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

Authors : Cihat ABDİOĞLU, Shakir ALI, Mohammad Salahuddin KHAN

Pages : 1237-1247

Doi:10.15672/hujms.776236

View : 20 | Download : 6

Publication Date : 2022-10-01

Article Type : Research Paper

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 $\zetainsert ignore into journalissuearticles values(x);x^{n} + x^{n}\zetainsert ignore into journalissuearticles values(x);=0$, $\forall$ $x\in S.$ 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 $finsert ignore into journalissuearticles values(x);x^* + x^*ginsert ignore into journalissuearticles values(x); \in Zinsert ignore into journalissuearticles values(A);$ for all $x \in A,$ then both $f$ and $g$ must be zero. As the applications of first main result insert ignore into journalissuearticles values(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.
Keywords : ideal, prime ring, C algebra, commuting mapping, n skew commuting mapping