Homomorphisms on infinite direct products of groups, rings and monoids

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We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ∏ᵢ 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.