Rings in which power values of K-Engels with derivations annihilate a certain element

TL;DR

This paper investigates conditions under which power values of derivations in a 2-torsion free semiprime ring lead to a decomposition into a commutative part and a part where the derivation acts trivially, revealing structural properties of such rings.

Contribution

It establishes a new criterion involving power values of derivations that results in a ring decomposition with a commutative component and a trivial derivation component.

Findings

Existence of an idempotent e in B such that eA is commutative

Derivation d induces zero derivation on (1-e)A

Structural decomposition of the ring based on derivation properties

Abstract

Let R be a 2 torsion free semiprime ring and d a nonzero derivation. Further let A = O(R) be the orthogonal completion of R and B = B(C) the Boolean ring of C where C be the extended centroid of R. We show that if a[[d(x),x]^n- [y, d(y)]^m]^t = 0 such that a in R for all x, y in R, where m, n, t > 0 are fixed integers, then there exists an idempotent e in B such that eA is a commutative ring and d induce a zero derivation on (1-e)A.