Classical S-Primary submodules and their decomposition

This paper introduces the concept of classical [Formula: see text]-primary submodules, a generalization of [Formula: see text]-primary submodules where [Formula: see text] is a given multiplicatively closed subset. It provides various examples, characterizations, and relationships with classical primary submodules. Additionally, it investigates the conditions under which a submodule qualifies as classical [Formula: see text]-primary. The paper proves the existence of classical [Formula: see text]-primary decomposition in [Formula: see text]-Noetherian modules, that is, every proper submodule [Formula: see text] of an [Formula: see text]-Noetherian [Formula: see text]-module [Formula: see text] with [Formula: see text] can be expressed as a finite intersection of classical [Formula: see text]-primary submodules. Also, it introduces two versions of uniqueness theorems of classical [Formula: see text]-primary decomposition. Finally, the behavior of the concept has been examined under homomorphisms, in factor modules, Cartesian products of modules, and amalgamated duplication of modules.