Posner's second theorem and an annihilator condition.
Posner's second theorem and annihilator conditions with generalized skew derivations
Let be a prime ring of characteristic different from 2, be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there exist and...
Prime and semiprime rings with symmetric skew n-derivations
Let n ≥ 3 be a positive integer. We study symmetric skew n-derivations of prime and semiprime rings and prove that under some certain conditions a prime ring with a nonzero symmetric skew n-derivation has to be commutative.
Prime rings with hypercommuting derivations on a Lie ideal
Products of derivations which act as Lie derivations on commutators of right ideals.