B121405021025

Completed Functor 𝑆⁻¹ ̂() of the Localization Functor 𝑆⁻¹(), Isomorphism and Adjunction
Abdoulaye Mane¹, Mohamed Ben Maaouia², Mamadou Sanghare³

¹Abdoulaye Mane, Department of Mathématiques, Université Gaston Berger, Saint-Louis, Senegal.

²Mohamed Ben Maaouia, Laboratory of Algebra, Codes And Cryptography Applications (LACCA), UFR-SAT, University Gaston Berger (UGB), Saint-Louis, Senegal.

³Mamadou Sanghare, Doctoral School of Mathematics-Computer – UCAD-Sénégal, University Cheikh Anta Diop of Dakar, Dakar, Senegal. 

Manuscript received on 05 September 2025 | First Revised Manuscript received on 13 September 2025 | Second Revised Manuscript received on 02 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025 | PP: 27-35 | Volume-5 Issue-2, October 2025 | Retrieval Number: 100.1/ijam.B121405021025 | DOI: 10.54105/ijam.B1214.05021025

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© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: This article serves as a continuation of our previous work 1, which remains our primary reference for investigating specific homological properties with completion. Let the rings not be necessarily commutative and the modules be the unitary left (resp. right) modules. Let (𝑮, (𝑮_𝒏)_𝒏∈ℕ) be a filtered normal group equipped with the group topology associated with the filtration (𝑮_𝒏)_𝒏∈ℕ formed of normal subgroups and 𝓒(𝑮) the set of Cauchy sequences with values in 𝑮. We define an equivalence relation 𝓡 on 𝓒(𝑮) by: (𝒙_𝒏)𝓡(𝒚_𝒏) ⇔ (𝒙_𝒏) − (𝒚_𝒏) = (𝒙_𝒏 − 𝒚_𝒏) converges to 0, noted by (𝒙_𝒏 − 𝒚_𝒏 ) → 𝟎. The quotient set 𝓒(𝑮)/𝓡: = {(𝒙_𝒏 ̂) ∣ (𝒙_𝒏) ∈ 𝓒(𝑮)} denoted 𝑮̂ is equipped with a group structure and is called the completed groupe of 𝑮. For any filtered ring (resp. left 𝑨-module) (𝑨, (𝑰_𝒏 )_𝒏∈ℕ) (resp. (𝑴, (𝑴_𝒏 )_𝒏∈ℕ) ), the completed group 𝑨̂ (resp. 𝑴̂ ) is equipped with a ring structure (resp. 𝑨̂-module) by (𝒂_𝒏 ̂) ×̂ (𝒃_𝒏 ̂) = (𝒂_𝒏𝒃_𝒏 ̂) (𝒓𝒆𝒔𝒑. (𝒂_𝒏) ⋅ (𝒎_𝒏 ̂) = (𝒂_𝒏 ⋅ 𝒎_𝒏 ̂ )) where (𝒂_𝒏 ̂), (𝒃_𝒏 ̂) ∈ 𝑨̂ (resp. (𝒎_𝒏 ̂) ∈ 𝑴̂) called completed ring (resp. module) of 𝑨 (resp. 𝑴). And for all saturated multiplicative subset 𝑺 of 𝑨 that satisfies the left Ore conditions, 𝑺̂ = {(𝒙_𝒏 ̂) ∈ 𝑨̂ ∣ (𝒙_𝒏 ̂) ≠ 𝟎̂ and ∃𝒏₀ ∈ ℕ, 𝒏 ≥ 𝒏₀, 𝒙_𝒏 ∈ 𝑺} is a saturated multiplicative subset of 𝑨̂ that satisfies the left Ore conditions 1. Among the main results of this article, we have : – the functors 𝑺̂^−𝟏 ̂() is isomorphic to 𝑺̂^−𝟏(𝑨̂)⊗_𝑨̂−. and 𝑺̂^−𝟏() is isomorphic to 𝑺̂^−𝟏(𝑨) ⊗_𝑨̂−. – the functors 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨 ⊗_𝑨̂ 𝑴̂,−) and 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨̂⊗_𝑨̂ 𝑴,−) are isomorphic. – the functors 𝑺̂^−𝟏𝑨⊗_𝑨̂ – and 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨̂,−) are adjoints. This Study Allows How Establish a Relationship Between Completion [2] and Localization [4] Under the Assumptions of a Topological Structure.

Keywords: Ring, Modules, Filtration, Completion, Ore Condition, Localization, Isomorphisms, Categories, Functors, Completed Functor, Adjunction.
Scope of the Article: Algebra