Results on prime and semi-prime rings with skew and generalized reverse derivations

Abstract

For this study, R represents a semiprime ring or a prime ring, as the case may be. The ring R is said to be semiprime if for any a ∈ R, aRa = {0}, implies a = 0. R is a prime ring if aRb = {0}, implies a = 0 or b = 0, ∀ a, b ∈ R. By assuming that d is a skew derivation with an automorphism β: R→R associated with it, we prove some results on skew-derivations for semi-prime rings. In particular, we show that for a skew-derivation, if d(a)d(b) ± ab = 0, ∀ a, b ∈ R then d = 0. Also, by introducing new differential identities, we establish that a prime ring with a generalized reverse derivation defined on it is commutative.