Hypercommuting Values In Associative Rings With Unity

Let K be a commutative ring with unity, R an associative K-algebra of characteristic different from 2 with unity element and no nonzero nil right ideal, and f(x(1,) . . . , x(n)) a multilinear polynomial over K. Assume that, for all x is an element of R and for all r(1), . . . , r(n) is an element of R there exist integers m = m(x, r(1), . . . , r(n)) >= 1 and k = k(x, r(1), . . . , r(n)) >= 1 such that [x(m), f(r(1), . . . , r(n))](k) = 0. We prove that: (1) if char(R) = 0 then f (x(1), . . . , x(n)) is central-valued on R; and (2) if char(R) = p > 2 and f (x(1), . . . , x(n)) is not a polynomial identity in p x p matrices of characteristic p, then R satisfies s(n+2)(x(1), . . . , x(n+2)) and for any r(1), . . . , r(n) is an element of R there exists t = t(r(1), . . . , r(n)) >= 1 such that f(pt)(r(1), . . . , r(n)) is an element of Z(R), the center of R.