Some results on ideals of semiprime rings with multiplicative generalized derivations

ABSTRACT

Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) $F\left(\left[u,v\right]\right)=\pm u^m\left[u,v\right]u^n,$ ii) $F\left(u\circ v\right)=\pm u^m(u\circ v)u^n,$ iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.

KEYWORDS:

• Centralizing mapping
• commuting mapping
• ideal
• multiplicative generalized derivation
• SCP map
• semiprime ring

2010 Mathematics Subject Classification:

• 13A99
• 16A12
• 16A70
• 16A72