Abdul Nadim KHAN, Shakir ALI, Nadeem Ahmad DAR

Herstein’s theorem for generalized derivations in rings with involution

Let $R$ be an associative ring. An additive map $F:R\toR$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $F(xy)=F(x)y+xd(y)$ for all $x,y\in R$. In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative. In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.

Keywords:

Prime ring involution, derivation, generalized derivation,

PDF

___

Ali, S. and Dar, N. A. On -centralizing mappings in rings with involution, Georgian Math. J. 21(1), 25–28, 2014.
Bell, H. E. and Rehman, N. Generalized derivations with commutattivity and anticommutativity conditions, Math. J. Okayama Univ. 49, 139–147, 2007.
Brešar, M. On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33, 89–93, 1991.
Daif, M. N. Commutativity results for semiprime rings with derivation, Int. J. Math. and Math. Sci. 21(3), 471–474, 1998.
Dar, N. A. and Ali, S. On -commuting mappings and derivaitons in rings with involution, Turkish J. Math. 40, 884–894, 2016.
Herstein, I. N. Rings with involution (The university of Chicago Press, 1976).
Herstein, I. N. A note on derivations, Canad. Math. Bull. 21, 369–370, 1978.
Hvala, B. Generalized derivations in rings, Comm. Algebra 26(4), 1147–1166, 1998.
Rehman, N. and De Filippis, V. Commutativity and skew-commutativity conditions with generalized derivations, Algebra Colloq. 17, 841-850, 2010.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv