Existence and uniqueness of S -primary decomposition in S -Noetherian modules

Let R be a commutative ring with identity, S⊆R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N:RM)∩S=∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules.