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Throughout this work, R will represent an associative ring and Z will denote the center of R. We shall write [r, y] for ry–yr. A mapping D (.,.): R x R–R is called symmetric if D (x, y)= D (y, x) holds for all r, ye R. A mappıng d-R–R defined by d (a)= D (a, r) ıs called trace of D (.,.), where D (.,.): R x R–R

Is a symmetric mapping It is obvious that, if D (,.): R x R–R is a symmetric mapping which is also bi-additive (ie, additive ın both arguments), then trace of D (.,.) satisfies the relation d (x+ y)= d (x)+ d (y)+ 2D (t, y) for all x, ye R.