The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

It has been observed by different authors that QTAG-modules behave very much like torsion abelian groups. In this paper, in section 3, we characterize quasi-essential submodules (Theorem 3.9) and further find a characterization for an h-pure submodule to be a direct summand (Theorem 3.11). In section 4, we obtained a necessary and sufficient condition for a submodule to be contained in a minima...

we first characterize $tau$-complemented modules with relative (pre)-covers. we also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

An R-module A is called GF-regular if, for each a ∈ A and r ∈ R, there exist t ∈ R and a positive integer n such that r(n)tr(n)a = r(n)a. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by M GF(A). Furthermore, the radical properties of A are investigated; we proved that if A is an R-module and K is a submodule of A, then MGF(K) = K∩M GF(A...

We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called φ-derivations) are shown to satisfy the commutation relations. Our main result states that for every so-called rig...

The Hardy space on the unit ball in C provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n = 1 the submodule has finite codimension. In this note we show that this phenomenon can only occur for modules over domains in C and for finitely-connected domains only for Hardy-like spaces, the bundle shifts. M...

The notions of pure ideals and pure submodules play an important role in the study of rings and their module categories. In this paper, we introduce right t-pure fuzzy ideals and pure fuzzy submodules of an R-module. We establish their basic properties, and use them to obtain fuzzy theoretic characterizations of right weakly regular rings. Among other results, we prove that a ring R is right we...