Isolated Submodules and Skew Fields

Temple H. Fay () and Stephan V. Joubert ()
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Temple H. Fay: University of Southern Mississippi, Department of Mathematics
Stephan V. Joubert: Technikon Pretoria, Department of Mathematical Technology

A chapter in Papers in Honour of Bernhard Banaschewski, 2000, pp 317-326 from Springer

Abstract: 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.

Keywords: isolated; honest; relatively divisible; skew field.; 16A90; 16A36 (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-2529-3_18

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DOI: 10.1007/978-94-017-2529-3_18

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