Ideal stable modules

Let R be a commutative ring with identity. We will say that an R-module M is ideal stable, if IM = M, where I is an ideal of R, implies that Ix = Rx for all x 2 M. In this paper, we will study ideal stable modules. Among other results, it is proved that if R is an Artinian ring, then every R-module is ideal stable and the converse is true if R is Noetherian.