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

TL;DR

The paper extends Hindman–Strauss refinements of Moreira’s theorem to large ideal domains by developing a polynomial framework for monochromatic sums and products. It proves: (i) a polynomial van der Waerden-style refinement for $IP$-sets over infinite rings, ensuring color-homogeneous polynomial configurations appear on an $IP^{\star}$-set; and (ii) a Moreira-type theorem for LIDs that yields piecewise syndetic monochromatic polynomial configurations formed from products and sums. The results rely on the Polynomial Hales–Jewett theorem, Stone–Čech compactifications, and a careful handling of piecewise syndeticity under dilation and unique finite products. These findings generalize known natural-number and field results to a broad class of integral domains, with potential implications for combinatorial number theory in algebraic settings.

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.

Theorems & Definitions (29)

• Theorem 1.2: Moreira
• Definition 1.3: Piecewise syndetic
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2: Hales--Jewett Theorem
• Theorem 2.3: Polynomial Hales--Jewett Theorem
• proof : Proof of Theorem \ref{['pvw ip in LID']}