Some Characterizations of Modules via Essentially Small Submodules
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In this paper, the structure of $e$-local modules and classes of modules via essentially small are investigated. We show that the following conditions are equivalent for a module $M$:\begin{description}\item(1) $M$ is $e$-local;\item(2) $\Rad_e(M)$ is a maximal submodule of $M$ and every proper essential submodule of $M$ is contained in a maximal submodule;\item(3) $M$ has a unique essential maximal submodule and every proper essential submodule of $M$ is contained in a maximal submodule. \end{description}
Abstract
In this paper, the structure of $e$-local modules and classes of modules via essentially small are investigated. We show that the following conditions are equivalent for a module $M$:\begin{description}\item(1) $M$ is $e$-local;\item(2) $\Rad_e(M)$ is a maximal submodule of $M$ and every proper essential submodule of $M$ is contained in a maximal submodule;\item(3) $M$ has a unique essential maximal submodule and every proper essential submodule of $M$ is contained in a maximal submodule. \end{description}
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