Herstein’s theorem for prime ideals in rings with involution involving pair of derivations

Abstract

Herstein proved that a prime ring R of $\mathit{char}(R)\ne 2$ is commutative if there is a nonzero derivation d of R such that $[d(x),d(y)]=0$ for all $x,y\in R.$ The aim of this paper is to prove the $*$-version of Herstein’s result with a pair of derivations on prime ideals of a ring with involution. Precisely, we prove the following result: let R be a ring with involution $*$ of the second kind, P a prime ideal of R such that $S(R)\cap Z(R)\cancel{\subseteq }P$ and $\mathit{char}(R/P)\ne 2.$ If d₁ and d₂ are derivations of R satisfying the condition $[d_1(x),d_2(x^{*})]\in P$ for all $x\in R,$ then one of the following holds: (a) $d_1(R)\subseteq P$

(b) $d_2(R)\subseteq P$

(c) R/P is a commutative integral domain.

Moreover, some related results are also discussed. As consequences of our main theorems, many known results can be either generalized or deduced.

Keywords:

• Associative ring
• derivation
• involution
• prime ideal

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W10
• 16W25

Acknowledgments

The authors are deeply indebted to the learned referees for their careful reading of the manuscript and constructive comments.

Additional information

Funding

This research is supported by SERB-DST MATRICS Project (Grant No. MTR/2019/000603), India.