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Isolated Submodules and Skew Fields

In: Papers in Honour of Bernhard Banaschewski

Author

Listed:
  • Temple H. Fay

    (University of Southern Mississippi, Department of Mathematics)

  • Stephan V. Joubert

    (Technikon Pretoria, Department of Mathematical Technology)

Abstract

We study the generally distinct concepts of isolated submodule, honest submodule, and relatively divisible submodule for unital right R-modules, where R is an associative ring with identity. This is accomplished by studying a certain subset called the Q-torsion subset relative to a subset Q (sometimes a right ideal but not always) of R. The Q-isolator turns out to always to be a categorical closure operator and the notion of Q-honest is an ‘operator’ but need not be a closure operator. It is shown that the notions of Q-isolated and Q-honest coincide precisely when the Q-honest operator is a closure operator and this happens precisely when all submodules are Q-honest. As a corollary, we obtain when Q = R, every submodule is honest if and only if every submodule is isolated if and only if R is a skew field. We also determine a new characterization of a right Ore domain.

Suggested Citation

  • Temple H. Fay & Stephan V. Joubert, 2000. "Isolated Submodules and Skew Fields," Springer Books, in: Guillaume Brümmer & Christopher Gilmour (ed.), Papers in Honour of Bernhard Banaschewski, pages 317-326, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-2529-3_18
    DOI: 10.1007/978-94-017-2529-3_18
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