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ON (?, ?)-DERIVATIONS OF PRIME RINGS
(Korean Soc Mathematical Education, 2006) Kaya, K.; Guven, E.; Soyturk, M.
Let R be a prime ring with characteristics not 2 and sigma, tau, alpha, beta be automorphisms of R. Suppose that d(1) is a (sigma, tau)-derivation and d(2) is a (alpha, beta)-derivation on R such that d(2)alpha = alpha d(2), d(2)beta = beta d(2). In this note it is shown that; (1) If d(1)d(2)(R) = 0 then d(1) -0 or d(2) = 0. (2) If [d(1)(R), d(2)(R)] = 0 then R is commutative. (3) If(d(1)(R), d(2)(R)) = 0 then R is commutative. (4) If [d(1)(R), d(2)(R)]sigma,tau = 0 then R is commutative.
On Lie ideals with generalized derivations
(Maik Nauka/Interperiodica/Springer, 2006) Goelbasi, Oe.; Kaya, K.
Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a is an element of R and [a, f (U)] = 0 then a is an element of Z or d(a) = 0 or U subset of Z; (ii) If f(2)(U) = 0 then U subset of Z; (iii) If u(2) is an element of U for all u is an element of U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U subset of Z.