Prime rings with generalized derivations on right ideals

Let K be a commutative ring with unit, R be a prime K-algebra with center Z(R), right Utumi quotient ring U and extended centroid C, and I a nonzero right ideal of R. Let g be a nonzero generalized derivation of R and f(X 1,?,Xn) a multilinear polynomial over K. If g(f(x1,?,xn)) f(x1,?,x n) ? C for all x1,?,xn ? I, then either f(x1,?,xn)xn+1 is an identity for I, or char(R)=2 and R satisfies the standard identity s 4(x1,?,x4), unless when g(x)=ax+[x,b] for suitable a, b ? U and one of the following holds: (i) a, b ? C and f(x1,?,xn)2 is central valued on R; (ii) a ? C and f(x1,?,xn) is central valued on R; (iii) aI=0 and [f(x1,?,xn), x n+1]xn+2 is an identity for I; (iv) aI=0 and (b-ß)I=0 for some ß ? C. © 2011 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University