On nilpotent homoderivations in prime rings

Abstract

Let R be an associative ring and let $s\ge 1$ be a fixed integer. An additive map h on R is called a homoderivation if $h(xy)=h(x)h(y)+h(x)y+xh(y)$ holds for all $x,y\in R.$ In 1978, 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 main objective of this paper is to prove the above mentioned result for homoderivations with nilpotency ‘s’ in prime rings. Moreover, we prove that if a prime ring admits homoderivations h1 and h2 such that $h_1°h_2=h_2°h_1$ and $h_1^{s_1}°h_2^{s_2}=0$ for positive integers s1 and s2, then at least one of the homoderivations must be nilpotent.

KEYWORDS:

• Homoderivation
• iterates of homoderivations
• Leibniz formula
• nilpotent homoderivation
• prime and semi-prime ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16N60
• Secondary: 16R60
• 16W10

Acknowledgments

The authors wish sincere thank to the anonymous referee(s) for their carefully reading the manuscript.

Additional information

Funding

The research of second named author is supported by SERB-DST MATRICS Project (grant no. MTR/2019/000603), India.