  A finitely generated module $M$ over a commutative Noetherian ring $R$ is
called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M),
\] where $I$ is a proper ideal of $R$. The aim of this paper is to study the
structure of this class of modules. It is discovered that $I$-Cohen Macaulay
modules enjoy many interesting properties which are analogous to those of Cohen
Macaulay modules. Also, various characterizations of $I$-Cohen Macaulay modules
are presented here.
