We study a modified version of the classical Ulrich modules, which we call
c-Ulrich. Unlike the traditional setting, c-Ulrich modules always exist. We
prove that these modules retain many of the essential properties and
applications observed in the literature. Additionally, we reveal their
significance as obstructions to Cohen-Macaulay properties of tensor products.
Leveraging this insight, we show the utility of these modules in testing the
finiteness of homological dimensions across various scenarios.Comment: 22 page