We address the problem of when the tensor product of two finitely generated
modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of
Goto et al., and in particular in the original sense from the 80's. As
applications, besides freeness criteria for modules, characterizations of
complete intersections, and an Ulrich-based approach to the long-standing
Berger's conjecture, we show that two celebrated homological conjectures,
namely the Auslander-Reiten and the Huneke-Wiegand problems, are true for the
class of Ulrich modules.Comment: 12 page