IHJPAS. 36 (4) 2023 359 This work is licensed under a Creative Commons Attribution 4.0 International License *Corresponding Author: khawlahahmmed@gmail.com Abstract In this work we discuss the concept of pure-maximal denoted by (Pr-maximal) submodules as a generalization to the type of R- maximal submodule, where a proper submodule of an R- module is called Pr- maximal if < 𝐻 ≀ π‘Š ,for any submodule of W is a pure submodule of W, We offer some properties of a Pr-maximal submodules, and we give Definition of the concept, near-maximal, a proper submodule of an R-module is named near (N-maximal) whensoever is pure submodule of such that then K= .Al so we offer the concept Pr-module, An R-module W is named Pr-module, if every proper submodule of is Pr-maximal. A ring is named Pr-ring if whole proper ideal of is a Pr-maximal ideal, we offer the concept pure local (Pr-local) module an R- module is named pure local (Pr-local) module. If it has only a Pr-maximal submodule which includes all proper submodule of . A ring is named pure local (Pr-local) ring, if is a Pr- local R-module. We give some relatio among Pr-maximal submodules and others related concept. Keywords: , R-submodule, Pr-module, Pr-maximal, Pr-local, N-maximal. Introduction In this work is commutative ring with identity, and all R-modules are left until. A proper submodule of an R-module is named a pure submodule β€œif for every ideal of , [1]. A proper submodule of an R-module is named maximal in [3] β€œif whenever is a submodule of R-Module with Implies . Abduljaleel and Yaseen in [2] offer the concept of large maximal submodules as a generalization of the concept maximal submodules, β€œwhere a proper submodule of an R- module is named large-maximal(L-maximal) if implies is an essential submodule of , where a submodule of R-module is named essential, if for every non-zero doi.org/10.30526/36.4.3139 Article history: Received 11 December 2022, Accepted 28 Februray 2023, Published in October 2023 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Pure Maximal Submodules and Related Concepts Khawla Ahmed* Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. Nuhad S. Al. Mothafar Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. https://creativecommons.org/licenses/by/4.0/ mailto:khawlahahmmed@gmail.com mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq IHJPAS. 36 (4) 2023 360 submodule of , [3]. Many authors studies module and submodule for example see [4] and [5]. In [11] B.H.AL-Bahrani generalization of the type of a purely extending modules, defined using Y-closed submodules, In,this discuss, we introduce, the concept of pure-maximal (Pr-maximal) submodules as a generalization of maximal submodule , where a proper submodule of an R-module is named Pr-maximal, if , for any submodule of , implies is a pure a submodule of , In section two we give several properties of this type of submodules as every multiplication module contains a Pr-maximal submodule. Also, if N, K are non-zero submodule of such that 𝑁 ≀ 𝐾 if N is Pr-maximal in W then K is Pr-maximal in W and if N is Pr-maximal submodule of an R- module W and I be ideal of R, if [π‘π‘Š: 𝐼] is a proper submodule of W then [π‘π‘Š: I] is Pr-maximal submodule. we study the relation, among Pr-maximal (submodules and other related module), In section three we study Pr-maximal submodule, under the multiplication module and we check some condition under which Pr-maximal submodules and maximal submodules are equivalent. Every multiplication module contains a Pr-maximal submodule. Also, we have every cyclic R- module has Pr-maximal submodule. We found if W is a F-regular module then every submodule of W is Pr-maximal. 2. Preliminaries This section is going to review some well-known definitions in a algebraic theory. Definition 2.1 [3] A proper submodule of an R-module i s named maximal if such that namely = Definition 2.2 [1] A submodule of an -module is named pure -submodule if for each ideal of Definition 2.3 [6] β€œA submodule of an R-module is called weak maximal if π‘Š is F-regular R-module”. Lemma (2.4) [7] β€œIf : is an epimorphism and is pure submodule of , then ( ) is pure in .” Definition 2.5 [8] β€œAn R-module is named pure simple if and it has no pure, submodule except and Definition 2.6 [4] An R-module is called faithful if = Definition 2.7 [9,10] β€œAn R-module is said to be multiplication if for each submodule of there exists an ideal of such that = ”. Equivalently is a multiplication R-module if and only if for each submodule of , = [ ] . Proposition 2.8 [10] If is an R-module and has an, unique maximal submodule , then is called local module. 3. Pr-Maximal Submodules: In this section, the basic definitions and facts related to this work are recalled, which starts with the following definition. IHJPAS. 36 (4) 2023 361 Definition (3.1) A sound R-submodule of an R-module is named pure-maximal ( -maximal) submodule of if there exists with then is pure - submodule of W (𝐻 ≀𝑃 π‘Š). Remark and Example (3.2) 1. Every maximal submodule of R-module is Pr-maximal. Proof: Impose maximal R- submodule of R-module there exist 0 H submodule of such that since is maximal then but is pure of therefore is Pr-maximal the convers is not true as the following example in W= + as a Z-module Let, =2𝑍4 and H= such that 2 since H= is a summand of = , hence is pure in , then is Pr-maximal submodule of but is not maximal since . 2. A subset of Pr-maximal -submodule need not be Pr-maximal -submodule as the breech example in as Z-module impose 3𝑍12 < 𝑍12 ≀ 𝑍12 implies 3 is Pr-maximal submodule of since , but 6 is not Pr-maximal submodule of since 6 and is not pure in . 3. 𝑍4 as Z-module we have {0Μ…, 2Μ… }is Pr-maximal since 𝑍4 is pure of 𝑍4 and {0Μ…,2Μ…}< 𝑍4≀ 𝑍4. 4. Z6= {0Μ…,3Μ…} ⨁ { 0,Μ… 2Μ…,4Μ…}, 3𝑍6is Pr-maximal of Z6 since {0Μ…, 3Μ…}