ON SEMIDERIVATIONS OF PRIME RINGS
Mohammad Ashraf, Nadeem ur Rehman · Demonstratio Mathematica · 2004
A semiderivation of a ring R is an additive mapping / : R -• R together with a function g : R -• R such that f(xy) = f{x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)), for all x,y G R. If / is a non-zero semiderivation of a prime ring R, then it is well known that g must necessarily be an endomorphism.Let R be a prime ring with center Z(R), f a non-zero semiderivation with associated endomorphism g which is one-one & onto, and a, r be two automorphisms of R such that fa = erf, fr = rf, go = ag, gr = rg.Suppose that U is a non-zero (cr, r)-Lie ideal of R and C(R)a,T = {c G R \ ca(x) = T(X)C, for all x € R}.In the present paper it is shown that (i) if char R 2 and f(U) C C{R)a,T, then R is commutative or U C C{R)A,T (ii) if char R / 2 and f 2 (U) = 0, then U C Z{R) (iii) if char R^ 2 and f(U) C Z(R), then U C Z{R) (iv) if char R^ 2,3 and f{U) C U, f 2 (U) C Z(R), then U C Z(R).1991