Submodules Satisfying the Uniformly Classical S ‐Primary Property

We define uniformly classical S ‐primary submodules, where S is a multiplicatively closed subset. A submodule W of an H ‐module E with ( W : H E )∩ S = ∅ is said to be a uniformly classical S ‐primary submodule if ∃ s ∈ S and such that whenever η γ ν ∈ W for η , γ ∈ H , ν ∈ E , then s η ν ∈ W or ( s γ ) k ν ∈ W . We investigate many properties of this new type of submodules and give relations with the other submodules. We provide various characterizations of this class of submodules in terms of other submodules and ideals. Moreover, we study the notion under homomorphisms, in factor modules, Cartesian product, localization, idealization, and amalgamation modules along an ideal with respect to a homomorphism.