Noetherian, Artinian Regular Modules and Injective Property

this article we provide that several relationships between some concepts and injective module. We investigate, if 𝑀 is a cyclic and regular module is injective


2-The Main Result
In this section, we present some new results about the relationship between Noetherian, Artinian and regular rings and injective module.We should start with the following definitions.Definition (2.1).[1]."Any module  is called regular if Ɐ ∊ , ∃ g∊(,R), then(mg)m=m".
More details of regular property can find it in [2].
In the beginning, we need a basic preliminary in order to proceed from it towards then main objective of the current paper.See the following lemma: Lemma (2.2).Every Noetherian or Artinian Regular module over abelian ring R with unity is injective module.

Proof.
Any module  fulfills all the conditions in theory, this means  is a finite direct sum of projective module have only two submodule are {0} and .Since R is a commutative ring with identity element, so  is a flat module iff it is injective and a finite direct sum of injective is also injective.(see[8]).

Definition (2.3
).An R-module  is called projective if and only if for any ℎ :C→V such that C,V are any R-modules and for any homomorphism g:→V ∃ a homomorphism h:→C such that f∘h=g [9].

Remark (2.4).
A ring R is called finite-dimensional if R have no infinite direct sums of ideals.[1] Theorem (2.5).Let R be a (QF) and perfect ring or finite-dimensional ring.If  is a regular R-module, then  is injective.

Proof.
Let R is a perfect ring.Let T=direct limits of projective-module.Then T projective (see [5]).But  is a direct limit of   ∋   f. submodules.So  is a projective.But from [4], "for a QF-ring.every projective module is an injective module".Now if R have no infinite direct sums of ideals.Let  be a regular and Г={ ∑ ⨁ Rm α : m α ∊} is a partially ordered and Hence  ⊆  and then =.Therefor  is a projective.Thus it is injective module [4].

Lemma (2.6)
. "Every f-generated regular R-module is a projective".[1] Recall that for all   ∊ , ∊ ∋  is a some of several generators of .So we can present a definition of finitely generated module  by the following way: Example (2.8).Any f-dimensional vector space is a f-generated over a field K.

Proposition (2.9).
Let  be a regular R-module.If ≅    , then  is injective module.

Proof.
Since  is regular R-module then it is f-generated and by (Lemma 2.6),  is injective.

Corollary (2.11).
Let  1 and  2 be an R-modules over (QF)-ring and let :  1 → 2 be an onto homomorphism such that  2 is a regular and  1 is f-generated, so  2 is injective module.

Proof.
Suppose that  1 and  2 are two modules over the ring R. Also suppose  1 is f-generated module.To prove that  2 is injective.
Theorem (2.12).Let  be regular module over P.I.D.If  is acyclic module, then it is a Noetherian and is injective module.

Proof.
Since  is a cyclic, then it is a f-generated.So  have a generators m 1 ,…...m k.Hence ∃ :   →  defined by

Corollary (2.13).
Let  be a regular R-module.If  and   are Noetherian ∋ N is a submodule of , so  is injective module.