A concept of largeness of monochromatic sums and products in large ideal domain

TL;DR

This paper extends a recent theorem by Hindman and Strauss from natural numbers and infinite fields to large ideal domains, introducing a polynomial extension to the existing results.

Contribution

It generalizes Hindman and Strauss's theorem to large ideal domains and includes a polynomial extension, broadening the scope of the original result.

Findings

The theorem applies to large ideal domains.

The result is extended to polynomial cases.

It unifies previous results with new generalizations.

Abstract

An infinite integral domain $R$ is called a large ideal domain (LID) if every nontrivial ideal of $R$ has finite index in $R$. Recently, N. Hindman and D. Strauss have established a refinement of Moreira's theorem for the set of natural numbers and infinite fields. In this article, we prove the same result of N. Hindman and D. Strauss for large ideal domains (LID) and a polynomial extension.