Systems of Parameters and the Cohen-Macaulay Property

Let $R$ be a commutative, Noetherian, local ring and $M$ an $R$-module. Consider the module of homomorphisms ${\mathrm{Hom}}_R(R\mathrm{/}\mathfrak{a},M\mathrm{/}\mathfrak{b}M)$ where $\mathfrak{b}\subseteq \mathfrak{a}$ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is always isomorphic to $R\mathrm{/}\mathfrak{a}$, and in particular, a free module over $R\mathrm{/}\mathfrak{a}$ of rank one. In this work, we study the structure of such modules of homomorphisms for general $M$.