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This article embodies a ring theoretic property that preserves the reversibility at non zero k-potents. The structure of k-potents in k-reversible and pseudo k-reversible rings are studied in connection with various kinds of ring extensions. A ring R is said to be k-reversible if and only if ab ∈ K(R) for a, b ∈ R implies ab = ba; and a ring R is called pseudo k-reversible if 0 ≠ ab ∈ K(R) for a, b ∈ K(R) implies ab = ba, where K(R) is the set of all k-potents in R. It is proved that k-reversible rings are pseudo k-reversible but the converse is not true in general. Under certain conditions the polynomial rings of pseudo k-reversible inherit the character of pseudo k-reversible rings. [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]