Preradicals of Associative Algebras and their Connections with Preradicals of Modules

  • M. Luísa Galvão
Part of the Trends in Mathematics book series (TM)


We study preradicals on an universal class \( \mathcal{D} \) of algebras and we present a process to construct preradicals over algebras from certain families of preradicals over modules. We also define a torsion Plotkin radical on the class of all associative algebras which satisfies dual properties of the Jacobson radical.


Preradical socle radical 


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  1. [1]
    T. Anderson, N. Divinsky and A. Sulinski, Hereditary radicals in associative and alternative rings, Canad. J. Math. 17 (1965) 594–603.MathSciNetzbMATHGoogle Scholar
  2. [2]
    V.A. Andrunakievch, Radical of Associative Rings I, Mat. Sb. 44(86) (1958) 179–212 (in Russian), English Translation: Amer. Math. Soc. Transl. 2(52) (1966) 95–128.MathSciNetGoogle Scholar
  3. [3]
    N. Bourbaki, Modules et anneaux semisimples, (Éléments de mathématique Livre II, Algèbre, Cap. 8, Hermann, Paris 1958).zbMATHGoogle Scholar
  4. [4]
    J. Dauns, Modules and Rings (Cambridge University Press, 1994).Google Scholar
  5. [5]
    N.J. Divinski, Rings and Radicals (Mathematical Expositions No14, University of Toronto Press, 1965).Google Scholar
  6. [6]
    M.L. Galvão, M.T. Nogueira and J.A. Green, Radicals and Socles of an Algebra without Identity, Commun. Algebra 31(6) (2003) 2883–2907.CrossRefzbMATHGoogle Scholar
  7. [7]
    B.J. Gardner and R. Wiegandt, Radical Theory of Rings, Pure and Applied Mathematics. A series of Monographs and Textbooks 261 (2004) (New York, Marcel Dekker).Google Scholar
  8. [8]
    N. Jacoson, Structure of Rings, Amer. Math. Soc. Colloquium Publ. 37 (1956) (Providence).Google Scholar
  9. [9]
    T.Y. Lam, A First Course in Noncommutative Rings, 2nd edition, Graduate Texts in Mathematics 131 (1991) (Springer-Verlag, New York).zbMATHGoogle Scholar
  10. [10]
    G. Michler, Radikale und Sockel, Math. Annalen 167 (1966) (1–48).CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    R. Mlitz and R. Wiegant, Radicals and subdirect decompositions of Ω-groups, J. Austral. Math. Soc., Ser A 48 (1990) 171–198.zbMATHGoogle Scholar
  12. [12]
    F. Raggi, J.R. Montes, H. Rincón, R. Fernández-Alonso and C. Signoret, The lattice structure of preradicals, Commun. Algebra 30(3) (2002) 1533–1544.zbMATHCrossRefGoogle Scholar
  13. [13]
    B. de la Rosa, S. Veldsman and R. Wiegandt, On the theory of Plotkin Radicals, Chinese J. Math. (Taiwan, R.O.C.) 21(1) (1993) 33–54.MathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • M. Luísa Galvão
    • 1
  1. 1.Centro de Álgebra da Universidade de LisboaLisboaPortugal

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