On products of ultrafilters

Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter; that the two ultrafilters commute in the tensor product; that for all cardinals $\lambda$, one of the ultrafilters is both $\lambda$-indecomposable and $\lambda^+$-indecomposable; that the ultrapower embedding associated to each ultrafilter restricts to a definable embedding of the ultrapower of the universe associated to the other.