Generalization of Quasi-modular Extensions

  • El Hassane FliouetEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 228)


Let K/k be a purely inseparable extension of characteristic \(p>0\). Let lm(K/k) and um(K/k) be the smallest extensions \(k \longrightarrow lm(K\)/\(k) \longrightarrow K\longrightarrow um(K\)/k) such that K/lm(K/k) and um(K/k)/k are modular. In this note, we continue to study the locus problem of lm(K/k) and um(K/k) relative to K/k. Thus improving ([3], Theorem 1.4), we show that lm(K/k) is nontrivial when K/k is of finite size, more precisely if K/k has a finite size and unbounded exponent, the same is true of K/lm(K/k). However, if K/k is of unbounded size, it may well be that we lose this property by obtaining lm(K/\(k)=K \). In the following, we will say that K/k is lq-modular (respectively, uq-modular) if lm(K/k)/k (respectively, um(K/k)/K) has an exponent. The first study of these two concepts devoted to the extensions of finite size is in [4, 6, 7]. However, the object of the present work consists to generalize the results of finite size to any extension. In particular, we treat the stability questions of the lq-modularity and the uq-modularity relative to inclusion, intersection, and product. Furthermore, we are interested by the questions about existence of the smallest extensions which preserve these concepts in the ascendant or descendant sense, and also to the questions of existence of the maximal subextensions (closures).


Purely inseparable q-finite modular extension Lq-modular extension Up-modular 


  1. 1.
    Bourbaki, N.: Eléments de Mathématique Théorie des Ensembles. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Chellali, M., Fliouet, E.: Sur les extensions purement inséparable. Arch. Math. 81, 369–382 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chellali, M., Fliouet, E.: Extensions purement inséparables d’exposant non borné. Arch. Math. 40(40), 129–159 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chellali, M., Fliouet, E.: Extension presque modulaire. Ann. Sci. Math Québec 28(1–2), 65–75 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chellali, M., Fliouet, E.: Sur la tour des clôtures modulaires. An. St. Univ. Ovidius Constanta 14(1), 45–66 (2006)Google Scholar
  6. 6.
    Chellali, M., Fliouet, E.: Théorème de la clôture \(lq\)-modulaire et applications. Colloq. Math. 122, 275–287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chellali, M., Fliouet, E.: Extensions i-modulaires. Int. J. Algebra 6(10), 457–492 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fliouet, E.: Absolutely \(lq\)-finite extensions (1917). Cited 19 Jan 2017
  9. 9.
    Kime, L.K.: Purely inseparable modular extensions of unbounded exponent. Trans. Amer. Math. Soc 176, 335–349 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mordeson, J.N., Vinograde, B.: Structure of Arbitrary Purely Inseparable Extension Fields. LNM, vol. 173. Springer, Berlin (1970)Google Scholar
  11. 11.
    Pickert, G.: Inseparable Körperweiterungen. Math. Z. 52, 81–135 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sweedler, M.E.: Structure of inseparable extensions. Ann. Math. 87(2), 401–410 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Waterhouse, W.C.: The structure of inseparable field extensions. Trans. Amer. Math. Soc. 211, 39–56 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Regional Center for the Professions of Education and TrainingAgadirMorocco

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