Purity relative to a cyclic module

We study relative projectivity and injectivity classes of exact sequences with respect to the classes of cyclic modules. A characterization of cyclic pure exact sequences has given in terms of exactness of a certain sequence of submodules of the modules appearing in the given sequence. We also study the concept of relative divisibility of elements in submodules (known as - purity). We give the generalization of the proposition of Stenstrom. We characterize the preservation of exactness by cyclic modules where be a left ideal of the ring . We also relate the purity in Quasi- projective module [9]. We also try to define Copure for a class of modules and co-relate Copure injective or Copure projective with it. We derive some results dualize certain results of R.B. Warfield.

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Introduction

The notion of purity plays a fundamental role in the theory of abelian groups as well as in module categories. In similar manner module-purity plays an important role in the study of module categories. The aim of the present paper is to study some aspects of purity relative to a fixed cyclic module | for a left ideal . We try to give the characterization of cyclic pure exact sequences has given in terms of exactness of a certain sequence of submodules of the modules appearing in the given sequence. We also study the concept of relative divisibility of elements in submodules (known as RDpurity). We give the generalization of the proposition of Stenstrom [13] . We characterize the preservation of exactness by cyclic modules | where be a left ideal of the ring . We also relate the purity in Quasi-projective module [9] . We also try to define

Copure for a class of modules and corelate Copure injective or Copure projective with it. We derive some results dualize certain results of R.B. Warfield [15] . In this paper R refers to a ring with identity, which need not be necessarily commutative. Also by an module we always mean a left module.

Definition1.1. An exact sequence is said to be M-pure M ↓ 0 → → → → 0 if given : ⟶ , there exists : ⟶ such that . Now we shall consider purity with respect to a fixed cyclic left module | . Where is a left module and is a left ideal of R [6] .

⊆ ⇒ ∃ ′ with 0 and ∈ . We define : | ⟶ by . We want to show it is well defined.

Conversely, given ∈ with ⊆ , define : | ⟶ | by , then is well defined as ∈ ⇒ ∈ . By | purity of the lower sequence,

Definition1.3: Let be a left module. For a two sided ideal . we define ∈ . . 0 . Then is an submodule of . The following theorem is analogue of the corresponding results on purity in abelian groups [12] .

Theorem1.4: Suppose that be a fixed two sided ideal of and is | pure in , where | is a fixed cyclic left module. Then the following conditions are equivalent: is proved by using of 3 3 lemma [6] .

We have called an exact sequence Copure for a class of modules if objects of are injective with respect to the exact sequence 0 ⟶ → → ⟶ 0, Where , , ∈ , and we call a module

Copure injective or Copure projective if it is respectively injective or projective relative to Copure sequences. The following results dualize certain results of R.B. Warfield [15] , and at the same time, they generalize certain results in purity in abelian group in Fuchs [12] to module categories.

Definition1.5: An

module is said to be Concyclic, if it is a submodule of , for some simple module . These are nothing but the subdirectly irreducible modules.

Proposition1.6: Let be a class of left -modules containing the modules where are a representative class of of all simple modules, such that there is a subclass * which is a set with the property that for any ∈ , there is an ∈ * with , then for any module , there is an copure sequence 0 → → → ′ → 0 such that is a direct product of copies of modles in .

Proof: Let Λ be the set of pairs , with ∈ * and ∈ , and for each ∈ Λ denote the corresponding and by and . Let ∏ ∈ and let : → be the product map of the maps . Since ∅ , then ∅ is injective because can be embedded into a direct product of .

Theorem1.7:

A module P is S-copure injective if and only if it is a direct summand of direct product of copies of modules in S.

Notes: Proof of this theorem is similar as in [8] .

Corollary1.8: Let be the class of all cocyclic modules, then for any module there is an copure sequence 0 → → → → 0 such that is the direct product of copies of cocyclic modules in . Notes: Proof of this theorem is similar as in [8] .

Corollary1.9: A left module is cocyclic copure injective if and only if it is a direct summand of a direct product of cocyclic modules. Notes: Proof of this theorem is similar as in [8] . Proposition1.10: A submodule of an -module is pure in if and only if given ∈ and ∈ : , there exists ∈ such that 0 and ∈ . Now we take vary over all left ideals and give the generalization of the following proposition of Stenstrom [13] . , then ⊆ and hence the condition holds. Conversely, given the condition holds and ∈ ,we take , then ⊆ and ⊆ and so, | is Quasi-projective.

Proposition1.15:

If is a two sided ideal, then | is a Quasiprojective module.

Remark1.16:

Recall that a ring is called * -ring if all cyclic modules are quasi-projective (Koehler [16]). We get that a ring * -ring if and only if given any ∈ and any left ideal , there exists ∈ such that ⊆ .

Remark1.17: All commutative and duo rings are * -rings.

Proposition1.18: If is a two sided ideal of a left * -ring , then | is a left * -ring. Proof: Given ∈ and a left ideal ⊆ , there exists ∈ such that ⊆ . Then | ⊆ | and hence the result follows.

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