Commuting Values of Generalized Derivations on Multilinear Polynomials

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.

Key Words:

• Generalized derivation
• Prime ring

2010 Mathematics Subject Classification:

• Primary 16N60
• Secondary 16W25

Notes

Communicated by V. A. Artamonov.