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.
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Fay, T.H., Joubert, S.V. Isolated Submodules and Skew Fields. Applied Categorical Structures 8, 317–326 (2000). https://doi.org/10.1023/A:1008622617846
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DOI: https://doi.org/10.1023/A:1008622617846