# Homomorphisms Between Rings with Infinitesimals and Infinitesimal Comparisons

## TL;DR

This paper compares different types of infinitesimals in various mathematical frameworks, showing that some hyperreal infinitesimals are smaller than nilpotent ones, revealing subtle differences in their structures.

## Contribution

It introduces a study of order-preserving homomorphisms between rings with infinitesimals, highlighting the differences and relationships among nilpotent and hyperreal infinitesimals.

## Key findings

- Existence of elements too small for Giordano's ring but nonzero in Henle's extension.
- Hyperreal infinitesimals can be smaller than nilpotent infinitesimals.
- Contradiction with Reeder's argument is due to nilpotent elements in Giordano's ring.

## Abstract

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of "non-nonstandard" analysis. In particular, we will show that there exists a nonzero element in Henle's ring that is "too small" to be registered as nonzero in Paolo Giordano's ring, while it is seen as a nonzero infinitesimal in ${^\ast\mathbb{R}}$. This result suggests that some hyperreal infinitesimals are smaller than the nilpotent infinitesimals. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ${^{\bullet}\mathbb{R}}$.