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Using the concept of prime submodule forM ∈ R-Mod,P ∈ Spec(M), andN ∈ σ[M], we define whenNisP-Mtame\ (Mtame) module. This concept generalizes the concept\ ofP-tame(tame) modules. ForM ∈ R-Mod and τ ∈M-tors, we use the concept of τM-Gabriel dimension and we study the relationship betweenMtamemodules and τM-Gabriel dimension. We find equivalent conditions for a moduleMprogenerator in σ[M] with τM-Gabriel dimension to have τM-Gabriel correspondence in terms of theP-Mtamemodules. This result extends the results by Albu et al. and Kim and Krause. [ABSTRACT FROM AUTHOR]

For M ∈ R-Moit anil τ ∈ M-tors, we define the concept of fully τ-hounded module as a generalization of the concept of fully x-hounded ring. We prove that for a x-noetherian module M with local τM-Gahrie! correspondence, which is a progenerator of σ[M] and with tau; is FIS-invariant, then M is fully x-hounded. Also, we show that if M is τ-noetherian and fully τ-hounded, then M has local τM-Gabriel correspondence. [ABSTRACT FROM AUTHOR]