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We demonstrate polarization-independent holographic gratings (PIHGs) based on azo-dye-doped polymer-dispersed liquid crystal films. The PIHGs are recorded by irradiation under an intensity-modulated interference field, generated by two linearly polarized green optical beams, and by simultaneously applying a suitable AC voltage to the sample. The photoexcited azo dyes are adsorbed onto the UV-cured polymer film with their long axes parallel to the normal of the substrate. When the applied voltage is switched off, the PIHGs are generated with periodic modulation of liquid-crystal structures with transparently homeotropic and randomly scattered alignments. Additionally, the generated PIHGs can be completely switched off by an applied voltage. OCIS codes: 050.1950, 160.3710, 160.5470.

Let R be a prime ring with the extended centroid C and L a noncentral Lie ideal of R. Suppose that a ∈ R , γ ∈ C and d is a derivation of R such that a d (u 2) s − γ u 2 t m ∈ C for all u ∈ L , where s , t , m are fixed positive integers. If γ ≠ 0 or d ≠ 0 , then a = 0 unless R ⊆ M 2 (F) , the 2 × 2 matrix ring over a field F. This result gives an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator conditions on Lie ideals in prime rings, J. Algebra Appl. 19 (2020) 2050025]. [ABSTRACT FROM AUTHOR]

Let R be a prime algebra over a commutative ring K, let I be a nonzero ideal of R and let f (X 1 , ... , X k) be a multilinear polynomial over K in k non-commuting indeterminates which is not central-valued on R. Suppose that a ∈ R and d is a derivation of R such that a (d (f (x 1 , ... , x k)) s − γ f (x 1 , ... , x k ) t ) m ∈ Z (R) for all x 1 , ... , x k ∈ I , where γ ∈ K , s, t, m are fixed positive integers and Z (R) is the centre of R. It is shown that a = 0 or d = 0 except when R ⊆ M 2 (F) , the 2 × 2 matrix ring over a field F. This result gives a natural generalization of several well-known theorems in the literature. Moreover, with this we give an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator conditions on Lie ideals in prime rings. J Algebra Appl. 2020;19:2050025]. [ABSTRACT FROM AUTHOR]

Let D be a division ring, let M be a right vector space over D and let End (M D ) be the ring of all D -linear transformations from M into M. Suppose that R is a dense subring of End (M D ) consisting of finite rank transformations and f : R → End (M D ) is an additive map. We show that if f (x) x m (x) = x m (x) f (x) for every rank-k transformation x ∈ R , where k is a fixed integer with 1 < k < dim M D and m (x) ≥ 1 is an integer depending on x, then there exist λ ∈ Z (D) and an additive map μ : R → Z (D) I such that f (x) = λ x + μ (x) for all x ∈ R , where I denotes the identity transformation on M. This gives a natural generalization of the recent results obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815; Liu and Yang, Power commuting additive maps on invertible or singular matrices. Linear Alg Appl. 2017;530:127–149] and can be regarded as an infinite-dimensional version of the Franca theorem obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815]. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that [ g (x m) x n − x s h (x t) , x r ] k = 0 for all x ∈ I , where m , n , s , t , r , k are fixed positive integers. Then, one of the following conditions holds: there exist α , β ∈ C such that g (x) = α x and h (x) = β x for all x ∈ R ; m = s, n = t and there exist a ∈ Q and α ∈ C such that g(x) = xa and h (x) = (a + α) x for all x ∈ R ; C is a finite field, R = Q ≅ M ℓ (C) , the ℓ × ℓ matrix ring over C for some integer ℓ ≥ 2 and there exist a , u ∈ R such that g(x) = xa and h(x) = ux for all x ∈ R. [ABSTRACT FROM AUTHOR]

Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime*-algebra. As an application, we prove that every triple homomorphism from a Banach⋆-algebra onto a prime semisimple idempotent Banach*-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [ y , x ]1 = [ y , x ] = yx − xy for x , y ∈ Q and inductively [ y , x ] k = [[ y , x ] k −1, x ] for k > 1. Suppose that f : R → Q is an additive map satisfying [ f (x), x ] n = 0 for all x ∈ R , where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f (x) = λ x + μ (x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996. [ABSTRACT FROM AUTHOR]

Let A be a complex Banach algebra, let be the set of all quasinilpotent elements in A and let be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that provided that A has the property . The class of Banach algebras with the property is quite large: it includes -algebras, group algebras of locally compact groups, commutative Banach algebras, Banach algebras of all bounded linear operators on Banach spaces and so on. Our theorems are natural generalizations of the recent results for derivations obtained by Alaminos et al. [Bull. London Math. Soc. 46:379-384, 2014]. [ABSTRACT FROM AUTHOR]

Let be a prime ring with the extended centroid , a noncommutative Lie ideal of and a nonzero -generalized derivation of . For , let . We prove that if for all , where are fixed positive integers, then there exists such that for all except when , the matrix ring over a field . The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.] [ABSTRACT FROM AUTHOR]

LetRbe a noncommutative prime ring with the extended centroidC,Ia nonzero ideal ofRandgab-generalized derivation ofR. We show that, iffor all, wherem, n, kare fixed positive integers, then there existssuch thatfor allunless, thematrix ring over the Galois fieldGF(2) of two elements. This gives a natural generalization of the results for derivations, generalized derivations and generalized-derivations with an X-inner automorphism. [ABSTRACT FROM PUBLISHER]

LetRbe a prime ring,La noncentral Lie ideal ofR, anda ∈ R. Set [x,y]1 = [x,y] = xy − yxforx,y ∈ Rand inductively [x,y]k = [[x,y]k−1,y] fork > 1. Suppose that δ is a nonzero σ-derivation ofRsuch thata[δ(x),x]k = 0 for allx ∈ L, where σ is an automorphism ofRandkis a fixed positive integer. Thena = 0 except when char R = 2 andR ⊆ M2(F), the 2 × 2 matrix ring over a fieldF. [ABSTRACT FROM AUTHOR]