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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]

On the basis of the concept of grades of a fuzzy point to belongingness (2) or quasi-coincident (q) or belongingness and quasi-coincident (∊ ⋁q) or belongingness or quasi-coincident (∊ ⋀q) in an intuitionistic fuzzy set of a ring, the notion of a (α,β)- intuitionistic fuzzy subring and ideal is introduced by applying the Lukasiewicz 3-valued implication operator. Using the notion of fuzzy cut set of an intuitionistic fuzzy set, the support and a-level set of an intuitionistic fuzzy set are defined and it is established that, for a ≢=∊ ⋀q, the support of a (a;b)-intuitionistic fuzzy ideal of a ring is an ideal of the ring. It is also established that the level sets of an intuitionistic fuzzy ideal with thresholds (s; t) of a ring is an ideal of the ring. We investigate that an intuitionistic fuzzy set A of a ring is a (∊;∊) (or (∊;∊ ⋀q ) or (∊ ⋀q;∊) )-intuitionistic fuzzy ideal of the ring if and only if A is an intuitionistic fuzzy ideal with thresholds (0; 1) (or (0;0:5) or (0:5; 1)) of the ring respectively. We also establish that A is a (∊;∊) (or (∊;∊ ⋀q ) or (∊ ⋀q;∊) )-intuitionistic fuzzy ideal of the ring if and only if for any a ∊ (0;1] (or a ∊ (0;0:5] or a ∊ (0:5;1] ), Aa is a fuzzy ideal of the ring. Finally, we investigate that an intuitionistic fuzzy set of a ring is an intuitionistic fuzzy ideal with thresholds (s; t) of the ring if and only if for any a ∊ (s; t], the cut set Aa is a fuzzy ideal of R. [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]