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Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

Filippis V. de, Dhara B.

Demonstratio Mathematica

2014

Vol. 47, nr 1

22--36

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f(x1,…,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f(r))f(r) – f(r)H(f(r)) € C, for all r = (r1,…,rn) € In. Then one of the following holds: 1. there exist a; b; p (…) ; 2. R satisfies s4, the standard identity of degree 4, and there exist a; (…) ; 3. R satisfies s4 and there exist a; (…) ; 4. R satisfies s4 and there exist a; (…) ; 5. there exists e2 = e (…) and one of the following holds: (a) (…) is an identity for I; (b) char(R) = 2 and s4(x1, x2, x3, x4)x5 is an identity for I; (c) (…) is an identity for I and there exist a, a’, b, b’ (…) , a derivation of R, such that G(x) = ax + xa’ + d(x), H(x) = bx + xb’ – d(x), for all x (…) R, with (a – b’ – α)I = (0) = (b – a’ – α)I.

Derivation with engel conditions on multilinear polynomials in prime rings

Dhara B.

Demonstratio Mathematica

2009

Vol. 42, nr 3

467-478

Let R be a prime ring with extended centroid C and characteristic different from 2, d a nonzero derivation of R, f(x1,..., xn) a nonzero multilinear polynomial over C such that [d2(f(x1,... , xn)),d(f(x1,... ,xn))]k = 0 for all x1,... ,xn in some nonzero right ideal [...] of R, where k is a fixed positive integer. If d(p) p [..]0, then pC = eRC for some idempotent e in the socle of RC and /(x1,..., xn) is central-valued on eRCe.

Generalized Jordan derivations on semiprime rings

Wei F., Xiao Z.

Demonstratio Mathematica

2007

Vol. 40, nr 4

789-798

It is shown that, given a 2-torsion-free semiprime ring with unit e, every generalized Jordan derivation on R is a generalized derivation. Let n be a fixed positive integer, R be a noncommutative (n+1)!-torsion-free prime ring with the center CR. It is proved that, if [..]: R -> R is a generalized Jordan derivation of R such that [..](x)xn + x [...].