Using the concept of prime submodule introduced by Raggi et. al. we extend the notion of
reduced rank to the module-theoretic context. We study the quotient category of $\sigma [M] …

For M∈ R-Mod and τ a hereditary torsion theory on the category σ [M] we use the concept of
prime and semiprime module defined by Raggi et al. to introduce the concept of τ-pure …

Let M be a right R-module,\cal M the class of all M-small modules, and P a projective cover
of M in σ M. We consider the torsion theories\cal M=(\cal T _\cal M,\cal F _\cal M),\tau_V=(\cal …

An R-module M is said to be an extending module if every closed submodule of M is a direct
summand. In this paper we introduce and investigate the concept of a type 2 τ-extending …

Relative to a hereditary torsion theory $ tau $ we introduce a dimension for a module $ M $,
called {em $ tau $-rank of} $ M $, which coincides with the reduced rank of $ M $ whenever …

Let $ R $ be an arbitrary ring with identity and $ M $ a right $ R $-module with $ S= $ End $
_R (M) $. Let $ Z_2 (M) $ be the second singular submodule of $ M $. In this paper, we …

Let τ be a hereditary torsion theory. The purpose of this paper is to extend results about
singular (resp. nonsingular) modules to τ-singular (resp. τ-nonsigular) modules. An R …

Let τ be a hereditary torsion theory on the category R-mod of left modules over an
associative unitary ring R. We will study the τ-injective hull of certain modules, giving an …

We consider the concept of prime submodule defined by Raggi et al.. We find equivalent
conditions for a module M progenerator in σ [M], with τ M-Gabriel dimension, to have a one …

With the notion of prime submodule defined by F. Raggi et. al. we prove that the intersection
of all prime submodules of a Goldie module $ M $, is a nilpotent submodule provided that …