Prime Rings with Generalized Derivations on Right Ideals


ALGEBRA COLLOQUIUM, vol.18, pp.987-998, 2011 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18
  • Publication Date: 2011
  • Doi Number: 10.1142/s1005386711000861
  • Title of Journal : ALGEBRA COLLOQUIUM
  • Page Numbers: pp.987-998


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), ... , X(n)) a multilinear polynomial over K. If g(f(x(1,) ... , x(n) ))f(x(1), ... , x(n)) is an element of C for all x(1), ... , x(n) is an element of I, then either f(x1, ... , ) x(n) (+ 1) is an identity for I, or char(R) = 2 and R satisfies the standard identity s(4)(x(1), ... , x(4)), unless when g(x) = ax [x, b] for suitable a, b is an element of U and one of the following holds: (i) a, b is an element of C and f, x)2 is central valued on R; (ii) a is an element of C and f(x(1), ... , x(n)) is central valued on R; (iii) aI = 0 and [f(x(1) , ... , x(n)), x(n + 1)]x(n +) (2) is an identity for I; (iv) aI = 0 and (b - beta)I = 0 for some beta is an element of C.