The Cartan matrix as an indicator of finite global dimension for Artinian rings
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- by W. D. Burgess, K. R. Fuller, E. R. Voss and B. Zimmermann-Huisgen PDF
- Proc. Amer. Math. Soc. 95 (1985), 157-165 Request permission
Abstract:
A left serial ring has finite global dimension if and only if its Cartan matrix has determinant equal to 1.References
- Philip J. Davis, Circulant matrices, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 543191
- Samuel Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310–319. MR 65544, DOI 10.1007/BF02566937
- Samuel Eilenberg, Masatoshi Ikeda, and Tadasi Nakayama, On the dimension of modules and algebras. I, Nagoya Math. J. 8 (1955), 49–57. MR 69157
- Kent R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248–260. MR 232795, DOI 10.1007/BF01110273
- K. R. Fuller and Joel Haack, Rings with quivers that are trees, Pacific J. Math. 76 (1978), no. 2, 371–379. MR 498683
- Robert Gordon and Edward L. Green, Modules with cores and amalgamations of indecomposable modules, Mem. Amer. Math. Soc. 10 (1977), no. 187, viii+145. MR 453815, DOI 10.1090/memo/0187 W. H. Gustafson, Global dimension in serial rings, 1983 (typescript).
- J. P. Jans and Tadasi Nakayama, On the dimension of modules and algebras. VII. Algebras with finite-dimensional residue-algebras, Nagoya Math. J. 11 (1957), 67–76. MR 86824
- G. J. Janusz, Some left serial algebras of finite type, J. Algebra 23 (1972), 404–411. MR 302703, DOI 10.1016/0021-8693(72)90139-1
- Herbert Kupisch, Beiträge zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung, J. Reine Angew. Math. 201 (1959), 100–112 (German). MR 104707, DOI 10.1515/crll.1959.201.100
- Ichiro Murase, On the structure of generalized uniserial rings. I, Sci. Papers College Gen. Ed. Univ. Tokyo 13 (1963), 1–22. MR 156875
- Tadasi Nakayama, Some studies on regular representations, induced representations and modular representations, Ann. of Math. (2) 39 (1938), no. 2, 361–369. MR 1503413, DOI 10.2307/1968792
- George V. Wilson, The Cartan map on categories of graded modules, J. Algebra 85 (1983), no. 2, 390–398. MR 725091, DOI 10.1016/0021-8693(83)90103-5
- Dan Zacharia, On the Cartan matrix of an Artin algebra of global dimension two, J. Algebra 82 (1983), no. 2, 353–357. MR 704760, DOI 10.1016/0021-8693(83)90156-4
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 157-165
- MSC: Primary 16A60; Secondary 16A35, 16A48
- DOI: https://doi.org/10.1090/S0002-9939-1985-0801315-7
- MathSciNet review: 801315