Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HEREDITARY AND SEMIHEREDITARY REPRESENTATIONS OF QUIVERS

  • Bennis, Driss (Department of Mathematics Mohammed V University in Rabat Faculty of Sciences) ;
  • Roudi, Adnane (Department of Mathematics Mohammed V University in Rabat Faculty of Sciences)
  • Received : 2020.08.15
  • Accepted : 2021.09.02
  • Published : 2021.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate hereditary and semihereditary representations of quivers over an arbitrary ring. As consequences hereditary and semihereditary category of representations of quivers over an arbitrary ring are characterized.

Keywords

Acknowledgement

Adnane Roudi's research reported in this publication was supported by a scholarship from the Graduate Research Assistantships in Developing Countries Program of the Commission for Developing Countries of the International Mathematical Union.

References

  1. J. Asadollahi and R. Hafezi, On the dimensions of path algebras, Math. Res. Lett. 21 (2014), no. 1, 19-31. https://doi.org/10.4310/MRL.2014.v21.n1.a2
  2. H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
  3. S. Crivei, D. Keskin Tutuncu, and R. Tribak, Split objects with respect to a fully invariant short exact sequence in abelian categories, To appear in Rend. Sem. Math. Univ. Padova, 2020.
  4. S. Eilenberg, H. Nagao, and T. Nakayama, On the dimension of modules and algebras. IV. Dimension of residue rings of hereditary rings, Nagoya Math. J. 10 (1956), 87-95. http://projecteuclid.org/euclid.nmj/1118799771 https://doi.org/10.1017/S002776300000009X
  5. E. Enochs and S. Estrada, Projective representations of quivers, Comm. Algebra 33 (2005), no. 10, 3467-3478. https://doi.org/10.1081/AGB-200058181
  6. E. Enochs, J. R. Garc'ia Rozas, L. Oyonarte, and S. Park, Noetherian quivers, Quaest. Math. 25 (2002), no. 4, 531-538. https://doi.org/10.2989/16073600209486037
  7. E. Enochs and L. Oyonarte, Covers, Envelopes and Cotorsion Theories, Nova Science, New York, 2002.
  8. E. Enochs, L. Oyonarte, and B. Torrecillas, Flat covers and flat representations of quivers, Comm. Algebra 32 (2004), no. 4, 1319-1338. https://doi.org/10.1081/AGB-120028784
  9. D. J. Fieldhouse, Regular rings and modules, J. Austral. Math. Soc. 13 (1972), 477-491. https://doi.org/10.1017/S144678870000923X
  10. D. A. Hill, Endomorphism rings of hereditary modules, Arch. Math. (Basel) 28 (1977), no. 1, 45-50. https://doi.org/10.1007/BF01223887
  11. I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327-340. https://doi.org/10.2307/1990759
  12. B. Pareigis, Categories and Functors, translated from the German, Pure and Applied Mathematics, Vol. 39, Academic Press, New York, 1970.
  13. J.-E. Roos, Locally Noetherian categories, Category theory and their applications II, Battelle instetute Conference 1968 (ed. P. Hilton), Lecture notes in Mathematics 92 (Springer, Berlin, 1969), 197-277.
  14. M. S. Shrikhande, On hereditary and cohereditary modules, Canadian J. Math. 25 (1973), 892-896. https://doi.org/10.4153/CJM-1973-094-2
  15. D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006.
  16. B. Stenstrom, Purity in functor categories, J. Algebra 8 (1968), 352-361. https://doi.org/10.1016/0021-8693(68)90064-1
  17. A. A. Tuganbaev, The structure of modules over hereditary rings, Math. Notes 68 (2000), no. 5-6, 627-639; translated from Mat. Zametki 68 (2000), no. 5, 739-755. https://doi.org/10.1023/A:1026675709016