Some results on ideals of semiprime rings with multiplicative generalized derivations

Generalized multiplicative α-skew derivations on rings

“…It is easy to see that the following result is a generalization of [9,Corollary 2.7]. This leads to G([α(x), α(y)]) = [α(x), α(y)] for all x, y ∈ U.…”

“…This theorem was obtained for generalized derivations by Quadri et al in [11]. It was further extended by Shang in [14] and by E. Koç in [9].…”

“…In fact, our theorems generalize, extend and complement several results viz. (Lemma 2.3, Theorem 2.4,Corollary 2.7, Theorem 2.8 and Corollary 2.10) in [9] concerning semiprime rings to a multiplicative generalized α-skew derivations.…”

“…[9, Corollary 2.7] Let R be a semiprime ring and U a nonzero ideal of R. Suppose that R admits a generalized derivation G associated with a nonzero additive map d. If G satisfies any one of the following conditions, then R contains a nonzero central ideal(i) G(y) = xy for all x, y ∈ U; (ii) G(xy) = −xy for all x, y ∈ U; (iii) For each x, y ∈ U, either G(xy) = xy or G(xy) = −xy. Let R be a semiprime ring, α : R → R be an automorphism and U a nonzero ideal of R. Suppose that R admits a multiplicative generalized α-skew derivation G associated with a nonzero additive map d and an automorphism α…”

There are no nontrivial two-sided multiplicative (generalized)-skew derivations in prime rings

“…For example one tries to extend to a more general setting Posner's second theorem for derivations of prime rings ([18, Theorem 2]), which states that a prime ring R is commutative when it has a derivation d = 0 such that xd(x) − d(x)x is central for every x ∈ R. These efforts have generated literature in abundance (e.g. [1,4,7,9,10,11,13,15,19]). g) A multiplicative (generalized) derivation ( [8]) is a map F : R → R, not necessarily additive, together with a map (not necessarily additive nor a derivation) d : R → R such that…”