UC Berkeley

We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ΠI Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → 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.