Completed Functor 𝑆 −1 ̂() of the Localization Functor 𝑆 −1 (), Isomorphism and Adjunction

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 27 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Completed Functor 𝑆−1()  of the Localization Functor 𝑆−1(), Isomorphism and Adjunction Abdoulaye Mane, Mohamed Ben Maaouia, Mamadou Sanghare 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,(𝑮𝒏)𝒏∈ℕ) be a filtered normal group equipped with the group topology associated with the filtration (𝑮𝒏)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: (𝒙𝒏)𝓡(𝒚𝒏)⇔(𝒙𝒏)−(𝒚𝒏)=(𝒙𝒏−𝒚𝒏) converges to 0, noted by (𝒙𝒏−𝒚𝒏)→𝟎. The quotient set 𝓒(𝑮)/𝓡:={(𝒙𝒏) ∣(𝒙𝒏)∈𝓒(𝑮)} denoted 𝑮 is equipped with a group structure and is called the completed groupe of 𝑮. For any filtered ring (resp. left 𝑨-module) (𝑨,(𝑰𝒏)𝒏∈ℕ) (resp. (𝑴,(𝑴𝒏)𝒏∈ℕ) ), the completed group 𝑨 (resp. 𝑴  ) is equipped with a ring structure (resp. 𝑨-module) by (𝒂𝒏) ×(𝒃𝒏) = (𝒂𝒏𝒃𝒏) (𝒓𝒆𝒔𝒑.(𝒂𝒏)⋅ (𝒎𝒏) =(𝒂𝒏⋅𝒎𝒏))  where (𝒂𝒏) ,(𝒃𝒏) ∈𝑨 (resp. (𝒎𝒏) ∈𝑴  ) called completed ring (resp. module) of 𝑨 (resp. 𝑴 ). And for all saturated multiplicative subset 𝑺 of 𝑨 that satisfies the left Ore conditions, 𝑺={(𝒙𝒏) ∈𝑨∣(𝒙𝒏) ≠𝟎 and ∃𝒏𝟎∈ℕ,𝒏≥𝒏𝟎,𝒙𝒏∈𝑺} is a saturated multiplicative subset of 𝑨 that satisfies the left Ore conditions 1. Among the main results of this article, we have : - the functors 𝑺−𝟏()  is isomorphic to 𝑺−𝟏(𝑨)⊗𝑨−. and 𝑺−𝟏() is isomorphic to 𝑺−𝟏(𝑨) ⊗𝑨−. - the functors 𝑯𝒐𝒎𝑨(𝑺−𝟏𝑨 ⊗𝑨𝑴 ,−) and 𝑯𝒐𝒎𝑨(𝑺−𝟏𝑨⊗𝑨𝑴 ,−) are isomorphic. - the functors 𝑺−𝟏𝑨⊗𝑨 - and 𝑯𝒐𝒎𝑨(𝑺−𝟏𝑨,−) 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. I. INTRODUCTION This article serves as a continuation of our previous work 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. *Correspondence Author(s) Abdoulaye Mane*, Department of Mathématiques, Université Gaston Berger, Saint-Louis, Senegal. Email ID: [email protected], ORCID ID: 0009-0006-8729-6431 Mohamed Ben Maaouia, Laboratory of Algebra, Codes And Cryptography Applications (LACCA), UFR-SAT, University Gaston Berger (UGB), Saint-Louis, Senegal Email ID: [email protected] Mamadou Sanghare, Doctoral School of Mathematics-Computer – UCAD-Sénégal, University Cheikh Anta Diop of Dakar, Dakar, Senegal Email ID: [email protected] © 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/ [1], which remains our primary reference for investigating specific homological properties with completion. In this paper, the rings are not necessarily commutative and the modules are the unitary left (resp. right) modules. Let (𝐺,(𝐺𝑛)𝑛∈ℕ) be a filtered normal group equipped with the group topology associated with the filtration (𝐺𝑛)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: (𝑥𝑛)ℛ(𝑦𝑛)⇔(𝑥𝑛)−(𝑦𝑛)= (𝑥𝑛−𝑦𝑛) converges to 0, noted by (𝑥𝑛−𝑦𝑛)→0. The quotient set 𝒞(𝐺)/ℛ:={(𝑥𝑛) ∣(𝑥𝑛)∈𝒞(𝐺)} denoted 𝐺 is equipped with a group structure and is called the completed groupe of 𝐺. For any filtered ring (𝐴,(𝐼𝑛)𝑛∈ℕ ) (resp. left 𝐴module (𝑀,(𝑀𝑛)𝑛∈ℕ) ), the completed group 𝐴󰆹 (resp. 𝑀 ) is equipped with a ring structure (resp. left 𝐴󰆹-module) by (𝑎𝑛) ×(𝑏𝑛) =(𝑎𝑛𝑏𝑛) ( resp. (𝑎𝑛)⋅ (𝑚𝑛) =(𝑎𝑛⋅𝑚𝑛) ) where (𝑎𝑛) ,(𝑏𝑛) ∈𝐴󰆹( resp. (𝑚𝑛) ∈𝑀) called completed ring (resp. module) of 𝐴( resp. 𝑀). In the commutative case, the localization functor 𝑆−1() and the functor 𝑆−1𝐴⊗𝐴 - have been studied by many authors [3]. However, in the non-commutative case, these functors have been addressed by few authors [4]. But, the completed functors 𝑆−1()  and 𝑆−1(𝐴) ⊗𝐴 - have not been explicitly studied in either the commutative or noncommutative case to our knowledge, which constitutes the main objective of this work. In this article, we study the completed functor 𝑆−1()  of the localization functor 𝑆−1() and the localization functor 𝑆󰆹−1(), then their relationships with the tensor product functors 𝑆−1(𝐴) ⊗𝐴−,𝑆−1(𝐴)⊗𝐴 −,𝑆󰆹−1𝐴󰆹⊗𝐴−and Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,−). We also study the adjunction between the functors and 𝑆−1() , where 𝑆 is a saturated multiplicative subset of 𝐴 that satisfies the left Ore conditions and 𝑆󰆹={(𝑥𝑛) ∈𝐴󰆹∣(𝑥𝑛) ≠0 and ∃𝑛0∈ℕ,𝑛≥ 𝑛0,𝑥𝑛∈𝑆} is the set of classes of Cauchy sequences in 𝐴 with values in 𝑆 that do not converge to 0 , which is a saturated multiplicative subset of 𝐴󰆹 that satisfies the left Ore conditions [1]. Thus, the main results in this article are: The section 1 consists of preliminary results. In section 2, we proove that: ▪The completion functor 𝑆−1()  of the functor 𝑆−1() is isomorphic to the localization functor 𝑆󰆹−1(). ▪The functors 𝑆󰆹−1() and 𝑆−1𝐴⊗𝐴−  are isomorphic. ▪The functors 𝑆󰆹−1() and 𝑆󰆹−1(𝐴󰆹)⊗𝐴 - are isomorphic. ▪The functor 𝑆−1()  is isomorphic to the functor 𝑆󰆹−1(𝐴󰆹)⊗𝐴-. ▪The functor 𝑆󰆹−1(𝐴󰆹)⊗𝐴 - is isomorphic to 𝑆−1𝐴⊗𝐴− . Completed Functor 𝑺−𝟏()  of the Localization Functor 𝑺−𝟏(), Isomorphism and Adjunction 28 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. ▪ The functor Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,−) is isomorphic to the functor Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,−). ▪ The functor 𝑆−1𝐴⊗𝐴−  is adjoint to the functor Hom𝐴(𝑆󰆹−1𝐴󰆹,−). II. DEFINITIONS AND PRELIMINARY RESULTS Proposition 1. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring, (𝑀,(𝑀𝑛)𝑛∈ℕ) a filtered left 𝐴-module and (𝑁,(𝑁𝑛)𝑛∈ℕ) a filtered right 𝐴-module. Then, the sequence of subgroups with general term (𝑀⊗𝐴𝑁)𝑛=∑ 𝑖+𝑗≤𝑛𝑀𝑖⊗𝐴𝑁𝑗 equips the tensor product group 𝑀⊗𝐴𝑁 with a filtered group structure. Proof. (𝑀⊗𝐴𝑁)𝑛∈ℕ is a sequence of subgroups of 𝑀⊗𝐴𝑁 by definition. Since (𝑀⊗𝐴𝑁)𝑛+1=∑ 𝑖+𝑗≤𝑛+1𝑀𝑖⊗𝐴𝑁𝑗= ∑ 𝑖≤𝑛+1𝑀𝑖⊗𝐴𝑁0+∑ 𝑖+𝑗≤𝑛+1𝑀𝑖⊗𝐴𝑁𝑗+1 and (𝑀⊗𝐴𝑁)𝑛=∑ 𝑖+𝑗≤𝑛𝑀𝑖⊗𝑅𝑁𝑗=∑ 𝑖≤𝑛𝑀𝑖⊗𝐴𝑁0+ ∑ 𝑖+𝑗≤𝑛𝑀𝑖⊗𝐴𝑁𝑗−1,then (𝑀⊗𝐴𝑁)𝑛⊆(𝑀⊗𝐴𝑁)𝑛+1. Theorem 1. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring, (𝑀,(𝑀𝑛)𝑛∈ℕ) a filtered left 𝐴 module and (𝑁,(𝑁𝑛)𝑛∈ℕ) a filtered right 𝐴-module. Then, the tensor product group 𝑀⊗𝐴𝑁 is equipped with a topological group structure associated with the filtration (𝑀⊗𝐴𝑁)𝑛∈ℕ. Proof. [1] Proposition 2. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered duo-ring and (𝑀,(𝑀𝑛)𝑛∈ℕ) a filtered left 𝐴-module. Then, the sequence of submodules with general term (𝐴⊗𝐴𝑀)𝑛=∑ 𝑖+𝑗≤𝑛𝐼𝑖⊗𝑀𝑗 equips the left tensor product 𝐴-module 𝐴⊗𝐴𝑀 with a filtered module structure. Proof. [2] According to proposition 1, (𝐴⊗𝐴𝑀)𝑛∈ℕ is a filtration of (𝐴⊗𝐴𝑀) as a group when considering 𝐴 as an 𝐴-module. Let 𝑦∈𝐼𝑛(𝐴⊗𝐴𝑀)𝑚, then there is 𝑥𝑙∈𝐼𝑛,𝑢𝑙𝑖∈𝐼𝑖 and 𝑣𝑙𝑗∈𝑀𝑗, where 𝑙∈𝐽 is a set of indices, such that 𝑦=∑ 𝑙∈𝐽 𝑥𝑙∑ 𝑖+𝑗≤𝑚 𝑢𝑙𝑖⊗𝑣𝑙𝑗=∑ 𝑙∈𝐽 (∑ 𝑖+𝑗≤𝑚 (𝑥𝑙𝑢𝑙𝑖)⊗𝑣𝑙𝑗)= ∑ 𝑖+𝑗≤𝑚 (∑ 𝑙∈𝐽 (𝑥𝑙𝑢𝑙𝑖)⊗𝑣𝑙𝑗) = ∑ 𝑖+𝑗+𝑛≤𝑛+𝑚 (∑ 𝑙∈𝐽 (𝑥𝑙𝑢𝑙𝑖)⊗𝑣𝑙𝑗) = ∑ 𝑖+𝑘≤𝑛+𝑚 (∑ 𝑙∈𝐽 (𝑥𝑙𝑢𝑙𝑖)⊗𝑣𝑙(𝑘−𝑛)), where 𝑗=𝑘−𝑛 and 𝑘≥𝑛 = ∑ 𝑖+𝑘≤𝑛+𝑚 (∑ 𝑙∈𝐽 (𝑥𝑙𝑢𝑙𝑖)⊗𝑣𝑙(𝑘−𝑛)), where 𝑗=𝑘−𝑛 and 𝑘≥𝑛 = ∑ 𝑖+𝑘≤𝑛+𝑚 (∑ 𝑙∈𝐽 𝑥𝑙𝑢𝑙𝑖)⊗(∑ 𝑙∈𝐽 𝑣𝑙(𝑘−𝑛)) = ∑ 𝑖+𝑘≤𝑛+𝑚 𝑤𝑖⊗𝑧𝑘 where 𝑤𝑖=∑ 𝑙∈𝐽 𝑥𝑙𝑢𝑙𝑖∈𝐼𝑖 and 𝑧𝑘=∑ 𝑙∈𝐽 𝑣𝑙(𝑘−𝑛)∈𝑀𝑘 = ∑ 𝑖+𝑘≤𝑛+𝑚 𝑤𝑖⊗𝑧𝑘∈ ∑ 𝑖+𝑘≤𝑛+𝑚 𝐼𝑖⊗𝑀𝑘=(𝐴⊗𝐴𝐴𝑀)𝑛+𝑚. Then 𝐼𝑛(𝐴⊗𝐴𝑀)𝑚⊆(𝐴⊗𝐴𝑀)𝑛+𝑚,∀𝑛,𝑚∈ℕ. Theorem 2. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the ring topology associated with (𝐼𝑛)𝑛∈ℕ and (𝑀,(𝐼𝑛𝑀)𝑛∈ℕ) a filtered left 𝐴-module. Then, the tensor product 𝐴-module 𝐴⊗𝐴𝑀 is equipped with a topological module structure associated with the filtration (𝐴⊗𝐴𝑀)𝑛∈ℕ. Proof. Let's show that the application ∙:𝐴×(𝐴⊗𝐴𝑀)→𝐴⊗𝐴𝑀 such that (𝑎,(𝑏⊗𝑥))↦𝑎(𝑏⊗𝑥) is continuous. For all 0≤ 𝑖,𝑗≤𝑛∈ℕ, we have: 𝐼𝑖⊗𝐼𝑗𝑀≅𝐼𝑗𝑀⇒∑ 𝑖+𝑗≤𝑛𝐼𝑖⊗𝐼𝑗𝑀≅∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛𝐼𝑗𝑀⇒(𝐴⊗𝐴𝑀)𝑛≅∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛𝐼𝑗𝑀, then for all open set 𝐾 of 𝐴⊗𝐴𝑀, there is an open set 𝐾′ of 𝑀 such that ∑ 𝑗=𝑛−𝑖 𝑖≤𝑛 𝐼𝑗𝑀⊆𝐾′⇔(𝐴⊗𝐴𝑀)𝑛⊆𝐾. Since (𝑎+𝐼𝑛)×(𝑏⊗𝑥+(𝐴⊗𝐴𝑀)) is a neighborhood of all (𝑎,𝑏⊗𝑥)∈∙−1(𝐾), then Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 29 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (𝑎+𝐼𝑛)(𝑥+∑ 𝑗≤𝑛−𝑖 𝑖≤𝑛 𝐼𝑗𝑀)=𝑎𝑥+𝐼𝑛𝑥+𝑎 ∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀+𝐼𝑛∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀 =𝑎𝑥+𝐼𝑛𝑥+ ∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀, since 𝐼𝑗 are ideals =𝑎𝑥+𝐼𝑛𝑥+𝐼𝑛𝑀+ ∑ 𝑗≤𝑛−𝑖,1≤𝑖≤𝑛 𝐼𝑗𝑀=𝑎𝑥+ ∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀, since 𝐼𝑛𝑥⊂𝐼𝑛𝑀 =𝑎𝑥+ ∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀⊆𝐾′ since the laws of 𝑀 are continuous ⇒(𝑎+𝐼𝑛)(𝑥+ ∑ 𝑗≤𝑛−𝑖, 𝐼𝑗𝑀)=𝑎𝑥+ ∑ 𝑗≤𝑛−𝑖,𝑖≤𝑛 𝐼𝑗𝑀⊆𝐾′ ⇒∙ ((𝑎+𝐼𝑛)×(𝑏⊗𝑥+(𝐴⊗𝐴𝑀)𝑛))=(𝑎+𝐼𝑛)(𝑏⊗𝑥+(𝐴⊗𝐴𝑀)𝑛)⊆𝐾 ⇒(𝑎+𝐼𝑛)×(𝑏⊗𝑥+(𝐴⊗𝐴𝑀)𝑛) ⊆.−1(𝐾). Remark 1. We denote by 𝐴⊗𝐴𝑀  the completed of the left tensor product 𝐴 module 𝐴⊗𝐴𝑀 equipped with the topology associated with the filtration (𝐴⊗𝐴𝑀)𝑛∈ℕ [1] . Lemma 1. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed, 𝑆 a saturated multiplicative subset of 𝐴 that satisfies the left Ore conditions and 𝑆󰆹 the set of classes of Cauchy sequences in 𝐴 with values on 𝑆 that do not converge to 0 . Then, 𝑆󰆹 is a saturated multiplicative subset of 𝐴󰆹 that satisfies the left Ore conditions. Proof. [1] Definition 1. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝑆 a saturated multiplicative subset of 𝐴 that satisfies the left Ore conditions, 𝑆−1() the localization functor [4] and 𝐹() the completion functor. We call the completed functor of 𝑆−1() the functor 𝑆−1() =𝐹∘𝑆−1() defined by 𝑆−1() =𝐹∘𝑆−1():𝐴𝐹−𝑀𝑜𝑑→ 𝑆−1𝐴 − Mod such that: (1) For all 𝑀∈𝑂𝑏(𝐴𝐹−𝑀𝑜𝑑), then 𝑆−1(𝑀) =𝐹∘𝑆−1(𝑀)∈𝑂𝑏(𝑆−1𝐴 𝐹−𝑀𝑜𝑑) (2) For all 𝑓∈Hom𝐴𝐹(𝑀,𝑀′), then 𝑆−1(𝑓) =𝐹∘𝑆−1(𝑓)∈Hom𝑆−1𝐴 𝐹(𝑆−1𝑀 ,𝑆−1𝑀′ ) Proposition 3. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝑆 a saturated multiplicative subset of 𝐴 that satisfies the left Ore conditions. Then, the functor 𝑆−1()  is covariant, additive and left exact. Proof. Use the functors completion 𝐹 and 𝑆−1() [4]. III. FUNCTORIAL ISOMORPHISM AND ADJUNCTION Theorem 3. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed. Then, the completed functor 𝑆−1()  of the functor 𝑆−1() is isomorphic to the localization functor 𝑆󰆹−1() (noted by 𝑆󰆹−1()≅ 𝑆−1() ). Proof. (1) 𝜗𝑀:𝑆−1𝑀 →𝑆󰆹−1𝑀,(𝑚𝑛 𝑠𝑛) ↦(𝑚𝑛)  (𝑠𝑛) is isomorphic by the theorem 9.6 [1]. (2) Let (𝑀′,(𝑀𝑛′)𝑛∈ℕ) a filtered left 𝐴-module equipped with the group topology associated with (𝑀𝑛′)𝑛∈ℕ and 𝑓:𝑀→𝑀′ a compatible modules morphism. Then, we have : 𝑆󰆹−1(𝑓󰆹):𝑆󰆹−1𝑀→𝑆󰆹−1𝑀′ ,(𝑚𝑛 𝑠𝑛) ↦(𝑓(𝑚𝑛))  (𝑠𝑛)  and 𝑆−1(𝑓)  : 𝑆−1𝑀 → 𝑆−1𝑀′ ,(𝑚𝑛 𝑠𝑛) ↦(𝑓(𝑚𝑛) 𝑠𝑛)  are modules morphisms. We have the commutative diagramm : Indeed : Completed Functor 𝑺−𝟏()  of the Localization Functor 𝑺−𝟏(), Isomorphism and Adjunction 30 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (a) 𝑆󰆹−1(𝑓󰆹)∘𝜗𝑀((𝑚𝑛 𝑠𝑛) )=𝑆󰆹−1(𝑓󰆹)((𝑚𝑛)  (𝑠𝑛) )=(𝑓(𝑚𝑛))  (𝑠𝑛)  (b) 𝜗𝑀′∘𝑆−1(𝑓) ((𝑚𝑛 𝑠𝑛) )=𝜗𝑀′((𝑓(𝑚𝑛) 𝑠𝑛) )=(𝑓(𝑚𝑛))  (𝑠𝑛)  Proposition 4. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed, 𝑆 a saturated multiplicative subset of 𝐴 that satisfies the left Ore conditions, (𝑀,(𝑀𝑛)𝑛∈ℕ) a filtered left 𝐴module equipped with the group topology associated with (𝑀𝑛)𝑛∈ℕ. Then, the correspondence defined by: is an isomorphism of left 𝐴󰆹-modules. Proof. (1) Show that 𝜙 is a well-defined map. Let∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) ,∑ 𝑖 (𝑏𝑛𝑖 𝑡𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ∈𝑆−1𝐴 ⊗𝐴𝑀 such that ∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) =∑ 𝑖 (𝑏𝑛𝑖 𝑡𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) . We have: ∑ 𝑖(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) =∑ 𝑖(𝑏𝑛𝑖 𝑡𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ⇒{(𝑎𝑛𝑖 𝑠𝑛𝑖) =(𝑏𝑛𝑖 𝑡𝑛𝑖)  (𝑚𝑛𝑖) =(𝑚𝑛𝑖 ′) ,∀𝑖 by definition of tensor product. Therefore, we have{(𝑎𝑛𝑖 𝑠𝑛𝑖−𝑏𝑛𝑖 𝑡𝑛𝑖)→0 (𝑚𝑛𝑖−𝑚𝑛𝑖 ′)→0,∀𝑖. Moreover, (𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖−𝑏𝑛𝑖 𝑡𝑛𝑖⊗𝑚𝑛𝑖 ′)=(𝑎𝑛𝑖 𝑠𝑛𝑖−𝑏𝑛𝑖 𝑡𝑛𝑖)⊗(𝑚𝑛𝑖−𝑚𝑛𝑖 ′)−(𝑎𝑛𝑖 𝑠𝑛𝑖)⊗(𝑚𝑛𝑖 ′− 𝑚𝑛𝑖)+(𝑎𝑛𝑖 𝑠𝑛𝑖−𝑏𝑛𝑖 𝑡𝑛𝑖)⊗(𝑚𝑛𝑖 ′) thus, (𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖−𝑏𝑛𝑖 𝑡𝑛𝑖⊗𝑚𝑛𝑖 ′)→0∀𝑖, ⇒(∑𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖−∑𝑖 𝑏𝑛𝑖 𝑡𝑛𝑖⊗𝑚𝑛𝑖 ′)→0. Therefore,(∑𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖)=(∑𝑖 𝑏𝑛𝑖 𝑡𝑛𝑖⊗𝑚𝑛𝑖 ′)⇒𝜙(∑𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) )=𝜙(∑𝑖 (𝑏𝑛𝑖 𝑡𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ). (2) Show that 𝜙 is a morphism of 𝐴󰆹-modules. (a) It is clear that 𝜙 is a group morphism. (b) Let (𝑎𝑛) ∈𝐴󰆹 and ∑𝑖 (𝑏𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) ∈𝑆−1𝐴 ⊗𝐴𝑀, we have: (3) Show that 𝜙 is bijective. (a) The surjectivity is evident. (b) Let ∑ 𝑖(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖)  and ∑ 𝑖(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ∈𝑆−1𝐴 ⊗𝐴𝑀 such that 𝜙(∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) )=𝜙(∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ), let us show that ∑ 𝑖(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) =∑ 𝑖(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) . We have: Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 31 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝜙(∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) )=𝜙(∑ 𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ) ⇒(∑ 𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖−∑ 𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖 ′)→0. Moreover, for all 𝑖∈𝐼,∑ 𝑖𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖=1 𝑠𝑛⊗∑ 𝑖𝑥𝑛𝑖𝑎𝑛𝑖𝑚𝑛𝑖 and ∑ 𝑖𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖 ′=1 𝑠𝑛⊗∑ 𝑖𝑥𝑛𝑖𝑎𝑛𝑖𝑚𝑛𝑖 ′,∀𝑛∈ℕ where 𝑠𝑛=∏ 𝑖𝑠𝑛𝑖 and 𝑥𝑛𝑖=𝑠𝑛−1𝑠𝑛𝑖∈𝑆 。 Thus, (∑ 𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖−∑ 𝑖 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖 ′)=(1 𝑠𝑛⊗∑ 𝑖 𝑥𝑛𝑖𝑎𝑛𝑖𝑚𝑛𝑖−1 𝑠𝑛⊗∑ 𝑖 𝑥𝑛𝑖𝑎𝑛𝑖𝑚𝑛𝑖 ′)=(1 𝑠𝑛⊗∑ 𝑖 𝑥𝑛𝑖𝑎𝑛𝑖[𝑚𝑛𝑖−𝑚𝑛𝑖 ′])→ 0 ⇒(∑𝑖 𝑥𝑛𝑖𝑎𝑛𝑖[𝑚𝑛𝑖−𝑚𝑛𝑖 ′])→0 ⇒(∑𝑖∈𝐼 𝑥𝑛𝑖𝑎𝑛𝑖[𝑚𝑛𝑖−𝑚𝑛𝑖 ′])=0 ⇒∑𝑖 (𝑥𝑛𝑖) ×(𝑎𝑛𝑖) ×(𝑚𝑛𝑖) =∑𝑖 (𝑥𝑛𝑖) ×(𝑎𝑛𝑖) ×(𝑚𝑛𝑖 ′) . Therefore, for all 𝑖∈𝐼,(𝑥𝑛𝑖) ×(𝑎𝑛𝑖) ×(𝑚𝑛𝑖) =(𝑥𝑛𝑖) ×(𝑎𝑛𝑖) ×(𝑚𝑛𝑖 ′) ,∀𝑛∈ℕ⇒(𝑎𝑛𝑖) ×(𝑚𝑛𝑖) =(𝑎𝑛𝑖) ×(𝑚𝑛𝑖 ′) ,∀𝑖. Thus, (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) =(𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) ,∀𝑖, then ∑𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) =∑𝑖 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖 ′) . Corollary 1. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring and (𝑀,(𝑀𝑛)𝑛∈ℕ) a filtered left 𝐴 module equipped with the group topology associated with (𝑀𝑛)𝑛∈ℕ. Then, 𝑆−1𝐴 ⊗𝐴𝑀 and 𝑆−1(𝐴)⊗𝐴𝑀  are isomorphic. Corollary 2. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed ring, 𝑆 a set of regular elements of 𝐴∖𝑃 and (𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) a filtered left 𝐴-module equipped with the 𝑃-adic topology. Then, 𝐴𝑃 ⊗𝐴𝑀𝑃  and 𝐴𝑃⊗𝐴𝑀𝑃  are isomorphic. Proposition 5. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring. Then, 𝑆󰆹−1𝐴󰆹⊗𝐴𝑀 is isomorphic to 𝑆−1(𝐴)⊗𝐴𝑀 . Proof. According to theorem 3, we have 𝑆󰆹−1𝐴󰆹≅𝑆−1(𝐴) , then 𝑆󰆹−1𝐴󰆹⊗𝐴𝑀≅𝑆−1(𝐴) ⊗𝐴𝑀≅𝑆−1(𝐴)⊗𝐴𝑀 . Corollary 3. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed duo-ring, 𝑆 a set of regular elements of 𝐴∖𝑃,(𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) a filtered left 𝐴-module equipped with the 𝑃-adic topology. Then, 𝐴󰆹𝑃⊗𝐴𝑀𝑃 is isomorphic to 𝐴𝑃 ⊗𝐴𝑀𝑃 . Proposition 6. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring. Then, the correspondence: Theorem 4. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then, the functors 𝑆󰆹−1() and 𝑆󰆹−1(𝐴󰆹)⊗𝐴− are isomorphic. Proof. There is a left 𝐴󰆹-module, according to proposition 6, defined by: according to the universal property of the tensor product. 𝜒‾𝑀 is indeed bijective, as follows: (1) According to 4, 𝑆−1𝑀 and 𝑆−1𝐴⊗𝐴𝑀 are isomorphic. Consequently, 𝑆−1𝑀  and 𝑆−1𝐴⊗𝐴𝑀  are also isomorphic. By the theorem 3, 𝑆󰆹−1𝐴󰆹⊗𝑀 and 𝑆−1𝐴 ⊗𝐴𝑀 are isomorphic. Since 𝑆−1𝐴 ⊗𝐴𝑀 and 𝑆−1(𝐴)⊗𝐴𝑀  are isomorphic, by Proposition 4, then 𝑆󰆹−1𝐴󰆹⊗𝑀 and 𝑆󰆹−1𝑀 are isomorphic. Therefore, there exists 𝜔𝑀:𝑆󰆹−1𝑀⟶𝑆󰆹−1𝐴󰆹⊗𝐴6 𝑀 Completed Functor 𝑺−𝟏()  of the Localization Functor 𝑺−𝟏(), Isomorphism and Adjunction 32 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (2) Consider (𝑀,(𝑀𝑛)𝑛∈ℕ) and (𝑀′,(𝑀𝑛′)𝑛∈ℕ) two left 𝐴-modules, and f : 𝑀→𝑀′ a compatible module morphism. Then, according to point 1 , the following diagram commutes: Indeed: Let (𝑚𝑛)  (𝑠𝑛) ∈𝑆󰆹−1𝑀, then: ∙𝜔𝑀′∘𝑆󰆹−1(𝑓󰆹)((𝑚𝑛)  (𝑠𝑛) ) =𝜔𝑀′ (𝑓((𝑚𝑛))  (𝑠𝑛) )=𝜔𝑀′ ((𝑓(𝑚𝑛) ) (𝑠𝑛) ) = 1 (𝑠𝑛) ⊗(𝑓(𝑚𝑛) ) ∙(1𝑆󰆹−1𝐴⊗𝑓󰆹)∘𝜔𝑀′ ((𝑚𝑛)  (𝑝𝑛) ) =(1𝑆󰆹−1𝐴⊗𝑓󰆹)(1 (𝑠𝑛) ⊗(𝑚𝑛) ) = 1 (s𝑛)⊗(𝑓(𝑚𝑛) ) Thus, the functors 𝑆󰆹−1() and are isomorphic. Corollary 4. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then, the functors 𝑆󰆹−1() and 𝑆−1𝐴⊗𝐴−  are isomorphic. Proof. By the theorem 3, we have 𝑆󰆹−1()≅𝑆−1() ≅𝑆−1(𝐴)⊗𝐴 . Corollary 5. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed duo-ring, 𝑆 a subset of regular elements of 𝐴∖𝑃,(𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) a filtered left 𝐴-module equipped with the 𝑃-adic topology. Then, the functors 𝑆󰆹−1() and 𝐴𝑃⊗𝐴−  are isomorphic. Theorem 5. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then, we have: (1) 𝑆󰆹−1()≅𝑆−1(𝐴) ⊗𝐴. (2) 𝑆󰆹−1(𝐴󰆹)⊗𝐴−≅𝑆−1𝐴⊗𝐴− . (3) 𝑆−1() ≅𝑆󰆹−1(𝐴󰆹)⊗𝐴−. Proof. (1) We have 𝑆󰆹−1𝐴󰆹 and 𝑆−1𝐴  are isomorphic (theorem 3). Thus, 𝑆󰆹−1()≅𝑆󰆹−1𝐴󰆹⊗𝐴−≅𝑆−1(𝐴) ⊗𝐴− by theorem 4 . (2) It suffices to observe that 𝑆−1() ≅𝑆−1𝐴⊗𝐴− ≅𝑆󰆹−1𝐴󰆹⊗𝐴−≅𝑆󰆹−1(), by theorem 4. (3) Follows from 1. and 2. Corollary 6. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed ring, 𝑆 a subset consisting of regular elements of 𝐴∖𝑃, (𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) a filtered left 𝐴-module equipped with the 𝑃-adic topology, and 𝑆󰆹={(𝑥𝑛) ∈𝐴󰆹∣(𝑥𝑛) ≠0 and ∃𝑛0∈ℕ,𝑛≥𝑛0,𝑥𝑛∈𝑆} the set of equivalence classes of Cauchy sequences in 𝐴 with values in 𝑆 that do not converge to 0 . Then, we have: (1) 𝑆󰆹−1()≅𝐴𝑃 ⊗𝐴. (2) 𝐴󰆹𝑃⊗𝐴−≅𝐴𝑃⊗𝐴−  (3) 𝑆−1() ≅𝐴󰆹𝑃⊗𝐴−. Proposition 7. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, and (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, there is an isomorphism 𝜓𝑀′:Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ ) → Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,𝑀′ ) 𝑓 ↦ 𝜓𝑀′(𝑓):𝑆−1𝐴⊗𝐴𝑀 ⟶𝑀′  (∑ 𝑖∈𝐼 𝑎𝑛𝑖 𝑠𝑛𝑖⊗𝑚𝑛𝑖 )⟼𝑓(∑ 𝑖∈𝐼 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) ) Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 33 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Proof. It is evident that 𝜓 is well-defined (existence) and is a morphism of 𝐴󰆹-modules. Let us show that 𝜓 is bijective. Consider the application 𝜓𝑀′ ′ is a morphism of 𝐴󰆹-modules. For every 𝑔∈Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ ), we have: ⇒𝜓𝑀′ ′∘𝜓𝑀′(𝑔)=𝑔⇒𝜓𝑀′ ′∘𝜓𝑀′=IdHom𝐴 (𝑆−1𝐴 ⊗𝐴 𝑀,𝑀′ ). Proposition 8. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, and (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ ) and Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′ ) are isomorphic. Proof. By the theorem 3, we have 𝑆󰆹−1𝐴󰆹≅𝑆−1(𝐴) , hence Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ )≅Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′ ) by the proposition 7. Corollary 7. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, and (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,𝑀′ ) and Hom 𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′ ) are isomorphic. Corollary 8. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed duo-ring, 𝑆 a subset of regular elements of 𝐴∖𝑃,(𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) and (𝑀′,(𝑃𝑛𝑀′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, we have: (1) Hom𝐴(𝐴𝑃 ⊗𝐴𝑀,𝑀′ )≅Hom𝐴(𝐴𝑃⊗𝐴𝑀 ,𝑀′ ). (2)Hom𝐴(𝐴𝑃 ⊗𝐴𝑀,𝑀′ )≅Hom𝐴(𝐴󰆹𝑃⊗𝐴𝑀,𝑀′ ). (3) Hom𝐴(𝐴𝑃⊗𝐴𝑀 ,𝑀′)≅Hom𝐴(𝐴󰆹𝑃⊗𝐴𝑀,𝑀′ ). Theorem 6. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, the functors Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,−) and Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,−) are isomorphic. Proof. Let 𝑀′  be the completion of a filtered left 𝐴-module (𝑀′,(𝑀𝑛′)𝑛∈ℕ). According to proposition 7) Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ ) and Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,𝑀′ ) are isomorphic. Let (𝑀′,(𝑀𝑛′)𝑛∈ℕ),(𝑀′′,(𝑀𝑛′′)𝑛∈ℕ) be two filtered left 𝐴-modules and 𝑓:𝑀′→𝑀′′ a compatible morphism of 𝐴-modules. Then, we have the following commutative diagram: Indeed, for all 𝑔∈Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,𝑀′ ), we have: ▪ 𝑓∗ ∘𝜓𝑀′(𝑔)=𝑓󰆹∘𝜓𝑀′(𝑔)=𝑓󰆹∘𝑔(∑ 𝑖∈𝐼 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) ). ▪ 𝜓𝑀′′∘𝑓󰆹∗=𝜓𝑀′′∘(𝑓󰆹(𝑔))=(𝑓󰆹∘𝑔)(∑ 𝑖∈𝐼 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) )=𝑓󰆹∘𝑔(∑ 𝑖∈𝐼 (𝑎𝑛𝑖 𝑠𝑛𝑖) ⊗(𝑚𝑛𝑖) ). Thus, 𝑓󰆹∗∘𝜓𝑀′=𝜓𝑀′′∘𝑓󰆹∗. Corollary 9. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, the functors: (1) Hom𝐴(𝑆−1𝐴 ⊗𝐴𝑀,−) and Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,−) are isomorphic. Completed Functor 𝑺−𝟏()  of the Localization Functor 𝑺−𝟏(), Isomorphism and Adjunction 34 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (2) Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,−) and Hom𝐴(𝑆−1𝐴⊗𝐴𝑀 ,−) are isomorphic. Proposition 9. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ,𝐴󰆹 its completed ring, (𝑀,(𝑀𝑛)𝑛∈ℕ),(𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′) and Hom𝐴(𝑀,Hom𝐴(𝑆󰆹−1𝐴󰆹,𝑀′)) are isomorphic. Proof. It suffices to show that the map defined by: 𝜙𝑀𝑀′:Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′ ) →Hom𝐴(𝑀,Hom𝐴(𝑆−1 𝐴󰆹,𝑀′ )) 𝑓 ↦𝜙𝑀𝑀′(𝑓):𝑀→Hom𝐴(𝑆−1 𝐴󰆹,𝑀′ ) (𝑚𝑛) ⟼𝜙𝑀𝑀′(𝑓)((𝑚𝑛) ) is an isomorphism of 𝐴󰆹-modules, reasoning similarly as in Proposition 7 Corollary 10. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ, 𝐴󰆹 its completed ring, (𝑀,(𝑀𝑛)𝑛∈ℕ) and (𝑀′,(𝑀𝑛′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, Hom𝐴(𝑆󰆹−1𝑀,𝑀′) and Hom𝐴(𝑀,Hom𝐴(𝑆󰆹−1𝐴󰆹,𝑀′)) are isomorphic. Proof. Since 𝑆󰆹−1𝑀 and 𝑆󰆹−1𝐴󰆹⊗𝐴𝑀 are isomorphic, Hom𝐴(𝑆󰆹−1𝑀,𝑀′) is isomorphic to Hom𝐴(𝑀,Hom𝐴(𝑆󰆹−1𝐴󰆹,𝑀′)) by the proposition 9 Corollary 11. Let (𝐴,(𝑃𝑛)𝑛∈ℕ) be a filtered duo-ring equipped with the 𝑃-adic topology, 𝐴󰆹 its completed ring, 𝑆 a subset of regular elements of 𝐴∖𝑃,(𝑀,(𝑃𝑛𝑀)𝑛∈ℕ) and (𝑀′,(𝑃𝑛𝑀′)𝑛∈ℕ) two filtered left 𝐴-modules. Then, Hom𝐴(𝐴󰆹𝑃⊗𝐴𝑀,𝑀′) and Hom𝐴(𝑀,Hom𝐴(𝐴󰆹𝑃,𝑀′)) are isomorphic. Theorem 7. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology defined by (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then, the functors 𝑆󰆹−1𝐴󰆹⊗𝐴− and Hom𝐴(𝑆󰆹−1𝐴󰆹,−) are adjoints. Proof. Since Hom𝐴(𝑆󰆹−1𝐴󰆹⊗𝐴𝑀,𝑀′)≅Hom𝐴(𝑀,Hom𝐴(𝑆󰆹−1𝐴󰆹,𝑀′)) by the Proposition 9, it suffices to verify that: (1) For every left 𝐴󰆹-module 𝑇 that is the completion of the filtered left 𝐴-module (𝑇,(𝑇𝑛)𝑛∈ℕ) and 𝑔:𝑇→𝑀, the following diagram: is commutative. (2) For every left 𝐴󰆹-module 𝑇′  that is the completion of the filtered left 𝐴 module (𝑇′,(𝑇𝑛′)𝑛∈ℕ) and 𝑔′:𝑀′ →𝑇, the following diagram: is commutative [5]. Proposition 10. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology defined by (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then, the functors 𝑆−1𝐴⊗𝐴−  and Hom𝐴(𝑆󰆹−1𝐴󰆹,−) are adjoints. Proof. Since 𝑆󰆹−1(𝐴󰆹)⊗𝐴−𝑆−1𝐴⊗𝐴 − , then, by the theorem 𝛾,𝑆−1𝐴⊗𝐴−  and Hom𝐴(𝑆󰆹−1𝐴󰆹,−) are adjoints. Theorem 8. Let (𝐴,(𝐼𝑛)𝑛∈ℕ) be a filtered ring equipped with the group topology associated with (𝐼𝑛)𝑛∈ℕ, and 𝐴󰆹 its completed ring. Then: (1) The functors 𝑆−1()  and Hom𝐴(𝑆󰆹−1𝐴󰆹,−) are adjoints. (2) The functors 𝑆󰆹−1() and Hom𝐴(𝑆−1𝐴 ,−) are adjoints. Proof. (1) By the the theorem 5 and the proposition 10, we have the result. (3) It is similar. IV. CONCLUSION This work is a continuation of our previous research on the link between the properties of topological completion and those of localization 1. We introduced the completion functor 𝑆−1()  of the Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 35 Retrieval Number:100.1/ijam.B121405021025 DOI: 10.54105/ijam.B1214.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. localization functor 𝑆−1()1 and studied its relationships with the Hom functor, the tensor product functor 𝑆−1𝐴 ⊗𝐴-, as well as the completed functor 𝑆−1𝐴⊗𝐴−  of the tensor product functor 𝑆−1𝐴⊗𝐴−, among others. The results obtained allowed us to highlight several functorial isomorphisms 5 and functorial adjunctions 7, 8. In particular, we have shown that the completion of a localization functor is naturally isomorphic to the localization of its topological group completion 3. These results contribute to clarifing the structural relationships between completion and localization in a topological and in the non necessarily commutative case, and thus open new perspectives for the study of homological functor 𝐻𝑛 properties, notably the derived functors Ext, Tor with the properties of completion in the non necessarily commutative case. DECLARATION STATEMENT After aggregating input from all authors, I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed equally to all participating individuals. REFERENCES 1. Mane A., Ben Maaouia M., Sanghare M., Completion Fractions Modules of Filtered Modules over Non-Necessarily Commutative Filtered Rings, Springer, 223(468), 119-146, 2024, DOI: https://doi.org/10.1007/978-3-031-66222-5_9 2. Yekutieli A., Flatness and Completion Revisited, Springer, DOI: https://doi.org/10.1007/s10468-017-9735-7,2017 3. Faye D., Maaouia M. B., Sanghare M., Functor (𝑆‾)−1() and adjoint isomorphism, Springer Nature Switzerland AG 2019, DOI: https://doi.org/10.1007/978-3-030-36237-9_2 4. Faye D., Maaouia M. B., Sanghare M., Functor 𝑆−1() and Adjoint Isomorphism, International Mathematical Forum, Vol. 11, 2016, no. 5, 227-237, DOI: https://dx.doi.org/10.12988/imf.2016.512101 5. Thiaw M., Maaouia M., Adjunction and Localization in the Category A-Alg of A-Algebras, ISSN 1307-5543 - www.ejpam.com, Vol. 13, No. 3, 472-482, 2020, DOI: https://doi.org/10. 29020/nybg.ejpam.v13i3.3742. AUTHOR’S PROFILE Abdoulaye Mané, born in 1993 in Gaghagha, Senegal, is a PhD holder in mathematics, specializing in algebra, topology, and homological algebra. As an affiliated researcher at the UFR of Applied Sciences and Technology (SAT) at Gaston Berger University (UGB) in Saint-Louis, his interests lie in the connections between topology and algebraic structures, as well as homological theory within a topological context. He earned his bachelor's degree in mathematics and computer science from the University of Thiès in 2016 and his master's degree from UGB in 2019. He defended his thesis in 2025 under the supervision of Professor Mohamed Ben Maaouia. Actively involved in both teaching and research, he leads mathematical seminars and collaborations on a national and international level. Mohamed Ben Faraj Ben Maaouia, also known as Mohamed Ben Maaouia (ben Faraj), is a full professor at Gaston Berger University (UGB), UFR of Applied Sciences and Technologies, Department of Applied Mathematics, Saint-Louis, Senegal. He was born in 1967 in Menzel Temine, Tunisia. He was a former assistant professor in the Department of Mathematics at Cheikh Anta Diop University (UCAD) before joining UGB at its inception around 1990. He earned his Master's degree in Mauritania in 1996, and his academic path led him to UCAD, where he obtained his DEA (Diploma of Advanced Studies) in 1998, his third-cycle thesis in 2003, and his doctorate in 2011. He is a founding member of the Laboratory of Algebra, Cryptography, Codes and Applications (LACCA), where he served as director. Mamadou Sangharé, born in 1951 in Thiès (Senegal), is a full professor of mathematics at Cheikh Anta Diop University of Dakar (UCAD). Trained in Morocco (Universities of Fez and Rabat) and later in Senegal, he earned a third-cycle doctorate in 1985 in Rabat and a State Doctorate in 1992 at UCAD. A specialist in algebra, geometry, and homological algebra, he is also a pioneer in teaching cryptology in Senegal. Former director of UCAD's Doctoral School of Mathematics and Computer Science, he has trained many researchers and supervised several theses. A member of the National Academy of Sciences of Senegal, he received a 2022 African Mathematical Union award for his major contributions to the development of mathematics in Africa. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.