Homomorphisms on infinite direct products of groups, rings and monoids

George M. Bergman

Vol. 274 (2015), No. 2, 451–495

DOI: 10.2140/pjm.2015.274.451

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 $B\to B∕Z\left(B\right)$ 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.

Keywords

homomorphism on an infinite direct product, ultraproduct, slender group, algebraically compact group, cotorsion abelian group

Mathematical Subject Classification 2010

Primary: 03C20, 08B25, 20A15, 20K25, 20M15, 17A01

Secondary: 20K40, 22B05, 16B70, 16P60

Milestones

Received: 7 June 2014

Revised: 2 October 2014

Accepted: 2 October 2014

Published: 1 April 2015