On ideals of prime rings involving n-skew commuting additive mappings with applications

Let n > 1 be a fixed positive integer and S be a subset of a ring R. A mapping zeta of a ring R into itself is called n-skew-commuting on S if ((x)x(n) + x(n )zeta(x) = 0, for all x is an element of S. The main aim of this paper is to describe n-skew-commuting mappings on appropriate subsets of R. With this, many known results can be either generalized or deduced. In particular, this solves the conjecture in [M. Nadeem, M. Aslam and M.A. Javed, On 2-skew commuting additive mappings of prime rings, Gen. Math. Notes, 2015]. The second main result of this paper is concerned with a pair of linear mappings of C*-algebras. We show that here, if C*-Algebra admits a pair of linear mappings f and g such that f(x)x*+x*g(x) is an element of Z(A) for all x is an element of A, then both f and g must be zero. As the applications of first main result (Theorem 2.1) and apart from proving some other results, we characterize the linear mappings on primitive C*-algebras. Furthermore, we provide an example to show that the assumed restrictions cannot be relaxed.