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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]