On the sparsity of representations of rings of pure global dimension zero
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- by Birge Zimmermann-Huisgen and Wolfgang Zimmermann
- Trans. Amer. Math. Soc. 320 (1990), 695-711
- DOI: https://doi.org/10.1090/S0002-9947-1990-0965304-0
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Abstract:
It is shown that the rings $R$ all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer $n$ there are only finitely many isomorphism types of (a) indecomposable left $R$-modules of length $n$; (b) finitely presented indecomposable right $R$-modules of length $n$; (c) indecomposable right $R$-modules having minimal projective resolutions with $n$ relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right $R$-modules entails finite representation type.References
- Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI 10.1080/00927877408548230
- Maurice Auslander, Large modules over Artin algebras, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 1–17. MR 0424874
- Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
- Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
- Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
- I. S. Cohen and I. Kaplansky, Rings for which every module is a direct sum of cyclic modules, Math. Z. 54 (1951), 97–101. MR 43073, DOI 10.1007/BF01179851
- K. R. Fuller and Idun Reiten, Note on rings of finite representation type and decompositions of modules, Proc. Amer. Math. Soc. 50 (1975), 92–94. MR 376768, DOI 10.1090/S0002-9939-1975-0376768-5
- Phillip Griffith, On the decomposition of modules and generalized left uniserial rings, Math. Ann. 184 (1969/70), 300–308. MR 257136, DOI 10.1007/BF01350858
- Laurent Gruson and Christian U. Jensen, Modules algébriquement compacts et foncteurs $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1651–A1653. MR 320112 —, Deux applications de la notion de $L$-dimension, C. R. Acad. Sci. Paris Ser. A 282 (1976), 23-24.
- Manabu Harada and Youshin Sai, On categories of indecomposable modules. I, Osaka Math. J. 7 (1970), 323–344. MR 286859
- Gottfried Köthe, Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring, Math. Z. 39 (1935), no. 1, 31–44 (German). MR 1545487, DOI 10.1007/BF01201343
- Claus Michael Ringel, Representations of $K$-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302. MR 422350, DOI 10.1016/0021-8693(76)90184-8
- Claus Michael Ringel and Hiroyuki Tachikawa, $\textrm {QF}-3$ rings, J. Reine Angew. Math. 272 (1974), 49–72. MR 379578
- Sverre O. Smalø, The inductive step of the second Brauer-Thrall conjecture, Canadian J. Math. 32 (1980), no. 2, 342–349. MR 571928, DOI 10.4153/CJM-1980-026-0
- R. B. Warfield Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719. MR 242885
- Robert B. Warfield Jr., Rings whose modules have nice decompositions, Math. Z. 125 (1972), 187–192. MR 289487, DOI 10.1007/BF01110928
- R. B. Warfield Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), no. 2, 187–222. MR 401836, DOI 10.1016/0021-8693(75)90074-5
- Maher Zayed, Indecomposable modules over right pure semisimple rings, Monatsh. Math. 105 (1988), no. 2, 165–170. MR 930435, DOI 10.1007/BF01501169
- Wolfgang Zimmermann, Einige Charakterisierungen der Ringe, über denen reine Untermoduln direkte Summanden sind, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. II (1972), 77–79 (1973) (German). MR 360701
- Wolfgang Zimmermann, Rein injektive direkte Summen von Moduln, Comm. Algebra 5 (1977), no. 10, 1083–1117 (German). MR 450327, DOI 10.1080/00927877708822211
- Birge Zimmermann-Huisgen, Rings whose right modules are direct sums of indecomposable modules, Proc. Amer. Math. Soc. 77 (1979), no. 2, 191–197. MR 542083, DOI 10.1090/S0002-9939-1979-0542083-3
- Birge Zimmermann-Huisgen and Wolfgang Zimmermann, Algebraically compact ring and modules, Math. Z. 161 (1978), no. 1, 81–93. MR 498722, DOI 10.1007/BF01175615
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 320 (1990), 695-711
- MSC: Primary 16A64; Secondary 16A53, 16A65, 16A90
- DOI: https://doi.org/10.1090/S0002-9947-1990-0965304-0
- MathSciNet review: 965304