Commutativity results with derivations on semiprime rings

Abstract

In this paper, let R be a 2-torsion free semiprime ring and U a non-zero ideal of R, d a derivation mapping. If R admitting

A derivation d satisfies one of the following .

(i)[d₍x₎,d₍y₎]=[x,y] for all x, y in U.

(ii)[d₍x)²,d₍y)²]=[x ,y] for all x, y  in  U.

(iii) [d₍x₎ ,d₍y₎]=[x²,y² ] for all x, y in U.

(iv)[d₍x)²,d₍y)²]=[x²,y²] for all x, y in U.

A non – zero derivation d satisfies one of the following:

(i)d([d₍x₎,d₍y₎ ])=[x,y] for all x, y in U.

(ii)d([d₍x₎,d₍y₎]=[d₍x₎,d₍y₎] for all x, y in U. Then R contains a non-zero central ideal.