Abstract
Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x
1
,..., x
n
) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x
1
,..., x
n
) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds:
1.
f(x
1
,..., x
n
)
2
is central valued on R;
2.
R satisfies s
4
, the standard identity of degree 4.