Tensor purities, cyclic quasi-projectives and cocyolic copurity

Choudhury, Diptaishik; Tewari, K.

doi:10.1080/00927877908822418

Cited by 9 publications

(5 citation statements)

References 7 publications

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“…(The idea of the proof (iii)=~(i) is due to [5].) By the above proposition and by Theorem 3.5 of [5] it follows that for a ring R the following are equivalent.…”

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confidence: 84%

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Copure submodules

Hiremath

1984

Acta Math Hung

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44(1--2) (1984), 3--12.P. M. Cohn [6] has defined the notion of a pure submodule. Subsequently C. P. Walker [21], B. StenstrSm [19] and D. P. Choudhury and K. Tewari [5] have introduced notions of copurity as dual to the notion of purity. We find that another notion of copurity for the category of modules using cofinitely related modules is interesting.A submodule A of a right R-module B is said to be copure if for every cofinitely related right R-module M, every R-homomorphism from A into M has an extension to B. Over a commutative (co-)noetherian ring R, every pure submodule of an Rmodule is also copure. If R is a commutative semi-local ring then every copure submodule of an R-module is pure. Hence over a commutative semi-local co-noetherian (equivalently noetherian) ring R purity and copurity are equivalent. We also prove this result for a Dedekind domain. In general neither every pure submodule of a right R-module is copure nor every copure submodute of a right R-module is pure. We compare our copurity with three copurities mentioned above. We generalize a few results for pure submodules over commutative (co-)noetherian rings to copure submodules.Throughout this paper by a ring R we mean .an associative ring with identity and by an R-module we mean a unitary right R-module. mod-R stands for the category of all right R-modules and all R-homomorphisms. D~IFINITIONS 1. (i) An R-module M is said to be finitely embedded [20] (or cofinitely generated [11] (ii) An R-module M is said to be cocyclic [10] ifM is isomorphic to a submodule of E(S) for simple R-module S.(iii) An R-module M is said to be cofree [10] if M is isomorphic to 1-I{E(S,): S~ is a simple R-module, a(I} where I is some index set.(iv) An R-module M is said to be cofinitely related [10] if there is an exact sequence O~M~N--K--O of R-modules with N cofinitely generated, cofree and K cofinitely generated.(v) A ring R is said to be right co-noetherian [11] if every factor module of a cofinitely generated R-module is cofinitely generated.The notion of a pure submodule was first defined by P. M. Cohn

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“…(The idea of the proof (iii)=~(i) is due to [5].) By the above proposition and by Theorem 3.5 of [5] it follows that for a ring R the following are equivalent.…”

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confidence: 84%

“…(iii) (D. P. Choudhury and K. Tewari [5].) (a) An R-module M is said to be cocyclically copresented if both E(M) and E(M)/M are cocyclic.…”

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confidence: 98%

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Copure submodules

Hiremath

1984

Acta Math Hung

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“…Pandey [1]. In this there is an attempt to relativize the usual Cohn [4] purity with respect to a torsion theory We also develope the theory of (M, σ)purity and (μ, σ)purity relative to a torsion theory with radical σ which is weaker than the usual purity and given a sufficient condition for these two coicide (when M is a left Rmodule and μ is an i × j matrix determined by M). In the second section of this present paper we relativize the concept of weak (M, σ)purities corresponding to direct products of matrices of left modules which are row finite or those of right modules are column finite.…”

Section: Introductionmentioning

confidence: 99%

M-purity and torsion purity in modules

Pandey¹,

Pathak²

2013

ija

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“…In this section, we study the co-cyclic co-purity, which is given by Choudhari and Tiwari [2] . But we study the concept as the dualization of c-purity.…”

Section: The Co-cyclic Copuritymentioning

confidence: 99%