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Distinguished submodules

  • Vlastimil Dlab (a1)

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Although there is no need for a ‘distinguished’ submodule to be given a formal definition in the present paper, we like to indicate the meaning attached to this concept here. Perhaps the shortest way of doing so is to say that a distinguished submodule is a (covariant idempotent) functor from the category of (left) R-modules into itself mapping each R-module into its R-submodule specified by a family of left ideals of R. If is a family left ideals of R, then all elements of an R-module M of orders belonging to , do not, of course, in general form a submodule of M; but, there are certain families such that all the elements of orders from form a submodule in any R-module (distinguished submodules defined by ). Consequently, no particular structural properties of the R-module are involved in the definition of such submodules. In this way we can define radicals (in the sense of Kuroš [4]) of a module. In particular, we feel that an application of this method is an appropriate way in defining the (maximal) torsion submodule of a module.

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References

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[1]Dlab, V., ‘The concept of rank and some related questions in the theory of modules’, Comment. Math. Univ. Carolinae 8 (1967), 3947.
[2]Dlab, V., ‘Distinguished families of ideals of a ring’, Czechoslovak Math. J. (to appear).
[3]Goldie, A. W., ‘Torsion-free modules and rings’, J. of Algebra I (1964), 268287.
[4]Kuroš, A. G., ‘Radicals of rings and algebras’, Mat. Sbornik N.S. 33(75) (1953), 1326.
[5]Ore, O., ‘Galois connections’, Trans. Amer. Math. Soc. 55 (1944), 493513.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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