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Bixin 1 is the major constituent of the reddish carotenoids present in the seed‐coat of Bixa orellana. The use of the extract of the seed‐coat of Bixa orellana in food, cosmetics and garments is well known. The nano‐sized long 24 C chain molecule has nine conjugated double bonds having extended conjugation with the '‐COOH' and '‐COOMe' groups present at the two ends of the molecule. Herein, we report the first self‐assembly of bixin in several aqueous liquids. The molecule undergoes spontaneous self‐assembly in several liquids yielding vesicular self‐assembly. Characterizations of the self‐assemblies of bixin were carried out by various microscopic techniques, X‐ray diffraction and FTIR studies. The critical vesicular concentrations (CVCs) of the compound carried out in DMSO‐water in three different solvent ratios as 2: 1 (v/v), 1: 1 (v/v) and 1: 4 (v/v) were determined to be 100 μM, 90 μM and 60 μM respectively indicating lower CVC values at higher proportion of water. Utilization of the vesicular self‐assemblies of bixin have been demonstrated in the entrapment and release of fluorophores including the anticancer drugs doxorubicin and curcumin. Self‐assembled bixin and curcumin loaded self‐assembled bixin showed significant antibacterial activity with both Gram positive as well as Gram negative bacteria. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the zero-divisor associate graph ΓD(R) over a finite commutative ring R. It is a simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices a, b are adjacent if and only if there exist non-zero zero-divisors z1, z2 in R such that az1 = bz2. We determine the necessary and sufficient conditions for connectedness and completeness of ΓD(R) for a unitary commutative ring R. The chromatic number of ΓD(R) is also studied. Next, we characterize the rings R for which ΓD(R) becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as ΓD(R), characterizing the associated ring R in each case. [ABSTRACT FROM AUTHOR]

Let G be a group. Then the order two element graph of G is the simple undirected graph whose vertex set consists of all elements of G and two distinct vertices u,v are adjacent if and only if either uvorvu ∈{t2 : t ∈ G}∪{a ∈ G : a2 = e}∖{e}, where e is the identity element of G. This graph is denoted by 핊2(G). In this paper, we first establish some beautiful results on Merrifield–Simmons index of order two element graph over a group. After that we characterize the product of two order two element graph over two groups. Moreover we study some important results on domination number of order two element graph over a group. [ABSTRACT FROM AUTHOR]

This paper represents the structures of generalized derivations satisfying an identity with n -Engel condition. To prove our result, we take R as a prime ring with char (R) ≠ 2 , and U as Utumi quotient ring of R. The center of U is C which is called extended centroid of R. Let L be a non-central Lie ideal of R. Suppose that F and G are two nonzero generalized derivations of R and n ≥ 1 , any fixed integer. If [ F (u) G (u) , u ] n = 0 for all u ∈ L , then one of the below-mentioned conclusions holds: (1) F (x) = x a and G (x) = b x for all x ∈ R and for some a , b ∈ U with a b ∈ C ; (2) R satisfies s 4 . [ABSTRACT FROM AUTHOR]

Let R be a prime or semiprime ring, λ a nonzero left-sided ideal of R and α , β be two automorphisms of R. Let F : R → R and G : R → R be two generalized (α , β) -derivations of R associated with (α , β) -derivations d : R → R and g : R → R , respectively. The purpose of this paper is to investigate the following identities: (1) G (x y) + d (x) F (y) + α (x y) = 0 ; (2) G (x y) + d (x) F (y) = 0 ; (3) G (x y) + d (x) F (y) + α (y x) = 0 ; (4) G (x y) + d (x) F (y) + α (y x) + α (x y) = 0 ; (5) G (x y) + d (x) F (y) + α (y x) − α (x y) = 0 ; (6) G (x y) + d (y) F (x) + α (y x) = 0 ; (7) G (x y) + d (y) F (x) + α (y x) + α (x y) = 0 ; (8) G (x y) + d (y) F (x) + α (y x) − α (x y) = 0 ; (9) G (x y) + F (x) F (y) = 0 ; (10) G (x y) + F (y) F (x) = 0 ; (11) F (x y) + G (x) α (y) + α (y x) = 0 ; (12) F (x) F (y) + G (x) α (y) + α (y x) = 0 , for all x , y ∈ λ. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Z(R) its center, Qr its right Martindale quotient ring, C its extended centroid, f (x 1 , ... , x n) a non-central multilinear polynomial over C, b ∈ Q r and F, G two b-generalized skew derivations of R. If there exists a ∈ R \ Z (R) such that [ a , F (f (r 1 , ... , r n)) f (r 1 , ... , r n) ] = f (r 1 , ... , r n) G (f (r 1 , ... , r n)) for any r 1 , ... , r n ∈ R , then f (x 1 , ... , x n) 2 is central valued on R and there exist p , q ∈ Q r such that F(x) = px, G(x) = xq for all x ∈ R with [ a , p ] = q . [ABSTRACT FROM AUTHOR]

In this paper, we show the way to get into some results of partially ordered (in short, po-) ternary semigroup based on quasi-ideals, bi-ideals and semiprime ideals. We extend some results of po-semigroup into po-ternary semigroup under certain methodology. In particular, we characterize some properties of regular po-ternary semigroup, left (resp. right) regular po-ternary semigroup, completely regular po-ternary semigroup and intra-regular po-ternary semigroup by using quasi-ideal, bi-ideal and semiprime ideal of po-ternary semigroup. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of m-clean and strongly m-clean ring as generalizations of clean ring and strongly clean ring respectively. We prove that if a ring R is m-clean then the matrix ring M n (R) is also m-clean. We provide a characterization of strongly m-clean ring in terms of quotient ring by its ideal which is contained in its Jacobson radical. In particular, we prove that a ring is strongly m-clean if and only if its quotient ring by a nil ideal is strongly m-clean under certain conditions. We also introduce the notion of m-local ring as a subclass of local ring. We establish that a ring is m-local if and only if it is m-clean and it has no non-trivial m-potent element. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S a multiplicatively closed subset of R. Hamed and Malek [7] defined an ideal P of R disjoint with S to be an S-prime ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ P, then sa ∈ P or sb ∈ P. In this paper, we introduce the notions of S-k-prime and S-k-semiprime ideals of semirings, S-k-m-system, and S-k-p-system. We study some properties and characterizations for S-k-prime and S-k-semiprime ideals of semirings in terms of S-k-m-system and S-k-p-system respectively. We also introduce the concepts of S-prime semiring and S-semiprime semiring and study the characterizations for S-k-prime and S-k-semiprime ideals in these two semirings. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, $$C=Z(U)$$ the extended centroid of R, and $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that $$F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0$$ for all $$x_1,\ldots ,x_n\in R$$ . Then $$f(x_1,\ldots ,x_n)^2$$ is central valued on R and one of the following holds: [ABSTRACT FROM AUTHOR]

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 ≠ a ∊ R such that a|F(un1), un2, . . . , unk ] = 0 for all u ∊ L, where n1, n2, . . . , nk ≥ 1 are fixed integers. Then either there exists λ ≥ ∊ C such that F(x) = ≥x for all x ∊ R, or R satisfies s4, the standard identity in four variables. [ABSTRACT FROM AUTHOR]

The notion of k-regularity in a semiring generalizes the notion of a regular ring introduced by J. Von Neumaan [15]. In semiring theory, the notion of k-regularity was extensively studied by Sen and Bhuniya [17]. In this paper, we study the concept of k-regularities of a semiring in fuzzy setting and characterize some k-regularities of semirings by using different fuzzy ideals of semirings. We also introduce the concept of power fuzzy semirings and study the notion of k-regularities of power fuzzy semirings with the help of regularities of semigroups. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and σ, τ two epimorphisms of R. A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)-(σ, τ)-derivation if F(xy) = F(x)σ(y) + τ(x)g(y) holds for all x, y ∈ R, where g is any mapping on R. The main object in this article is to study the situations (1) F(xy) - F(x)F(y) = 0, (2) F(xy) + F(x)F(y) = 0, (3) F(xy) - F(y)F(x) = 0 and (4) F(xy) + F(y)F(x) = 0; for all x, y in some nonzero subsets of prime and semiprime rings, where F is a multiplicative (generalized)-(σ, σ)-derivation of R. [ABSTRACT FROM AUTHOR]

We introduce the notion of (i-v) semiprime (irreducible) fuzzy ideals of semigroups and investigate its different algebraic properties. We study the interrelation among (i-v) prime fuzzy ideals, (i-v) semiprime fuzzy ideals, and (i-v) irreducible fuzzy ideals and characterize regular semigroups by using these (i-v) fuzzy ideals. [ABSTRACT FROM AUTHOR]

In this paper we study some interesting properties of regular ternary semigroups, completely regular ternary semigroups, intra-regular ternary semigroups and characterize them by using various ideals of ternary semigroups. [ABSTRACT FROM AUTHOR]

This paper is a sequel of our previous paper [1]. In this paper, we introduce the notions of regular ideal and partial ideal (p-ideal) in a ternary semiring and using these two notions we characterize regular ternary semiring. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study of the square element graph which is defined as follows: if R is a ring then the square element graph 핊q(R) is the simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices u, v are adjacent if and only if u + v = x2 for some x ϵ R - {0}. Here we mainly consider the connectedness of 핊q(R) over different rings. We particularly look at 핊q(R) taken over some infinite rings which contain ℤ as a subring. For a ring R and an ideal I of R, the relation between the connectedness of 핊q(R) and 핊q(R/I) is studied. Finally, connectedness of 핊q(R) over various polynomial rings and matrix rings are considered. [ABSTRACT FROM AUTHOR]