Approximately Cohen-Macaulay modules

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is a variety of nice results about approximately Cohen-Macaulay rings. These results were done by Goto. In this paper we prove some these results for modules and generalize the concept of approximately Cohen-Macaulay rings to approximately Cohen-Macaulay modules. It is seen that when $M$ is an approximately Cohen-Macaulay module, for any proper ideal $I$ we have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially when $M$ is $R$ itself, we obtain an interval for $grade(I,R)$. We also give a definition for these modules in case that $R$ is not necessarily local and show that approximately Cohen-Macaulay modules are in close relationship with perfect modules. Finally we consider the behaviour of these modules under faithful flat extensions.