Extended S-supplement submodules

Yılmaz Durğun

doi:10.3906/mat-1907-107

Abstract

In this paper, we introduced and studied extended S-supplement submodules. A submodule U of a module V is called extended S-supplement submodule in V if there exists a submodule T of V such that V = T + U and U ∩ T is Goldie torsion. Extended S-supplement submodule is a dual notion of extended S-closed submodule. The class of extended S-supplement sequences is a proper class which is generated by nonsingular modules injectively. We studied coinjective objects of this class. Moreover, extended S-supplemented modules are also investigated. We present new characterizations of Z2(RR) -semiperfect rings and SI-rings by extended S-supplement submodules.

Highlights

In what follows, rings are associative with unit elements, and all modules are unitary right modules
The proper class of extended S-closed exact sequences is projectively generated by Goldie torsion modules, i.e. it is the largest proper class for which each Goldie torsion module is projective, [3]
For a proper class P, a module A is called P -coprojective ( P -coinjective) if every short exact sequence ending at A belongs to P, ([9, 11, 13])

Summary

Introduction

Rings are associative with unit elements, and all modules are unitary right modules. Recall from [3], a submodule X of module V is called extended S-closed in V if there exists S ≤ V such that S ∩ X = 0 and V /(S ⊕ X) is nonsingular. We will call a submodule X of module B extended S-supplement if there exists S ≤ B such that B = S + X and S ∩ X is Goldie torsion. Every Goldie torsion submodule of a module is extended S-supplement. Proposition 2.2 An exact sequence E : 0 → X → H → Z → 0 is extended S-supplement if and only if Hom(H, F ) → Hom(X, F ) → 0 for each nonsingular module F .

Objectives

Results

Conclusion