A NOTE ON CERTAIN SUBRINGS AND IDEALS OF PRIME-RINGS

Let R be a noncommutative prime ring with symmetric ring of quotients Q, with extended centroid C and with derivation D, let n be a positive integer. Given x, y is an element of R, we set [y,x](1) = [y,x] = yx - xy, [y,x](k+1) = [[y,x]k,x], k = 1,2,.... Suppose that D not equal a d(a) for any a is an element of Q such that (a + c)(2) = 0 for some c is an element of C, and either char(R) greater than or equal to n + 1 greater than or equal to 3, or char(R) = 0. We show that the subring o f R generated by {[x(D),x](n-1)/x is an element of R}, contains a nonz ero ideal of R.