2530:
2529:
2509:
2503:
2502:
2501:
2496:
2485:
2482:
2481:
2458:
2452:
2451:
2450:
2445:
2430:
2418:
2415:
2414:
2386:
2378:
2375:
2374:
2357:
2356:
2354:
2351:
2350:
2334:
2331:
2330:
2328:Noetherian ring
2324:
2319:
2285:
2284:
2269:
2264:
2263:
2261:
2258:
2257:
2240:
2239:
2237:
2234:
2233:
2204:
2203:
2188:
2183:
2182:
2180:
2177:
2176:
2159:
2158:
2156:
2153:
2152:
2151:In fact, every
2129:
2128:
2113:
2112:
2110:
2107:
2106:
2086:
2085:
2083:
2080:
2079:
2063:
2060:
2059:
2043:
2040:
2039:
2007:
2006:
1991:
1986:
1985:
1983:
1980:
1979:
1959:
1958:
1956:
1953:
1952:
1935:
1934:
1932:
1929:
1928:
1908:
1907:
1898:
1897:
1895:
1892:
1891:
1875:
1872:
1871:
1854:
1853:
1851:
1848:
1847:
1844:
1820:
1806:
1792:
1782:
1781:
1776:
1775:
1773:
1770:
1769:
1747:
1733:
1731:
1728:
1727:
1726:. Its socle is
1711:
1709:
1706:
1705:
1677:
1674:
1673:
1653:
1649:
1631:
1627:
1619:
1599:
1591:
1588:
1587:
1571:
1568:
1567:
1536:
1533:
1532:
1506:
1505:
1494:
1491:
1490:
1487:
1466:
1463:
1462:
1436:
1432:
1424:
1418:
1414:
1406:
1403:
1402:
1382:
1378:
1363:
1359:
1347:
1346:
1344:
1341:
1340:
1323:
1319:
1313:
1309:
1294:
1290:
1275:
1271:
1269:
1266:
1265:
1248:
1242:
1241:
1240:
1235:
1230:
1227:
1226:
1206:
1200:
1199:
1198:
1192:
1188:
1180:
1177:
1176:
1159:
1153:
1152:
1151:
1146:
1141:
1138:
1137:
1120:
1119:
1114:
1109:
1106:
1105:
1085:
1084:
1079:
1062:
1059:
1058:
1041:
1040:
1038:
1035:
1034:
1018:
1015:
1014:
1011:
961:
935:
918:over the field
874:
786:
775:
774:
773:
771:
768:
767:
738:
729:
651:
650:
648:
645:
644:
622:
615:
614:
610:
608:
605:
604:
587:quotient module
583:Dedekind domain
552:integral domain
548:
395:
390:
164:
156:category theory
127:Noetherian ring
119:Injective hulls
17:
12:
11:
5:
4775:
4765:
4764:
4759:
4745:
4744:
4716:
4688:
4671:(9): 989–999,
4660:
4630:
4591:Osofsky, B. L.
4587:
4544:
4497:
4462:
4443:(2): 416–425,
4427:
4387:
4348:Baer, Reinhold
4342:
4339:
4338:
4337:
4324:
4316:Academic Press
4307:
4294:
4269:
4256:
4239:
4226:
4201:
4189:
4168:
4165:
4162:
4161:
4152:
4131:
4117:
4107:Vogan, David.
4099:
4082:
4064:
4039:
4038:
4036:
4033:
4031:
4028:
4024:pure submodule
4012:Main article:
4009:
4006:
3962:Main article:
3959:
3956:
3890:
3876:
3853:Main article:
3850:
3847:
3845:
3842:
3809:that is not a
3786:
3783:
3769:The textbook (
3683:
3625:
3612:-module is an
3570:
3567:
3564:
3561:
3558:
3553:
3548:
3545:
3542:
3537:
3534:
3529:
3525:
3461:
3458:
3455:
3452:
3449:
3365:
3329:be rings, and
3308:quotient rings
3299:
3296:
3269:
3247:
3237:
3224:
3185:
3184:
3178:
3169:
3163:
3157:
3151:
3136:uniform module
3117:indecomposable
3112:
3109:
3099:) = 0 for all
3088:
3033:. If a module
2992:
2991:
2968:exact sequence
2955:
2952:
2940:injective hull
2934:injective hull
2932:Main article:
2929:
2926:
2875:
2813:Main article:
2810:
2807:
2713:
2710:
2663:
2660:
2647:
2625:
2619:
2596:
2576:
2554:
2548:
2541:
2537:
2517:
2512:
2506:
2499:
2495:
2492:
2489:
2466:
2461:
2455:
2448:
2444:
2441:
2438:
2433:
2429:
2425:
2422:
2399:
2396:
2393:
2385:
2382:
2360:
2338:
2323:
2320:
2318:
2315:
2302:
2299:
2296:
2293:
2288:
2283:
2280:
2277:
2272:
2243:
2221:
2218:
2215:
2212:
2207:
2202:
2199:
2196:
2191:
2162:
2137:
2132:
2127:
2124:
2121:
2116:
2089:
2067:
2058:-vector space
2047:
2024:
2021:
2018:
2015:
2010:
2005:
2002:
1999:
1994:
1962:
1938:
1916:
1911:
1906:
1901:
1879:
1857:
1843:
1840:
1827:
1823:
1819:
1816:
1813:
1809:
1805:
1802:
1799:
1795:
1791:
1785:
1757:
1754:
1750:
1746:
1743:
1740:
1736:
1714:
1693:
1690:
1687:
1684:
1681:
1661:
1656:
1652:
1648:
1645:
1642:
1639:
1634:
1630:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1602:
1598:
1595:
1575:
1555:
1552:
1549:
1546:
1543:
1540:
1520:
1517:
1514:
1509:
1504:
1501:
1498:
1486:
1483:
1470:
1450:
1447:
1444:
1439:
1435:
1431:
1421:
1417:
1413:
1410:
1390:
1385:
1381:
1377:
1374:
1371:
1366:
1362:
1358:
1355:
1350:
1326:
1322:
1316:
1312:
1308:
1305:
1302:
1297:
1293:
1289:
1286:
1283:
1278:
1274:
1251:
1245:
1238:
1234:
1214:
1209:
1203:
1195:
1191:
1187:
1184:
1162:
1156:
1149:
1145:
1123:
1117:
1113:
1093:
1088:
1082:
1078:
1075:
1072:
1069:
1066:
1044:
1022:
1010:
1007:
995:injective hull
989:, just as for
957:
931:
890:characteristic
873:
870:
789:
782:
779:
734:
725:
707:prime spectrum
654:
629:
625:
618:
613:
598:indecomposable
547:
544:
493:-module). The
485:The rationals
394:
393:First examples
391:
389:
386:
378:
377:
370:abelian groups
346:
345:
344:
343:
333:
332:
265:
238:
166:A left module
163:
160:
134:rings such as
131:indecomposable
64:direct summand
15:
9:
6:
4:
3:
2:
4774:
4763:
4762:Module theory
4760:
4758:
4755:
4754:
4752:
4742:
4738:
4734:
4730:
4726:
4722:
4717:
4714:
4710:
4706:
4702:
4698:
4694:
4689:
4686:
4682:
4678:
4674:
4670:
4666:
4661:
4658:
4654:
4650:
4646:
4642:
4638:
4637:
4631:
4628:
4624:
4620:
4616:
4611:
4606:
4602:
4598:
4597:
4592:
4588:
4585:
4581:
4577:
4573:
4568:
4563:
4559:
4555:
4554:
4549:
4545:
4542:
4538:
4534:
4530:
4525:
4520:
4516:
4512:
4511:
4506:
4502:
4498:
4495:
4491:
4486:
4481:
4477:
4473:
4472:
4467:
4463:
4460:
4456:
4451:
4446:
4442:
4438:
4437:
4432:
4428:
4425:
4421:
4417:
4413:
4408:
4403:
4399:
4395:
4394:
4388:
4385:
4381:
4377:
4373:
4368:
4363:
4359:
4355:
4354:
4349:
4345:
4344:
4335:
4331:
4327:
4321:
4317:
4313:
4308:
4305:
4301:
4297:
4291:
4287:
4283:
4279:
4275:
4270:
4267:
4263:
4259:
4253:
4248:
4247:
4240:
4237:
4233:
4229:
4223:
4219:
4215:
4211:
4207:
4202:
4192:
4186:
4182:
4178:
4177:
4171:
4170:
4156:
4149:
4145:
4141:
4135:
4127:
4121:
4110:
4103:
4096:. p. 10.
4092:
4086:
4078:
4071:
4069:
4054:
4050:
4044:
4040:
4027:
4025:
4021:
4015:
4005:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3971:
3965:
3955:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3913:
3909:
3906:
3902:
3898:
3893:
3888:
3884:
3879:
3874:
3870:
3866:
3862:
3856:
3841:
3839:
3835:
3831:
3827:
3822:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3782:
3780:
3776:
3772:
3767:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3738:
3734:
3730:
3726:
3722:
3718:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3686:
3681:
3677:
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3633:
3628:
3623:
3619:
3615:
3611:
3607:
3603:
3598:
3596:
3592:
3588:
3584:
3565:
3562:
3559:
3551:
3535:
3532:
3527:
3523:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3479:
3475:
3459:
3453:
3450:
3447:
3438:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3383:
3381:
3377:
3373:
3368:
3363:
3360:, the set of
3359:
3355:
3351:
3347:
3343:
3340:
3336:
3332:
3328:
3324:
3319:
3317:
3313:
3309:
3305:
3295:
3293:
3289:
3285:
3281:
3276:
3272:
3267:
3263:
3259:
3255:
3250:
3246:
3240:
3236:
3232:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3195:
3191:
3182:
3179:
3177:
3176:cyclic module
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3145:
3144:
3142:
3138:
3137:
3132:
3129:
3124:
3122:
3118:
3108:
3106:
3102:
3098:
3091:
3084:such that Ext
3083:
3079:
3074:
3072:
3068:
3064:
3060:
3057:such that id(
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3029:greater than
3028:
3024:
3020:
3016:
3012:
3007:
3005:
3001:
2997:
2989:
2985:
2981:
2977:
2973:
2972:
2971:
2969:
2965:
2961:
2958:Every module
2951:
2947:
2945:
2941:
2935:
2925:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2892:) is a right
2891:
2887:
2883:
2878:
2873:
2869:
2865:
2860:
2858:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2816:
2806:
2804:
2800:
2796:
2793:
2789:
2785:
2781:
2776:
2773:
2769:
2765:
2761:
2760:abelian group
2757:
2752:
2750:
2746:
2742:
2739:
2736:defined on a
2735:
2731:
2727:
2723:
2719:
2709:
2707:
2703:
2698:
2697:, Th. 3.22).
2696:
2692:
2688:
2684:
2681:
2677:
2673:
2669:
2659:
2645:
2623:
2594:
2574:
2552:
2539:
2535:
2510:
2497:
2493:
2487:
2477:
2459:
2446:
2442:
2436:
2431:
2427:
2423:
2420:
2411:
2394:
2383:
2380:
2336:
2329:
2314:
2297:
2294:
2278:
2270:
2216:
2213:
2197:
2189:
2148:
2122:
2103:
2065:
2045:
2035:
2019:
2016:
2000:
1992:
1976:
1877:
1870:over a field
1839:
1817:
1814:
1811:
1803:
1800:
1797:
1755:
1752:
1744:
1741:
1738:
1688:
1685:
1682:
1654:
1650:
1646:
1643:
1640:
1637:
1632:
1628:
1620:
1613:
1610:
1607:
1596:
1593:
1573:
1550:
1544:
1541:
1538:
1515:
1512:
1502:
1499:
1482:
1468:
1445:
1442:
1437:
1433:
1419:
1415:
1411:
1408:
1383:
1379:
1375:
1372:
1369:
1364:
1360:
1353:
1324:
1314:
1310:
1306:
1303:
1300:
1295:
1291:
1284:
1281:
1276:
1272:
1249:
1236:
1232:
1207:
1193:
1189:
1185:
1160:
1147:
1143:
1115:
1111:
1080:
1076:
1070:
1067:
1064:
1020:
1006:
1005:, §3G, §3J).
1004:
1000:
996:
992:
988:
987:Artinian ring
983:
981:
978:-module. For
977:
973:
969:
965:
960:
955:
951:
947:
943:
939:
934:
929:
925:
921:
917:
913:
908:
906:
902:
899:
898:group algebra
895:
891:
888:a field with
887:
883:
879:
869:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
824:
820:
816:
811:
809:
805:
787:
777:
766:
762:
758:
754:
750:
746:
742:
737:
733:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
684:
681:
676:
674:
670:
643:
627:
623:
611:
603:
602:localizations
599:
595:
591:
588:
584:
580:
576:
572:
568:
564:
560:
556:
553:
543:
541:
538:-module, but
537:
534:
530:
526:
522:
519:
515:
511:
507:
503:
499:
496:
492:
488:
483:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
425:
421:
417:
414:
410:
406:
403:
398:
385:
383:
375:
371:
367:
363:
359:
355:
352:
351:contravariant
348:
347:
341:
337:
336:
335:
334:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
270:
266:
263:
259:
255:
251:
247:
243:
239:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
179:
178:
176:
173:
169:
159:
157:
153:
149:
145:
141:
140:finite groups
137:
132:
128:
124:
120:
116:
112:
108:
103:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
29:module theory
26:
22:
4724:
4720:
4696:
4692:
4668:
4664:
4640:
4634:
4600:
4594:
4557:
4551:
4548:Matlis, Eben
4514:
4508:
4478:(2): 75–78,
4475:
4469:
4440:
4434:
4397:
4391:
4357:
4351:
4311:
4273:
4245:
4209:
4194:, retrieved
4175:
4155:
4138:This is the
4134:
4120:
4102:
4085:
4076:
4056:. Retrieved
4052:
4043:
4017:
3998:flat modules
3993:
3989:
3985:
3981:
3977:
3973:
3967:
3951:
3947:
3943:
3939:
3935:
3931:
3927:
3923:
3919:
3915:
3911:
3907:
3905:monomorphism
3900:
3896:
3891:
3886:
3883:ringed space
3877:
3858:
3823:
3788:
3775:localization
3768:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3736:
3732:
3728:
3724:
3720:
3716:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3684:
3679:
3675:
3674:-module. If
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3646:= 0 for all
3643:
3639:
3635:
3631:
3626:
3621:
3617:
3613:
3609:
3605:
3601:
3599:
3594:
3590:
3586:
3582:
3512:-module the
3509:
3505:
3501:
3497:
3489:
3488:over itself
3481:
3477:
3476:into a left-
3473:
3439:
3434:
3430:
3426:
3422:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3384:
3379:
3375:
3371:
3366:
3357:
3353:
3349:
3338:
3334:
3330:
3326:
3322:
3320:
3301:
3291:
3287:
3283:
3279:
3274:
3270:
3265:
3261:
3257:
3253:
3248:
3244:
3238:
3234:
3230:
3225:
3221:
3217:
3213:
3209:
3205:
3201:
3197:
3193:
3186:
3180:
3171:
3165:
3159:
3153:
3147:
3140:
3134:
3125:
3114:
3104:
3100:
3096:
3089:
3081:
3077:
3075:
3070:
3066:
3062:
3058:
3054:
3046:
3042:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3010:
3008:
3002:such as the
2995:
2993:
2987:
2983:
2979:
2975:
2970:of the form
2959:
2957:
2948:
2937:
2921:
2917:
2913:
2909:
2902:flat modules
2893:
2889:
2885:
2881:
2876:
2871:
2867:
2863:
2861:
2856:
2852:
2848:
2844:
2840:
2836:
2824:
2820:
2818:
2802:
2798:
2794:
2792:prime ideals
2777:
2763:
2755:
2753:
2748:
2744:
2740:
2733:
2729:
2725:
2721:
2717:
2715:
2699:
2665:
2479:
2413:
2325:
2150:
2105:
2037:
1978:
1845:
1488:
1012:
984:
975:
971:
967:
963:
958:
953:
949:
945:
937:
932:
927:
922:with finite
919:
914:is a unital
911:
909:
904:
900:
885:
882:finite group
877:
875:
861:
857:
853:
849:
845:
841:
837:
833:
830:Prüfer group
825:
822:
818:
814:
812:
807:
803:
760:
756:
744:
740:
735:
731:
726:
722:
718:
714:
710:
702:
698:
694:
677:
672:
642:prime ideals
593:
589:
578:
574:
570:
566:
562:
558:
554:
549:
539:
535:
532:
528:
524:
520:
517:
513:
509:
506:circle group
501:
497:
495:factor group
490:
486:
484:
479:
475:
471:
467:
463:
459:
455:
451:
447:
439:
435:
427:
423:
419:
415:
413:vector space
408:
404:
399:
396:
381:
379:
365:
357:
324:
320:
316:
312:
308:
304:
300:
296:
288:
284:
280:
276:
272:
268:
257:
256:→ 0 of left
253:
249:
245:
234:
230:
226:
222:
218:
214:
210:
202:
198:
194:
190:
186:
182:
174:
167:
165:
121:are maximal
104:
83:
79:
75:
67:
55:
47:
43:
39:
32:
18:
4643:: 311–327,
4603:: 405–413,
4560:: 511–528,
4517:: 363–370,
4466:Eckmann, B.
3791:free module
3771:Rotman 1979
3666:that is an
3190:Matlis 1958
3004:Ext functor
2862:For a left
2827:. It is an
2775:divisible.
2676:direct sums
749:local rings
687:Eben Matlis
680:commutative
446:a subspace
360:) from the
354:Hom functor
136:group rings
90:to that of
21:mathematics
4751:Categories
4384:0024.14902
4148:Chase 1960
4075:Eisenbud.
4058:2020-02-25
4030:References
3865:categories
3826:Noetherian
3795:projective
3597:-modules.
3496:as a left
3397:is a flat
3393:such that
3348:as a left-
3333:be a left-
3162:is uniform
3017:such that
2994:where the
2964:resolution
2859:-modules.
2788:Vámos 1983
2784:Smith 1981
2738:left ideal
2702:Noetherian
2691:hereditary
2683:semisimple
930:, then Hom
765:completion
669:zero ideal
581:. For any
319:such that
279:-modules,
201:such that
162:Definition
4705:0030-6126
4649:0033-3883
4619:0008-4395
4576:0030-8730
4533:0008-414X
4167:Textbooks
4144:Papp 1959
3972:. Here a
3779:Dade 1981
3528:∗
3457:→
3437:-module.
2912:, a left
2768:divisible
2607:then the
2428:⨁
2424:≅
2384:∈
2120:↪
2038:for some
1812:⋅
1804:⊕
1798:⋅
1753:⋅
1745:⊕
1739:⋅
1416:⊕
1373:…
1325:∙
1304:…
1277:∙
924:dimension
781:^
293:injective
275:are left
260:-modules
170:over the
96:Baer 1940
60:submodule
27:known as
4699:: 1–18,
4503:(1963),
4208:(2000),
3976:-module
3938: :
3926: :
3910: :
3838:Lam 1999
3834:Artinian
3824:A right
3815:quotient
3799:Lam 1999
3658:-module
3642: :
3492:is also
3480:, right-
3356:-module
3344:that is
3342:bimodule
3337:, right-
3316:Lam 1999
3304:subrings
3123:, §3F).
3121:Lam 1999
3051:Lam 1999
2866:-module
2801:-module
2728: :
2720:-module
2706:Lam 1999
2695:Lam 1999
2680:Artinian
2672:Lam 1999
1003:Lam 1999
985:For any
970:) where
836:-module
817:-module
691:Lam 1999
565:-module
504:and the
407:, every
400:Given a
388:Examples
364:of left
362:category
329:commutes
311: :
299: :
283: :
189:-module
100:Lam 1999
46:-module
4741:0733337
4713:0078966
4685:0614468
4657:0121390
4627:0166227
4584:0099360
4541:0147509
4494:0055978
4459:0617087
4424:0120260
4416:1993382
4376:0002886
4334:0538169
4304:1653294
4266:1201818
4236:1753146
4196:30 July
4146:) and (
3873:sheaves
2831:in the
2786:) and (
2668:product
1975:-module
942:duality
940:) is a
763:is the
685:due to
217:, i.e.
205:is the
102:, §3).
50:of all
4739:
4711:
4703:
4683:
4655:
4647:
4625:
4617:
4582:
4574:
4539:
4531:
4492:
4457:
4422:
4414:
4382:
4374:
4332:
4322:
4302:
4292:
4264:
4254:
4234:
4224:
4187:
3634:) = {
3472:makes
3268:to Hom
3233:, and
3045:). If
3011:length
2480:where
2317:Theory
751:. The
717:as an
701:where
667:. The
585:, the
561:, the
356:Hom(-,
291:is an
262:splits
237:= {0}.
144:fields
70:, any
37:module
4412:JSTOR
4112:(PDF)
4094:(PDF)
4035:Notes
3946:with
3817:of a
3811:field
3502:P = R
3260:) of
3128:local
3103:>
2990:→ ...
2966:: an
2874:= Hom
926:over
880:is a
828:(the
432:basis
402:field
374:exact
142:over
58:is a
35:is a
31:, an
4701:ISSN
4645:ISSN
4615:ISSN
4572:ISSN
4529:ISSN
4320:ISBN
4290:ISBN
4252:ISBN
4222:ISBN
4198:2016
4185:ISBN
4140:Bass
3875:of O
3719:and
3494:flat
3486:free
3417:its
3346:flat
3325:and
3321:Let
3294:)).
3200:for
3009:The
2974:0 →
2938:The
2666:Any
1339:and
936:(−,
884:and
852:for
523:for
462:and
454:and
444:span
349:The
271:and
240:Any
229:and
213:and
172:ring
88:dual
4729:doi
4673:doi
4605:doi
4562:doi
4519:doi
4480:doi
4445:doi
4402:doi
4380:Zbl
4362:doi
4282:doi
4214:doi
3918:in
3863:in
3781:).
3650:in
3638:in
3364:Hom
3306:or
3212:of
3095:(–,
2946:).
2743:of
2704:, (
2693:, (
2388:Mod
2266:Hom
2185:Hom
1988:Hom
1778:Hom
1426:Hom
1013:If
910:If
876:If
806:at
802:of
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