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Öğe CENTRALIZERS OF GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS(Maik Nauka/Interperiodica/Springer, 2017) Albas, E.; Argac, N.; De Filippis, V.Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x (1),..., x (n) ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r (1),..., r (n) ): r (i) is an element of R} be the set of all evaluations of f(x (1),..., x (n) ) in R, while A = {[G (f(r (1),..., r (n) )), f(r (1),..., r (n) )]: r (i) is an element of R}, and let C (R) (A) be the centralizer of A in R; i.e., C (R) (A) = {a is an element of R: [a, x] = 0, for all (x) is an element of A }. We prove that if A not equal (0), then C (R) (A) = Z(R).Öğe An engel condition with generalized derivations on lie ideals(Mathematical Soc Rep China, 2008) Argac, N.; Carini, L.; De Filippis, V.Let R be a prime ring, with extended centroid C, g a non-zero generalized derivation of R, L a non-central Lie ideal of R, k >= 1 a fixed integer. If [g(u), u](k) = 0, for all u, then either g(x) = ax, with a is an element of C or R satisfies the standard identity 84, Moreover in the latter case either char(R) = 2 or char(R) not equal 2 and g(x) = ax + xb, with a, b is an element of Q and a - b is an element of C. We also prove a more generalized version by replacing L with the set [I, I], where I is a right ideal of R.Öğe An Engel condition with generalized skew derivations on multilinear polynomials(Taylor & Francis Ltd, 2018) Albas, E.; Argac, N.; De Filippis, V.; Demir, C.Let R be a prime ring of characteristic different from 2 with right Martindale quotient ring Q and extended centroid C. Let further k = 1 be a fixed integer, f (x1,..., xn) a multilinear polynomial over C which is not central-valued on R. If F : R. R is a nonzero generalized skew derivation of R such that [ F(f (r1,..., rn)), f (r1,..., rn)] k = 0 for all r1,..., rn. R, then either there exists.. C such that F(x) =.x for all x. R, or one of the following holds: (a) char(R) = p > 0 and f (x1,..., xn) ps is central-valued on R for a suitable s = 0; (b) there exist a, b. Q with a -b. C such that F(x) = ax + xb for all x. R, and f (x1,..., xn) 2 is central-valued on R.Öğe Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals(C E D A M Spa Casa Editr Dott Antonio Milani, 2008) Argac, N.; De Filippis, V.; Inceboz, H. G.Let K be a commutative ring with unity, R a prime K-algebra, with extended centroid C and right Utumi quotient ring U, g a non-zero generalized derivation of R. Suppose that f(x(1), . . . . ,x(n)) is a multilinear polynomial over K, I is a non-zero right ideal of R and m >= 1, a fixed integer. We prove the following results: If g(f(r(1), . . . . , r(n)))(m) = 0, for all r(1), . . . . , r(n) is an element of I, then one of the following holds: (1) [f(r(1), . . . . . , X(n)), X(n+1)] X(n+2) is an identity for I; (2) g(x) = ax for all X is an element of R, where a is an element of U such that aI = 0; (3) g(x) = ax + [q, x] for all X is an element of R, where a, q is an element of U such that aI = 0 and [q, I] I = 0. If there exist a(1), . . . , a(n) is an element of I such that g(f(a(1), . . . . , a(n)))(m) not equal 0 and g(f (r(1), . . . , r(n)))(m) is an element of Z(R), for all r(1), . . . . , r(n) is an element of I, then one of the following holds: (1) f(x(1), . . . . , x(n))x(n+1) is an identity for I; (2) f(x(1), . . . . , x(n)) is central valued on R; (3) g(x) = ax for a is an element of C and f(x(1), . . . . , x(n)) is power central valued on R; (4) R satisfies s(4), the standard identity in four variables.Öğe GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT IDEALS OF PRIME RINGS(Hacettepe Univ, Fac Sci, 2014) Albas, E.; Argac, N.; De Filippis, V.; Demir, C.Let R be a prime ring, f(x(1),...,x(n)) a multilinear polynomial over C in n noncommuting indeterminates, I a nonzero right ideal of R, and F : R -> R be a nonzero generalized skew derivation of R. Suppose that F(f(r(1),...,r(n)))f(r(1),...,r(n)) is an element of C, for all r(1),...,r(n) is an element of I. If f(x(1),...,x(n)) is not central valued on R, then either char(R) = 2 and R satisfies s(4) or one of the following holds: (i) f (x(1),...,x(n))x(n+1) is an identity for I; (ii) F(I)I = (0); (iii) [f(x(1),...,x(n)),x(n+1)]x(n+2) is an identity for I, there exist b,c,q is an element of Q with q an invertible element such that F(x) = bx - qxq(-1) c for all x is an element of R, and q(-1)cI subset of I. Moreover, in this case either (b - c)I = (0) or b c is an element of C and f(x(1),...,x(n))(2) is central valued on R.Öğe GENERALIZED SKEW DERIVATIONS WITH INVERTIBLE VALUES ON MULTILINEAR POLYNOMIALS(Taylor & Francis Inc, 2012) Demir, C.; Albas, E.; Argac, N.; De Filippis, V.Let R be a prime ring, f(X-1, ..., X-n) a multilinear polynomial which is not central-valued on R, and G a nonzero generalized skew derivation of R. Suppose that G(f(x(1), ..., x(n))) is zero or invertible for all x(1), ..., x(n) is an element of R. Then it is proved that R is either a division ring or the ring of all 2 x 2 matrices over a division ring. This result simultaneously generalizes a number of results in the literature.Öğe Identities with inverses on matrix rings(Taylor & Francis Ltd, 2020) Argac, N.; Eroglu, M. P.; Lee, T. -K.; Lin, J. -H.Motivated by [1, 2], the goal of the paper is to study certain identities with inverses on matrix rings. Given D a division ring, we characterize additive maps f,g:D -> D satisfying the identity f(x)x-1+xg(x-1)=0 for all invertible x is an element of D. Let R be a matrix ring over a division ring of characteristic not 2. We also characterize additive maps f,g:R -> R satisfying the identity f(x)x-1+xg(x-1)=0 for all invertible x is an element of R. Precisely, there exist an element q is an element of R and a derivation d of R such that f(x)=xq+d(x) and g(x)=-qx+d(x) for all x is an element of R. This affirmatively answers the question below Theorem 4 in [1] due to L. Catalano.Öğe Power-Central Values and Engel Conditions in Prime Rings with Generalized Skew Derivations(Springer Basel Ag, 2021) Argac, N.; De Filippis, VLet R be a prime ring of characteristic different from 2 with extended centroid C, n >= 1 a fixed positive integer, F, G : R -> R two non-zero generalized skew derivations of R. (I) If (F(x)x)(n) is an element of C for all x is an element of R, then the following hold: (a) if F is an inner generalized skew derivation, then either R subset of M-2(C) or R is commutative; (b) if F is not an inner generalized skew derivation, then R is commutative. (II) If [F(x)x, G(y)y](n) = 0 for all x, y is an element of R, then R is commutative unless when char(R) = p > 0, G is an inner generalized skew derivation and R subset of M-2(C).Öğe Prime Rings with Generalized Derivations on Right Ideals(World Scientific Publ Co Pte Ltd, 2011) Demir, C.; Argac, N.Let K be a commutative ring with unit, R be a prime K-algebra with center Z(R), right Utumi quotient ring U and extended centroid C, and I a nonzero right ideal of R. Let g be a nonzero generalized derivation of R and f(X(1), ... , X(n)) a multilinear polynomial over K. If g(f(x(1,) ... , x(n) ))f(x(1), ... , x(n)) is an element of C for all x(1), ... , x(n) is an element of I, then either f(x1, ... , ) x(n) (+ 1) is an identity for I, or char(R) = 2 and R satisfies the standard identity s(4)(x(1), ... , x(4)), unless when g(x) = ax [x, b] for suitable a, b is an element of U and one of the following holds: (i) a, b is an element of C and f, x)2 is central valued on R; (ii) a is an element of C and f(x(1), ... , x(n)) is central valued on R; (iii) aI = 0 and [f(x(1) , ... , x(n)), x(n + 1)]x(n +) (2) is an identity for I; (iv) aI = 0 and (b - beta)I = 0 for some beta is an element of C.
