### ON LIE IDEALS AND GENERALIZED DERIVATIONS OF PRIME RINGS

#### Abstract

Let R be a ring and S a nonempty subset of R. An additive mapping

F : R?R is called a generalized derivation on S if there exists a

derivationd: R?R such that F(xy)=F(x)y+xd(y), for all x, y?S.

Suppose that U is a Lie ideal of R with the property that u^2 \in U, for all u \in U. In the present paper, we prove that if Ris a prime ring

with characteristic different from 2 admitting a generalized derivation F

satisfy any one of the properties: (i)F(uv)?uv?Z(R), (ii)F(uv)+uv?Z(R), (iii) F(uv)?vu?Z(R)and(iv) F(uv)+vu?Z(R), for all u, v \in U, then U must be central.

F : R?R is called a generalized derivation on S if there exists a

derivationd: R?R such that F(xy)=F(x)y+xd(y), for all x, y?S.

Suppose that U is a Lie ideal of R with the property that u^2 \in U, for all u \in U. In the present paper, we prove that if Ris a prime ring

with characteristic different from 2 admitting a generalized derivation F

satisfy any one of the properties: (i)F(uv)?uv?Z(R), (ii)F(uv)+uv?Z(R), (iii) F(uv)?vu?Z(R)and(iv) F(uv)+vu?Z(R), for all u, v \in U, then U must be central.

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