A note on semiprime rings with derivation

Motoshi Hongan

doi:10.1155/s0161171297000562

Abstract

Let R be a 2‐torsion free semiprime ring, I a nonzero ideal of R, Z the center of R and D : R → R a derivation. If d[x, y] + [x, y] ∈ Z or d[x, y] − [x, y] ∈ Z for all x, y ∈ I, then R is commutative.

Highlights

Throughout, R will represent a ring, Z the center of R, I a nonzero ideal of R, and d R R a derivation
I [ 1, Daif and Bell showed that a semiprime ring R must be commutative if it admits a derivation d such that (i) d[;r.,y] [z,y] for all z, y E R, or (ii) d [z, y] + [x, y] 0 for all x, y e R
As mentioned in 1, our present objective is to prove the following theorem which generalizes [1, Theorem 3]

Summary

Introduction

Throughout, R will represent a ring, Z the center of R, I a nonzero ideal of R, and d R R a derivation. Dedicated to the memory of Professor H. Let R be a 2-torsion free semiprime ring, I a nonzero ideal of R, Z the center of R and d" R R a derivation. KEY WORDS AND PHRASES: Derivation, semiprime ring, 2-torsion free ring.

Results

Conclusion