Herstein’s theorem for generalized derivations in rings with involution

Year 2017, Volume: 46 Issue: 6, 1029 - 1034, 01.12.2017

Shakir Ali Abdul Nadim Khan , Nadeem Ahmad Dar

Abstract

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

References

• 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.