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

Abstract

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 $\zeta(x)x^{n} + x^{n}\zeta(x)=0$, $\forall$ $x\in S.$ 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 $f(x)x^* + x^*g(x) \in Z(A)$ for all $x \in A,$ 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.