Acta Scientific Computer Sciences

Review Article Volume 5 Issue 1

(FG,σ)- Purity and Semi-simple Modules

Ashok Kumar Pandey*

Department of Mathematics, Ewing Christian Post Graduate College (An Autonomous College of University of Allahabad, Prayagraj), Allahabad, India

*Corresponding Author: Ashok Kumar Pandey, Department of Mathematics, Ewing Christian Post Graduate College (An Autonomous College of University of Allahabad, Prayagraj), Allahabad, India.

Received: October 04, 2022; Published: December 13, 2022


The torsion sub-module of A⊆M is denoted by σ(A). Since it was proved by Walker [18] that the class of I- pure (J- copure) sequences form a proper class whenever I(J) is closed under homomorphic images (sub-modules) of a R- module M and if I(J) is closed under factors (sub-modules) then for any I- pure (J- copure) sequence E:0->A ->B ->C ->0 if E ∈π^(-1) (I) (E ∈i^(-1) (I)) and hence in this case Walker’s I- purity (J- copurity) coincides with the earlier notion of purity. We also study about class of R-modules dual to the modules of B. A sequence E:0⟶A ⟶B ⟶C ⟶0 is I- pure (J- copure) if and only if given C^'≤C∈ I, there existsB'≤B such that B^'≅C' and A∩B^'=0 ;we consider another notion of purity stronger than the Cohn’s purity [13]. If FG denotes the class of all finitely generated R-modules, since, this class is closed under factors. We shall try to give some characterizations of FG-purity and to determine its relationship with the FG-flat modules. We relativist this concept and also relate it with that of finite projectivity of Azumaya [10] with respect to a torsion theory and to study the inter-relationship between these concepts. We also try to consider finite σ-projectivity or (FG,σ)- pure flatness, cyclically σ- pure projectivity and cyclically σ- pure flatness, the concept of locally σ- projectivity and locally σ- splitness and study its inter-relationship with (FG,σ)- purity and semi-simple module.

Keywords: R- Modules; (FG,σ)- Purity; σ- Pure Projective; R-Modules; I- Pure (J- copure; FG-flat Modules; Cyclically σ- Pure Projectivity; σ- Pure Infectivity; Locally σ- Splitness; Semi-Simple Module. Subject classification: 16D99


  1. Ashok Kumar Pandey. “σ-Projectivity and σ-Semi- Simplicity in modules”. International Research Journal of Pure Algebra6 (2021): 08-14.
  2. Ashok Kumar Pandey. “Divisibility and Co-divisibility in modules”. International Journal of Research in Computer applications and Robotics 5 (2021): 12-22.
  3. Ashok Kumar Pandey. “T-Purity and F-Co-purity in modules”. Journal of Engineering Mathematics and statistics 1 (2021): 74 - 84.
  4. Ashok Kumar Pandey. “σ-Purity and σ-Regular rings and modules”. International Research Journal of Pure Algebra 10.8 (2020): 26-31.
  5. Ashok Kumar Pandey. “(FG, σ)-Pure flatness and locally σ-projectivity in modules”. Sambodhi Journal (UGC Care Journal)2 (2020): 240-244.
  6. Ashok Kumar Pandey. “Purity Relative to a Cyclic Module”. International Journal of Statistics and Applied Mathematics3 (2020): 55-58.
  7. Ashok Kumar Pandey. “Some problems in ring theory”. Ph. D. thesis, University of Allahabad, (2003).
  8. Ashok Kumar Pandey and M Pathak. “M-Purity and Torsion Purity in Modules”. International Journal of Algebra9 (2013): 421-427.
  9. Ashok Kumar Pandey and M Pathak. “Torsion Purity in Ring and Modules”. International Journal of Algebra8 (2013): 391-398.
  10. Garo Azumaya. “Finite splitness and finite projectivity”. Journal of Algebra 106 (1972): 114-134.
  11. BB Bhattacharyya and DP Chaudhury. “Purities Relative to a Torsion Theory”. Indian Journal of Pure and Applied Mathematics 4 (1983): 554-564.
  12. D P Chaudhury. “Relative Flatness via Stenstrom's Purity”. Indian Journal of Pure and Applied Mathematics2 (1984): 131-134.
  13. PM Cohn. “Free Products of Associative Rings”. Math. Z., 71 (1959): 380-398.
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  15. FW Anderson and KR Fuller. “Rings and Categories of Modules”. 2nd Edition, Springer- Verlag, New York, (1992).
  16. J Lambek. “Torsion theories, additive semantics and rings quotients, lecture”. Notes in Mathematics, No. 177, Springer Verlag, (1971).
  17. B Stenstrom. “Pure sub modules”. Mathematics 7 (1967): 159- 171.
  18. ML Teply and JS Golan. “Torsion free covers”. Israel Journal of Mathematics 15 (1973): 237-256.
  19. R B Warfield Jr. “Purity and algebraic compactness for modules”. Pacific Journal of Mathematics 28 (1969): 699-719.


Citation: Ashok Kumar Pandey. “(𝐅𝐆, 𝜎) − Purity and Semi-simple Modules".Acta Scientific Computer Sciences 5.1 (2023): 79-85.


Copyright: © 2023 Ashok Kumar Pandey. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.


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