Abstract
Let A be a prime ring with char(A) not 2, and with extended centroid C and Q be the maximal right quotient ring of A. Let F be a subring with 1 of the field C. Let f be a nonzero multilinear polynomial of F<X>. Let S be a ring (or, F-algebra) and let a be an additive homomorphism of groups (or, F-modules respectively) from S to Q, and c belong to C, such that a(f(s1,s2,...,sm))=cf(a(s1),a(s2),...,a(sm)). Let R=Im(a). If R is a 2m-free subset of Q, then there exists a homomorphism or an anti-homomorphism of rings (or, F-algebras) b from S to RC+C, an additive map v from S to C, and t belong to C, such that a(s)=tb(s)+v(s), and ct^(m-1)=1 or -1. Moreover, if R is a left ideal of A and deg(R)>2m+2, then we also have the same conclusion.