On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Alveera Mehdi,Firdhousi Begum,Fahad Sikander

doi:10.1155/2013/873193

Abstract

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

Highlights

All the rings R considered here are associative with unity, and right modules M are unital QTAG modules
We investigate the submodules N ⊆ M such that M/K is h-divisible for every h-pure submodule K ⊆ M, containing K
We find that the Ulm-Kaplansky invariants are same for all h-pure envelopes

Summary

Introduction

All the rings R considered here are associative with unity, and right modules M are unital QTAG modules. A submodule N of M is quasi-h-dense in M, if for every h-pure submodule K ⊆ M, containing N, M/K is h-divisible. There exists a h-pure submodule Q of M such that Soc(Q) = S2. A submodule N of a QTAG module M is quasi-h-dense in M if and only if for all integers k ≥ 0, N + Hk+1(M) ⊇ Soc (Hk(M)). By Lemma 2, there exists a proper submodule K ⊆ M containing N and a bounded submodule Q of M such that M/K ≅ Q This contradiction proves that N + Hk+1(M) ⊇ Soc(Hk(M)), for all integers k ≥ 0. (ii) N + Hk+1(M) ⊇ Soc (Hk(M)), ∀k > ω; (iii) for every h-pure submodule K containing N, M/K is h-divisible.

KK K

This implies that