Let R be an associative ring
with center C, and ∂ be a derivation on R. The authors consider the commutativity
of R which satisfies the property that ∂nx − ∂mx ∈ C, where n > m are fixed
nonnegative integers. An example is given to show that if m ≧ 2, R may not be
commutative. For 0 ≦ m ≦ 2, suppose R is either r-torsion free with large r
or torsion free. It is shown that (i) if ∂nx ± x ∈ C for all x ∈ R then all
commutators of R are central; (ii) if ∂nx ± ∂x ∈ C for all x ∈ R and n is
even then ∂x∂y − ∂y∂x ∈ C for all x,y ∈ R; (iii) if ∂nx ± ∂2x ∈ C for all
x ∈ R and if n is odd then ∂2x∂2y − ∂2y∂2x ∈ C for all x,y ∈ R. In all
these cases, if one assumes further that R is prime, then ∂ must be trivial.
Examples are also given to illustrate that some of these assumptions on evenness
of n, and that r being large are essential. Finally, those integral domains
which have ∂nx central for all x are also studied. They are shown to be
commutative.
