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

Abstract

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

References

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Citation

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

Copyright

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.