On commutativity of rings involving certain polynomial constraints

Let m greater than or equal to 0 and n > 1 be fixed integers. Let R be a ring with unity 1 satisfying the condition that, for every y in R, there exist polynomials f(x) epsilon X(2)Z[X] and g(X), h(X) epsilon Z[X] depending on y such that x(m)[x(n),y] = g(y)[x, f(y)]h(y) for all a in R. The main result of the present paper asserts that R is commutative if R has the property Q(n), i.e., for all x,y in R, n[a,y] = 9 implies [x,y] = 0.