Systems of parameters and the Cohen-Macaulay property

Let R be a commutative, Noetherian, local ring and M a finitely generated R-module. Consider the module of homomorphisms HomR(R/a,M/bM) where b ⊆ a are parameter ideals of M. When M = R and R is Cohen- Macaulay, Rees showed that this module of homomorphisms is isomorphic to R/a, and in particular, a free module over R/a of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay R-module M.