PS-hollow representations of modules over commutative rings

Let [Formula: see text] be a commutative ring and [Formula: see text] a nonzero [Formula: see text]-module. We introduce the class of pseudo-strongly[Formula: see text]PS[Formula: see text]-hollow submodules of [Formula: see text]. Inspired by the theory of modules with secondary representations, we investigate modules which can be written as finite sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of minimal PS-hollow strongly representations of modules over Artinian rings.