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Erucic acid, a nano-sized long-chain ω-9-cis-mono-unsaturated 22C fatty acid, is isolable from various plants, especially from the seeds of Brassicaceae (e.g., rape seed or mustard seed). Isolation of erucic acid has recently been reported by us from the pseudobulb of Crepidium acuminatum.Herein, we report the first self-assembly of erucic acid in different aqueous binary liquids yielding vesicles. Several microscopic techniques such as scanning electron microscopy, high-resolution transmission electron microscopy, atomic force microscopy, optical microscopy, and X-ray diffraction studies were carried out to characterize the vesicular morphology. Encapsulation of various fluorophores as guests inside the self-assemblies of erucic acid was also demonstrated. The examples of the guests include cationic fluorophores rhodamine B, anionic fluorophore carboxyfluorescein, and cationic fluorescence active anticancer drug doxorubicin. Release of the entrapped fluorophores was demonstrated by treating with a vesicle-rupturing agent Triton X-100. On treatment with a composite of self-assembled erucic acid and curcumin, antibacterial activity was observed for both Gram-positive as well as Gram-negative bacteria. Therefore the vesicular self-assemblies generated from the renewable nano-sized erucic acid are useful for drug delivery applications as well as antibacterial activities with poorly water-soluble drugs.

Let Rbe a prime ring of char(R)≠ 2, Uits Utumi ring of quotients and center C= Z(U) its extended centroid, Ia both sided ideal of R, f(x1,…,xn) a multilinear polynomial over C, that is noncentral-valued on R, F, Gbe two generalized derivations of Rand dbe a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial f(x1,…,xn) in I. If @@@ for all u∈ f(I), then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity [F2(u)u,G2(v)v] = 0 for all u,v∈ f(I), where Fand Gare two generalized derivations of R.

In this paper, we discover Anjaneyulu’s pseudo-symmetricity with the help of interval-valued fuzzy membership function. In semigroups, we propose the concept of (i–v) pseudo-symmetric (semipseudo-symmetric, semiprimary) fuzzy ideals and cultivate their different properties. Moreover, we mention some relationships among three (i–v) fuzzy radicals. Lastly, we verify that the (i–v) fuzzy ideals, namely, (i–v) completely prime, (i–v) completely semiprime, (i–v) pseudo-symmetric and (i–v) primary fuzzy ideals are equivalent in some specific semigroups.

In this paper, our main objective is to introduce and investigate the interval-valued (in short, (i-v)) prime fuzzy hyperideal in semihypergroups in detail. We notice that every (i-v) semiprime fuzzy hyperideal may not be an (i-v) prime fuzzy hyperideal and we produce a counter example to illustrate this result. Moreover, we define (i-v) fuzzy hyper radical of an (i-v) fuzzy hyperideal of a semihypergroup. Finally, we study some interesting properties regarding this radical.

In this paper, we highlight an analogous study to characterize the k-regular, k-intra-regular, k-completely regular semigroups in terms of interval-valued (in short, (i-v)) fuzzy ideals with interval-valued fuzzy k-closure.

In this paper we establish the primary decomposition of an (i-v) fuzzy ideal of a non-commutative semigroup. To set up the main results we initiate the notions of (i-v) left (right) primary fuzzy ideals on the basis of (i-v) fuzzy setting. We deduce some fascinating properties and characterizations for these ideals. Finally, we prove that in a duo Noetherian semigroup, a class of (i-v) fuzzy ideals has always an (i-v) fuzzy primary decomposition.

Let Rbe a prime ring of characteristic not equal to 2, Uthe Utumi quotient ring of R, $$C=Z(U)$$ C=Z(U) the extended centroid of R, and $$f(x_1,\ldots ,x_n)$$ f(x1,…,xn) be a multilinear polynomial over C, not central valued on R. Suppose that Fand Gare two nonzero generalized derivations of Rsuch that $$F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0$$ F(f(x1,…,xn))G(f(x1,…,xn))+G(f(x1,…,xn))F(f(x1,…,xn))=0 for all $$x_1,\ldots ,x_n\in R$$ x1,…,xn∈R . Then $$f(x_1,\ldots ,x_n)^2$$ f(x1,…,xn)2 is central valued on Rand one of the following holds:(1)there exist $$0\ne \lambda \in C$$ 0≠λ∈C and $$p\in U-C$$ p∈U-C , such that $$F(x)=\lambda x$$ F(x)=λx for all $$x\in R$$ x∈R and $$G(x)=[p,x]$$ G(x)=[p,x] for all $$x\in R$$ x∈R ;(2)there exist nonzero $$a, b \in U$$ a,b∈U , such that $$F(x)=xa$$ F(x)=xa for all $$x\in R$$ x∈R and $$G(x)=bx$$ G(x)=bx for all $$x\in R$$ x∈R with $$ab=-ba\in C$$ ab=-ba∈C ;(3)there exist nonzero $$a, b \in U$$ a,b∈U , such that $$F(x)=ax$$ F(x)=ax for all $$x\in R$$ x∈R and $$G(x)=xb$$ G(x)=xb for all $$x \in R$$ x∈R with $$ab=-ba\in C$$ ab=-ba∈C ;(4)there exist $$0\ne \lambda \in C$$ 0≠λ∈C and $$a\in U-C$$ a∈U-C , such that $$F(x)=[a,x]$$ F(x)=[a,x] for all $$x\in R$$ x∈R and $$G(x)=\lambda x$$ G(x)=λx for all $$x\in R$$ x∈R .

Let Rbe a prime ring with center Z(R) and with extended centroid C, da derivation of Rand f(x1,…,xn) a nonzero multilinear polynomial over C, m≥ 1 and p≥ 1 two integers. In the present paper, we study the situations (i) ((d(f(x1,…,xn)))m− f(x1,…,xn))p= 0; (ii) ((d(f(x1,…,xn)))m− f(x1,…,xn))p∈ Z(R) for all x1,…,xnin some subsets of R.