  Let R be a Cohen-Macaulay local ring. In this paper we study the structure of
Ulrich $R$-modules mainly in the case where R has minimal multiplicity. We
explore generation of Ulrich R-modules, and clarify when the Ulrich R-modules
are precisely the syzygies of maximal Cohen-Macaulay R-modules. We also
investigate the structure of Ulrich R-modules as an exact category.
