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Covering Coalgebras and Dual Non-singularity

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Abstract

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843–2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coalgebra colocalisation. Codense covers will dualise dense (or rational) extensions; a maximal codense cover construction for coalgebras with projective covers is proposed. We also look at a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. As a corollary we deduce that hereditary coalgebras and hence path coalgebras are non-singular in the above sense. We also look at coprime coalgebras and Hopf algebras which are non-singular as coalgebras.

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References

  1. Beidar, K., Kasch, F.: Good conditions for the total. In: International Symposium on Ring Theory. Trends in Mathematics, pp. 43–65. Birkäuser, Cambridge, MA (2001)

  2. Brzezinski, T., Wisbauer, R.: Corings and Coalgebras. Lecture Notes Ser. 309. London Mathematical Society, London, UK (2003)

  3. Chin, W.: Hereditary and path coalgebras. Comm. Algebra 30(4), 1829–1831 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chin, W., Montgomery, S.: Basic coalgebras. In: Modular interfaces. Modular Lie algebras, quantum groups, and Lie superalgebras. AMS/IP Stud. Adv. Math. 4, 41–47 (1997)

    MathSciNet  Google Scholar 

  5. Courter, R.C.: The maximal co-rational extension by a module. Canad. J. Math. 18, 953–962 (1966)

    MATH  MathSciNet  Google Scholar 

  6. Cuadra, J., Gmez-Torrecillas, J.: Serial coalgebras. J. Pure Appl. Algebra 189(1–3), 89–107 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cuadra, J., Gómez-Torrecillas, J.: Idempotents and Morita–Takeuchi theory. Comm. Algebra 30(5), 2405-2426 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Findlay, G.D., Lambek, J.: A generalized ring of quotients. I, II. Canad. Math. Bull. 1(77–85), 155–166 (1958)

    MathSciNet  Google Scholar 

  9. Generalov, A.I.: A radical dual to Goldie torsion. Abelevy Gruppy Moduli 11–12, 70–77 (1994)

    MathSciNet  Google Scholar 

  10. Jara, P., Merino, L., Navarro, G., Ruiz, J.F.: Localization in coalgebras, stable localizations and path coalgebras. Comm. Algebra 34(8), 2843–2856 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lam, T.Y.: Lectures on Modules and Rings, GTM vol. 189. Springer, Berlin Heidelberg New York (1999)

    Google Scholar 

  12. Lomp, C.: On dual Goldie dimension. MSc Thesis, University of Glasgow (1997)

  13. Lomp, C.: The splitting of the dual Goldie torsion theory. In: Algebra and its applications, pp. 377–386. Contemp. Math., vol. 259. American Mathematical Society, Providence, RI (2000)

  14. Matlis, E.: Injective modules over noetherian rings. Pacific. J. Math. 8(3), 511–528 (1958)

    MATH  MathSciNet  Google Scholar 

  15. Nastasescu, C., Torrecillas, B.: Colocalization on Grothendieck categories with applications to coalgebras. J. Algebra 185, 203–220 (1994)

    Google Scholar 

  16. Nastasescu, C., Torrecillas, B., Zhang, Y.H.: Hereditary coalgebras. Comm. Algebra 24(4), 1521–1528 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pareigis, B.: Radikale und kleine Moduln (Radicals and small modules). Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 1965, 185–199 (1966)

    MathSciNet  MATH  Google Scholar 

  18. Ramamurthi, V.S.: The smallest left exact radical containing the Jacobson radical. Ann. Soc. Sci. Brux. Ser. 1 96(4), 201–206 (1982)

    MathSciNet  MATH  Google Scholar 

  19. Talebi, Y., Vanaja, N.: Copolyform modules. Comm. Algebra 30(3), 1461–1473 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach, Reading, MA (1991)

    MATH  Google Scholar 

  21. Wisbauer, R.: Modules and Algebras: bimodule structure and group actions on algebras. Longman, Harlow (1996)

    MATH  Google Scholar 

  22. Zelmanowitz, J.: Representation of rings with faithful polyform modules. Comm. Algebra 14, 1141–1169 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zöschinger, H.: Kosinguläre und kleine Moduln. Comm. Algebra 33(10), 3389–3404 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Departamento de Matemática Pura, Universidade do Porto, Porto, Portugal

    Christian Lomp

  2. Departamento de Matemática, Universidade Federal de Santa Catarina, Florianopolis, Santa Catarina, Brazil

    Virgínia Rodrigues

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  1. Christian Lomp
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  2. Virgínia Rodrigues
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Correspondence to Christian Lomp.

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Lomp, C., Rodrigues, V. Covering Coalgebras and Dual Non-singularity. Appl Categor Struct 16, 195–211 (2008). https://doi.org/10.1007/s10485-006-9059-y

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  • DOI: https://doi.org/10.1007/s10485-006-9059-y

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