Homomorphisms on infinite direct products of groups, rings and monoids

TL;DR

This paper investigates conditions under which homomorphisms from infinite direct products of algebraic structures factor through finitely many ultraproducts, revealing structural properties and limitations of such mappings.

Contribution

It introduces criteria for when homomorphisms from infinite products factor through finitely many ultraproducts and examines the nature of ultrafilters involved.

Findings

Homomorphisms factor through finitely many ultraproducts under certain conditions.

Ultrafilters involved in factorizations are often principal.

The paper identifies open questions and directions for future research.

Abstract

We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions, and topics for further investigation, are noted.