Abstract
In this paper, there are two main objectives. The first objective is to study the relationship between the density property and some modules in detail, for instance; semisimple and divisible modules. The Addition complement has a good relationship with the density property of the modules as this importance is highlighted by any submodule N of M has an addition complement with Rad(M)=0. The second objective is to clarify the relationship between the density property and the essential submodules with some examples. As an example of this relationship, we studied the torsion-free module and its relationship with the essential submodules in module M.
Highlights
We say N is a dense in M if for all 0≠x, y ∈ M, ∃ r ∈ R ∋xr≠{0} and yr ∈ N
“It is clear that there is a strong relationship between the injective module and the density property, for more information about the injective module” [3]
In the lemma, we introduce the relationship between a semisimple module and a density property
Summary
We say N is a dense in M if for all 0≠x, y ∈ M, ∃ r ∈ R ∋xr≠{0} and yr ∈ N. An R-module M is an injective iff any morphism I→M, where I is an ideal of R, can be extended to a morphism R→M [1]. Recall that the singular submodule Z(M) of a module M is the set of m ∈ M such that mI = 0 for some essential right ideal I of R, or equivalently, rR(m) ≤ess RR. For any module M there is defined a submodule Z(M) which consists of singular elements in M, i.e. elements annihilated by essential right ideals. The focus was on showing basic and important results about the property of density and it’s a relationship with other concepts in module theory
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