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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]

We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent. [ABSTRACT FROM PUBLISHER]

This article is concerned with the study of atomic dimension defined on the category of left modules over a ring R . A module theoretic as well as torsion theoretic characterization of rings with atomic dimension is provided. The atomic and Gabriel dimensions are compared and necessary and sufficient conditions for these two dimensions to coincide are established. [ABSTRACT FROM AUTHOR]