Homomorphisms on infinite direct products of groups, rings and monoids

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.

26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. Version 2 has minor revisions in wording etc. from version 1

References in corpus (3)

• Homomorphisms on infinite direct product algebras, especially Lie algebras
• Finitely generated infinite simple groups of infinite commutator width
• Linear maps on k^I, and homomorphic images of infinite direct product algebras