Quadratic Central Differential Identities on a Multilinear Polynomial

Abstract

Let K be a commutative ring with unity, R a prime K-algebra, Z(R) the center of R, d and δ nonzero derivations of R, and f(x 1,…, x n ) a multilinear polynomial over K. If [d(f(r 1,…, r n )), δ (f(r 1,…, r n ))] ∈ Z(R), for all r 1,…, r n ∈ R, then either f(x 1,…, x n ) is central valued on R or {d, δ} are linearly dependent over C, the extended centroid of R, except when char(R) = 2 and dim C RC = 4.