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We study the structure of the Rudin-Frolik order on countably completeultrafilters under the assumption that this order is directed. This assumption,called the Ultrapower Axiom, holds in all known canonical inner models. Itturns out that assuming the Ultrapower Axiom, much more about the Rudin-Frolikorder can be determined. Our main theorem is that under the Ultrapower Axiom, acountably complete ultrafilter has at most finitely many predecessors in theRudin-Frolik order. In other words, any wellfounded ultrapower (of theuniverse) is the ultrapower of at most finitely many ultrapowers.

The Rudin-Frolik order and the Ultrapower Axiom