Lie Ideals and Symmetric Bi-Derivations of Prime Rings

Abstract

Throughout this work, R will be an associative ring and Z(R) the center of R, We shall write (x,y)=xy+yx, [x,y]=xy−yx for all x,y∈R. A mapping B(...):RxR→R is said to be symmetric if B(x,y)=B(y,x) holds for all pairs x,y∈R. A mapping f: R→R denoted by f (x)=B(x,x) is called the trace of B, where B(...): R×R→R is a symmetric mapping. It is obvious that, in case B(...): R×R→R is a symmetric mapping which is also bi-additive (i.e. additive in both arguments), the trace of B satisfies the relation f(x+y)=f(x)+f(y)+2B(x,y) for all x,y∈R. We shall use also the fact that the trace of a symmetric bi-additive mapping is an even function. D(...): R×R→R is called a symmetric bi-derivation if D(xy,z)= D(x,z)y+xD(y,z) is fulfilled for all x,y,z∈R. Of course the relation D(x,yz)=D(x,y)z+yD(x,z) is also fulfilled for all x,y,z∈R.