55 research outputs found
Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras
No full textLet R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G, the two nonzero generalized derivations of R, I an ideal of R and f(x(1),..., x(n)) a multilinear polynomial over C which is not central valued on R. If F(G(f (x(1),..., x(n))) f (x(1),..., x(n))) = 0 for all x(1),..., x(n). I, then one of the followings holds: (1) there exist a, b epsilon U such that F(x) = ax and G(x) = bx for all x. R with ab = 0; (2) there exist a, b, p epsilon U such that F(x) = ax + xb and G(x) = px for all x epsilon R with F(p) = 0 and f (x(1),..., x(n))(2) is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra.INSA, India; TUBA, TurkeyTurkish Academy of SciencesThe authors would like to thank the referee for providing shortened proof of Lemma 2.1 in the paper. This work was done when the first author visited Ege University, TURKEY, from the 9th June 2014 to the 15th June 2014 under the INSA-TUBA Exchange of Scientists Programme. The first author is grateful to INSA, India and TUBA, Turkey for the financial support provided for this visit
ON GENERALIZED (sigma, tau)-DERIVATIONS II
No full textThis paper continues a line investigation in [1] Let A be a K-algebra and M an A/K-bimodule In [5] Hamaguchi gave a necessary and sufficient condition for gDer(A,M) to be isomorphic to BDer(A, M) The main aim of this paper is to establish similar relationships for generalized (sigma, tau)-derivation
A RESULT ON GENERALIZED DERIVATIONS WITH ENGEL CONDITIONS ON ONE-SIDED IDEALS
No full textLet R be a non-commutative prime ring and I a non-zero left ideal of R Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers If there exists a generalized derivation g of R such that [g(x(m))x(n), x(r)](k) = 0 for all x is an element of I, then there exists a is an element of U such that g(x) = xa for all x is an element of R except when R congruent to M(2)(GF(2)) and I[I, I] = 0Scientific and Technological Research Council of Turkey, TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [108T257]This research has been supported by The Scientific and Technological Research Council of Turkey, TUBITAK, No 108T25
Generalized derivations of prime rings on multilinear polynomials with annihilator conditions
No full textLet K be a commutative ring with unity, R be a prime K-algebra with characteristic not 2, U be the right Utumi quotient ring of R, C the extended centroid of R, I a nonzero right ideal of R and a a fixed element of R. Let g be a generalized derivation of R and f(X-1,..., X-n) a multilinear polynomial over K. If ag(f(x(1),...,x(n)))f(x(1),...,x(n)) = 0 for all x1,..., x(n) is an element of I, then one of the following holds: (1) aI = ag(I) --= 0; (2) g(x) = bx [c,x] for all x is an element of R, where b,c is an element of U. In this case either [c,I]I = 0 = abI or aI = 0 = a(b + c)I; (3) [f,(X-1,...,X-n),Xn+1]Xn+2 is an identity for I
Cocentralizing Derivations And Nilpotent Values On Lie Ideals
No full textLet R be a prime ring with charR ±= 2, L a non-central Lie ideal of R, d, g non-zero derivations of R, n?1 a fixed integer. We prove that if (d(x)x - xg(x))n = 0 for all x?L, then either d = g = 0 or R satisfies the standard identity s4 and d,g are inner derivations, induced respectively by the elements a and b such that a + b?Z(R). © Indian National Science Academy
Annihilator condition of a pair of derivations in prime and semiprime rings
No full textLet n be a fixed positive integer, R be a prime ring, D and G two derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 not equal a a R such that a(D(u)u (n) -u (n) G(u)) = 0 for all u a L, where n a parts per thousand yen 1 is a fixed integer. Then one of the following holds: D = G = 0, unless R satisfies s (4); char (R) not equal 2, R satisfies s (4), n is even and D = G; char (R) not equal 2, R satisfies s (4), n is odd and D and G are two inner derivations induced by b, c respectively such that b + c a C; char (R) = 2 and R satisfies s (4). We also investigate the case when R is a semiprime ring.National Board for Higher Mathematics (NBHM), India [NBHM/R.P. 26/2012/Fresh/1745]; Scientific and Technological Research Council of Turkey, TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [110T586]This work is supported by a grant from National Board for Higher Mathematics (NBHM), India. Grant No. is NBHM/R.P. 26/2012/Fresh/1745 dated 15.11.12 and second author is supported by The Scientific and Technological Research Council of Turkey, TUBITAK, No. 110T58
On Identities with Composition of Generalized Derivations
No full textLet R be a prime ring with extended centroid C, Q maximal right ring of quotients of R, RC central closure of R such that dim(C) (RC) > 4, f(X-1,...,X-n) a multilinear polynomial over C that is not central-valued on R, and f (R) the set of all evaluations of the multilinear polynomial f(X-1,...,X-n) in R. Suppose that G is a nonzero generalized derivation of R such that G(2) (u) u is an element of C for all u is an element of f (R). Then one of the following conditions holds: (i) there exists a is an element of Q such that a(2) = 0 and either G(x) = ax for all x is an element of R or G(x) = xa for all x is an element of R; (ii) there exists a is an element of Q such that 0 not equal a(2) is an element of C and either G(x) = ax for all x is an element of R or G(x) = xa for all x is an element of R and f(X-1,...,X-n)(2) is central-valued on R; (iii) char(R) = 2 and one of the following holds: (a) there exist a, b is an element of Q such that G(x) = ax + xb for all x is an element of R and a(2) = b(2) is an element of C; (b) there exist a, b is an element of Q such that G(x) = ax + xb for all x is an element of R, a(2), b(2) is an element of C and f(X-1,...,X-n)(2) is central-valued on R; (c) there exist a is an element of Q and an X-outer derivation d of R such that G(x) = ax + d(x) for all x is an element of R, d(2) = 0 and a(2) + d (a) = 0; (d) there exist a is an element of Q and an X-outer derivation d of R such that G(x) = ax + d(x) for all x is an element of R, d(2) = 0, a(2) + d(a) is an element of C and f(X-1,...,X-n)(2) is central-valued on R. Moreover, we characterize the form of nonzero generalized derivations G of R satisfying G(2) (x) = lambda x for all x is an element of R, where lambda is an element of C.Scientific and Technological Research Council of Turkey, TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [2211-A, 110T586]First and second authors are supported by The Scientific and Technological Research Council of Turkey, TUBITAK, No:2211-A and No:110T586, respectively
Investigation of a combined continuous flow system for the removal of Pb and Cd from heavily contaminated soil
No full textKöleli, Nurcan (Arel Author)In this study, a combined continuous flow system was designed to remove Pb and Cd from heavily contaminated mine tailing soils. 0.05 M Na(2)EDTA was used as a chelating agent to remove Pb and Cd from polluted soil, taken from the vicinity of Kayseri CINKUR, Turkey. The initial concentrations of Pb and Cd were 16381 +/- 643 and 34347 +/- 1310 mg kg(-1), respectively. The electrochemical treatment process was applied to the waste washing solution, which emerged after being extracted from soil column and contained Pb and Cd. Metal ions were transformed to the metallic form by applying the electrochemical treatment process to the washing solution, containing Pb2+ and Cd2+
A note on generalized Lie derivations of prime rings
No full textLet R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x) (y-y) K(x)-T(y) (x + x) D(y) for all x, y E A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.Ege University, Scientific Research Project (BAP)Ege University [2014FEN036]This work was supported by Ege University, Scientific Research Project (BAP) (No. 2014FEN036)
DERIVATIONS OF PRIME AND SEMIPRIME RINGS
No full textLet R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer. (i) If (d(x)y+xd(y)+d(y)x+yd(x))(n) = xy+yx for all x, y is an element of I, then R is commutative. (ii) If char R not equal 2 and (d(x)y + xd(y) + d(y)x + yd(x))(n) - (xy + yx) is central for all x, y is an element of I, then R is commutative. We also examine the case where R is a semiprime ring
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