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Let R be a commutative ring with unity and M be an R-module with TΓ(M) be its total graph. The subject of this article is the investigation of the properties of the corresponding line graph L(TΓ(M)). In particular, we determine the girth and clique number of L(TΓ(M)). In addition to that we find a condition for L(TΓ(M)) to be Eulerian. [ABSTRACT FROM AUTHOR]

Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and sufficient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions. Finally, a relation between small L-submodules and Jacobson L-radical is established. [ABSTRACT FROM AUTHOR]