Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a sub- hypermodule of M. We introduce the concept of an essential sub- hypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts.
