Throughout this note R will denote a discrete valuation ring with unique prime ideal (p) ==Rp. Q will denote the quotient field of R and R* will denote the completion of R. For basic definitions related to R-modules see [1], [2], or [5]. By rank, we shall mean torsion-free rank. If M is an R-module and xEM, UM(x) will denote the Ulm sequence (ao, al, * , an, * * * ) where, for each n, a,, is the height of pnx. Two such sequences (ao, a,, * * ) and (3o, /31, * * * ) are said to be equivalent if there is an m and an n such that an+i =1m+i for all nonnegative integers i. If M has rank one, then UM(x) and UM(y) are equivalent whenever x and y are elements of M having zero order ideal. Consequently, if M has rank one, we associate with M an equivalence class U(M) of Ulm sequences. Mt will denote the torsion submodule of M. This note is devoted to establishing the following