Hurwitz series as formal functions

In this paper we continue the study, begun in [7], of properties of the ring HA of Hurwitz series over a commutative ring A with identity. In particular, we show that with its “natural” topology, HA is a complete metric space, and all of the natural mappings are continuous. Further, there is a composition which satisfies the usual properties, i.e., it is continuous, associative, has a two-sided identity, is additive and multiplicative in the left (outer) factor, and, of particular interest from the point of view of differential algebra, satisfies the chain rule. We also show that HA provides formal solutions to homogeneous linear ordinary differential equations, using Picardʹs method of successive approximations. It is noteworthy that these results hold independently of the characteristic of the ring A.