Hindman's finite sums theorem and its application to topologizations of algebras

TL;DR

This paper explores Hindman's finite sums theorem and applies it to demonstrate that the Zariski topology on broad classes of algebraic structures, like polyrings, is always non-discrete, using ultrafilter techniques.

Contribution

It introduces a novel application of Hindman's theorem to prove non-discreteness of Zariski topologies on polyrings and related algebraic structures.

Findings

Zariski topology on polyrings is always non-discrete

Infinite polyrings have maps that are closed nowhere dense in the Zariski topology

Multidimensional Hindman's theorem underpins the main topological results

Abstract

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups. The second, main part of the paper is devoted to the topologizability problem of a wide class of algebraic structures called polyrings; this class includes Abelian groups, rings, modules, algebras over a ring, differential rings, and others. We show that the Zariski topology of such an algebra is always non-discrete. Actually, a much stronger fact holds: if $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$. The proof essentially uses a multidimensional version of Hindman's finite sums theorem. The third part of the paper lists several problems concerning topologization of various algebraic structures, their Zariski topologies, and related questions.