Homomorphisms between rings with infinitesimals and infinitesimal comparisons

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $\,\!^{\bullet}\mathbb{\mathbb{R}}$ are smaller than any infinitesimal hyperreal number of Abraham Robinson's nonstandard extension of the real numbers $\,\!^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in $\,\!^{\bullet}\mathbb{\mathbb{R}}$ and $\,\!^\ast\mathbb{R}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of ``non-nonstandard'' analysis. The existence of a nonzero element in Henle's ring that is mapped to $0$ in $\,\!^{\bullet}\mathbb{\mathbb{R}}$ while it is seen as a nonzero infinitesimal in $\,\!^\ast\mathbb{R}$ suggests that some infinitesimals in $\,\!^\ast\mathbb{R}$ are smaller than the infinitesimals in $\,\!^{\bullet}\mathbb{\mathbb{R}}$. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in $\,\!^{\bullet}\mathbb{\mathbb{R}}$.