Derivations and Leibniz differences on rings

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most $$n+m$$. If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly $$n+m$$. As in the previous paper, the results are proved using Leibniz difference operators.