A note on power values of derivation in prime and semiprime rings

TL;DR

This paper investigates derivations in prime and semiprime rings satisfying a power condition, showing that such derivations are trivial or central, with implications for the structure of these rings.

Contribution

It establishes that derivations satisfying a power identity are either zero or induce central mappings in prime and semiprime rings, extending understanding of ring derivations.

Findings

In prime rings, derivation is zero or the ring is commutative.

In semiprime rings, derivation maps into the center.

Existence of an idempotent e such that eA is commutative and derivation is zero on (1-e)A.

Abstract

Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fixed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to its center. Moreover, in semiprime case let A = O(R) be the orthogonal completion of R and B = B(C) be the Boolian ring of C, where C is the extended centroid of R, then there exists an idempotent e in B such that eA is commutative ring and d induce a zero derivation on (1-e)A.