ABSTRACT
Let (Γ,a) ∈ Qdec(R,A) × A for some complete lattice A. Let M be a left R-module and let M′ be a submodule of M. We say that a set {Ni | i ∈ Ω} of submodules of M gives a (Γ,a)-decomposition of M′ in M if and only if the following conditions are satisfied:
M′ = ∩i∈Ω Ni and M′ ≠ ∩i∈Λ Ni for any proper subset Λ of Ω.
M/Ni is (Γ,a)-coirreducible for each i ∈ Ω.
Γ(M/Ni) ≠ Γ(M/Nj) for all i ≠ j ∈ Ω.
Γ(M/M′) = Γ(Ni/M′) ∪ Γ(M/Ni) for each i ∈ Ω.
Γ(0) = Γ(Ni/M′) ∩ Γ(M/Ni) for each i ∈ Ω.
Γ(M/M′) = ∪i∈Ω Γ(M/Ni).
