  Let (R,m,k) be a one-dimensional analytically unramified local ring with
minimal prime ideals P_1,...,P_s. Our ultimate goal is to study the direct-sum
behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by
the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the
structure of this monoid reveals, for example, whether or not every maximal
Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when
uniqueness does not hold, invariants of this monoid give a measure of how badly
this property fails. The key to understanding the monoid C(R) is determining
the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical
result shows that if R/P_1 has infinite Cohen-Macaulay type and the residue
field k is infinite, then there exist |k| pairwise non-isomorphic
indecomposable maximal Cohen-Macaulay R-modules of rank (r_1,...,r_s) provided
r_1 is greater than or equal to r_i for all i. This result allows us to
describe the monoid C(R) when all analytic branches of R have infinite
Cohen-Macaulay.
