Microsoft Word - 137-147 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 137          Fuzzy Semimaximal Submodules Abstract Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if M/N is a semisimple R-module. The main object of this work is to fuzzy this concept, study the basic properties and we investigate the sufficient conditions of F- submodules to be semimaximal. Also, the concepts of (simple, semisimple) F- submodules and quotient F- modules are introduced and given some properties. Keywords: fuzzy simple (semisimple) modules, fuzzy quotient modules, fuzzy semimaximal submodule. 1.Introduction Hatem in [1] introduced and studies semimaximal ideals of a ring and semimaximal submodules, where an ideal J of a ring R is called semimaximal if J is a finite intersection of maximal ideals [2]. We add many other results. Maysoun in [3] introduced the definition of fuzzy simple modules and fuzzy semisimple modules. Some properties of these concepts which are useful in next sections are given. Moreover, a submodule N of an ℛ module ℳ is said semimaximal if ℳ 𝑁 ⁄ is semismiple R- module. It is clear that every maximal ideal (submodule) is a semimaximal. In [4] , we fuzzify the concept semimaximal ideal where a fuzzy ideal ℋ is called a fuzzy semimaximal ideal if 𝐻 𝑖𝑠 s a finite intersection of fuzzy maximal ideals also we study many properties this concept. In this paper, we fuzzify the concept semimaximal submodules in to fuzzy semimaximal submodules. Also, we give many basic properties of this notion. Finally, (shortly fuzzy set, fuzzy submodule, fuzzy ideal and fuzzy module is F-set, F- submodule, F-ideal , and F-module). Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Maysoun A. Hamel Department of mathematics, Ibn-Al- Haitham College of Education, Baghdad University, Baghdad-Iraq mathmaysoon@gmail.com   Hatam Y. Khalaf Department of mathematics, Ibn-Al- Haitham College of Education, Baghdad University, Baghdad-Iraq dr.hatamyahya@yahoo.com Doi: 10.30526/33.4.2518 Article history: Received 5 December 2019, Accepted 12 January 2020, Published in October 2020   138  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 1. Preliminaries This section contains some definitions and properties of fuzzy set and fuzzy module. Definition 1.1 [5] Let S be a non-empty set and I be the closed interval 0,1 of the real line (real numbers). A F- set A in S (a F- subset of S) is a function from S in to I". Definition 1.2[6] Let 𝑥 : 𝑆 → 0,1 be are two F- set in S, where 𝑥 ∈ 𝑆 , 𝑡 ∈ 0,1 , defined by: 𝑥 𝑦 𝑡 𝑖𝑓 𝑥 𝑦 0 𝑖𝑓 𝑥 𝑦 for all 𝑦 ∈ 𝑆. , Then 𝑥 is called F- singleton . Definition 1.3[6] If 𝐴 an𝑑 𝐴 F sets in S , then: 1- 𝐴 𝐴 if and only if 𝐴 𝑥 𝐴 𝑥 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆 . 2- 𝐴 𝐴 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐴 𝑥 𝐴 𝑥 , 𝑓𝑜𝑟 𝑎𝑙𝑙𝑥 ∈ 𝑆.. 𝐼𝑓 𝐴 𝐴 𝑎𝑛𝑑 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑥 ∈ 𝑆 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴 𝑥 𝐴 𝑥 , 𝑡ℎ𝑒𝑛 𝐴 𝑖𝑠 𝑎 𝑝𝑟𝑜𝑝𝑒𝑟 𝐹 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵 𝑎𝑛𝑑 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝐴 𝐵 . By part (2), we can deduce that 𝑥 𝐴 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐴 𝑥 𝑡". Definition 1.4 [6],[7] Let ℳ be an R-modul𝑒 . A F- set ℋ of ℳ is F- module of an R-module M if : 1- ℋ 𝑥 𝑦 min ℋ 𝑥 , ℋ 𝑦 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ ℳ 2-ℋ 𝑟𝑥 ℋ 𝑥 𝑓𝑜𝑟𝑎𝑙𝑙 𝑥 ∈ ℳ 𝑎𝑛𝑑 𝑟 ∈ 𝑅. 3-ℋ 0 1" Definition 1.5[6] Let 𝑋 𝑎𝑛𝑑 𝑋 , be two F- modules of R-module ℳ . 𝑋 is said a F- submodule of 𝑋 , if 𝑋 𝑋 . Proposition 1.6 [6] Let A be a F- set of an R-module ℳ . Then the level subset A x ∈ M, A x t , t ∈ 0,1 is a submodule of ℳ if and only if A is a F- submodule of ℋ where ℋ is a F- module of an R-module ℳ Now, we shall give some properties of F- submodules , which are used in the next sections. Definition 1.7[ 6] Let A be a F-set of an R-module ℳ , then the submodule A 𝑜𝑓 ℳ is called the level submodule of ℳ , where 𝑡 ∈ 0,1 . " Proposition 1.8[7],[8] Let A be a F- module in ℳ , then we define 𝐴∗ 𝐴 𝑥 ∈ ℳ , 𝐴 𝑥 1 𝐴 0 = 1" Proposition 1.9[11] Let A be a fuzzy module of an R- module ℳ ,then 𝐴∗ is a submodule of ℳ ." . We add the following results : Proposition 1.10 If ℋ is a F- module of an R-module ℳ and ℕ ℋ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ℋ 0 1, 𝑡ℎ𝑒𝑛 ℕ 0 1. Proof: it is clear by the definition of F- module .   139  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Remark 1.11[11] If A and B are F- submodules of F- module X such that 𝐴 𝐵 𝑡ℎ𝑒𝑛 𝐴∗ 𝐵∗ Proof: Let 𝑥 ∈ 𝐴∗ , 𝑡ℎ𝑒𝑛 𝐴 𝑥 𝐴 0 . But B(x) 𝐴 𝑥 , ∀𝑥 ∈ 𝑀, ℎ𝑒𝑛𝑐𝑒 B x 𝐴 𝑥 𝐴 0 𝐵 𝑂 . 𝑇ℎ𝑢𝑠 𝑥 ∈ 𝐵 𝐵∗ Remark 1.12[11] The convers of the above Remark is not true in general as the following example shows : Let 𝑋: 𝑍 → 0,1 , 𝑑𝑒𝑓𝑖𝑛𝑒 𝑏𝑦 ∶ 𝑋 𝑥 1 , ∀ 𝑥 ∈ 𝑍 , Let 𝐴 𝑥 1 𝑖𝑓 𝑥 ∈ 4𝑍, 0.9 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝐵 𝑥 1 𝑖𝑓 𝑥 ∈ 2𝑍, 1 2 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 A and B are F- submodules X and 𝐴∗ 4𝑍 , 𝐵∗ 2Z Hence , 𝐴∗ 𝐵∗ . 𝐵𝑢𝑡 𝐴 𝐵 , 𝑠𝑖𝑛𝑐𝑒 𝐴 3 0.9 , 𝐵 3 1/2 0.5.x Remark We assume that if 𝐴∗ = 𝐵∗ . Then , 𝐴 𝐵 is called Condition (*) Maysoon in[11] introduced the following definition: Definition 1.13[11] Let X be a F- module of an R – module M , let A be F- submodule of X 𝐷𝑒𝑓𝑖𝑛𝑒 𝑋 𝐴:⁄ 𝑀 /𝐴∗ → 0,1 𝑏𝑦| : 𝑋 𝐴 𝑎 𝐴∗⁄ 1 𝑖𝑓 𝑎 ∈ 𝐴∗ sup 𝑋 𝑎 𝑏 𝑖𝑓 𝑏 ∈ 𝐴∗, 𝑎  𝐴∗ For all coset 𝑎 𝐴∗ ∈ 𝑀 𝐴∗" . P𝐫𝐨𝐩𝐨𝐬𝐢𝐭𝐢𝐨𝐧 𝟏 . 𝟏𝟒 𝟏𝟏 𝐼𝑓 𝑋 𝑖𝑠 𝑎 𝐹 𝑚𝑜𝑑𝑢𝑙𝑒 𝑜𝑓 𝑎𝑛 𝑅𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 𝑎𝑛𝑑 𝐴 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 𝑜𝑓 𝑋 , 𝑡ℎ𝑒𝑛 𝑋 𝐴⁄ 𝑖𝑠 𝑎 𝐹 𝑚𝑜𝑑𝑢𝑙𝑒 𝑜𝑓𝑀 /𝐴∗ . However , in [12 ] there exists a definition of quotient fuzzy module which is an equivalent to Definition 3.1 where 𝑋 0 1, as follows: Proposition 1.15[12] Let X be a F- module of an R- module ℳ and A be a F- submodule of X. Define 𝑋 𝐴: 𝑀 𝐴∗ ⁄ → 0,1 ⁄ , such that 𝑋 𝐴 a 𝐴∗ sup 𝑋 𝑎 𝑏 , 𝑎 ∈ ℳ, 𝑏 ∈ 𝐴∗ ⁄ Lemma 1.16[11] If A be F- submodule of F- module X , then 𝑋∗ ⁄ 𝐴∗ X/A ∗ . Proposition 1.17[11] Let X be a F- module of an R- module ℳ such that 𝑋 𝑥 1, ∀𝑥 ∈ ℳ. 𝑇𝑡ℎ𝑒𝑛 𝑋 𝐴⁄ ∗ 𝑋∗|𝐴 ∗ , 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝐴 𝑋. Proposition 1.18[11] If A , B are F- submodules of F- module X . Then 𝑋 𝐴 ∩ 𝐵⁄ 𝐹 submodules in 𝑋 𝐴 ⨁ 𝑋 𝐵⁄⁄ .   140  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Definition 1.19 [3] A F- submodule of F- module X is called pure if for each F-ideal K of R , 𝐾𝑋 ∩ 𝐴 𝐾𝐴. Defintion1.20 [3] A F-module X of an R-module M is called F-regular if every F-submodule of X is pure . Definition 1.21[13] Let A be a F- submodule of fuzzy module X is called an essential if 𝐴 ∩ 𝐵 0 , 𝑓𝑜𝑟 𝑛𝑜𝑛 𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝐹 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 𝐵 𝑜𝑓 𝑋 . Definition 1.22 [3] Let X be a F-module of an R- module M . X is called a multiplication F- module if and only if for each F- submodule A of X ,there exists a F- ideal K of R such that A= KX. 2.Fuzzy Simple (Semisimple) Modules Recall that an R-module M is called simple if and only if has no proper non trivial submodules [2] and " M is called semisimple if and only if M is sum of simple submodules of M [2]. Maysoon in[3] introduced the definition of fuzzy simple modules and fuzzy semisimple modules . Some properties of these concepts which are useful in next sections are given. Moreover we add many other results . Definition 2.1[3] A F- module X is called simple if X has no nontrivial F- submodules . In other words , X is simple if whenever 𝐴 𝑋 ,either 𝐴 𝑋 𝑜𝑟 𝐴 0 Moreover, let 𝐴 𝑋 , A is a F- simple submodule of X if A is a F- simply module . Remarks 2.2 [3] If X is a F- module , then the following are held: 1) Every simple F- module is F_ regular F- module ,where is F_ regular is every F- submodule of X is pure . 2) If X is a simple F- module, then 𝑋 is a simple module , ∀ t 0,1 . 3)If X is a simple module , ∀ t ∈ 0,1 ,then is not necessarily that X is a simple F module ." Proposition 2.3 [11] Let X be a F- module of an R-module M and A be a F- submodule of X if A is simple ,then A∗ is simple submodule in X∗. " Remark 2.4[11] If A∗ is a simple submodule in X∗, 𝑡ℎ𝑒𝑛 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝐹 𝑠𝑖𝑚𝑝𝑙𝑒 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 ". 𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 𝟐. 𝟓 [2] 𝐴 F module 𝑋 is called semisimple if 𝑋 is sum of simple F submodule of 𝑋 . Next , we need the following lemma: Lemma 2.6[11] If X is a F- module of an R- module M and A is a F -direct summand in X ,then 𝐴∗ 𝑖𝑠 𝑎 direct summand in 𝑋∗. Proposition 2.7[11] If 𝑋 is a F semisimple module , then X∗ is a semisimple module . Proposition 2.8[11] If X∗ is semisimple , then X semisimple when Condition ∗ hold."   141  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Corollary 2.9[11] Any F- submodule of semisimple fuzzy submodule is semisimple. Proposition 2.10[11] Ift X is a F- module of an R-module M ,where Condition (*) holds Then , the following statements are equivalent : 1- X is semisimple 2 - X has no proper essential F- submodule . 3- Every F- submodule of X is a direct summand of X ". Proposition 2.11 [11] If X is a F- module of an R –module M . Then , the following are equivalent: 1. Every fuzzy submodule of X is sum of fuzzy simple submodules. 2. X is a direct sum of fuzzy simple submodules of X . 3. Every fuzzy submodule of X is a direct summand of X Proof: It is easy . Proposition 2.12 [11] The following are equivalent: 1. Every F- submodle of semisimple F- module is semisimple. 2. Every epimorphic image of semisimple F- module is semisimple . 3. Every sum of semisimple F- modules is semisimple . Proof: It is easy . 3. Fuzzy Semimaximal Submodules Definition 3 .1 If A is a F- submodule of F- module X , then A is called semimaximal if and only if 𝑋 𝐴⁄ is a semisimple F- module. Proposition 3. 2 If A is a semimaximal fuzzy submodule of fuzzy module X , then 𝐴∗ is a semimaximal submodule of 𝑋∗ . Proof: Since A is semimaxmal F submodule , so X /A is semisimple . Hence ,  ∈^ 𝐶 | 𝐴 , where 𝐶 | 𝐴 is simple F- submodules ∀ 𝑖 ∈ ^ Which implies (XlA ∗  ∈^ 𝐶𝑖|𝐴 ∗  ∈^ 𝐶𝑖 |𝐴 ∗ 𝐵𝑢𝑡 𝐶 | 𝐴 𝑖𝑠 𝐹 𝑠𝑖𝑚𝑝𝑙𝑒 , 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝐶𝑖|𝐴 ∗ 𝑖𝑠 𝑠𝑖𝑚𝑝𝑙𝑒 , 𝑏𝑦 𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜 2 .3 ∀ 𝑖 ∈ ^ .Hence, (𝑋𝑙𝐴 ∗ 𝑖𝑠 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒 . 𝐴𝑠 𝑋∗|𝐴∗ (𝑋𝑙𝐴 ∗ 𝑏𝑦 𝐿𝑒𝑚𝑚𝑎 1.16 . 𝑇ℎ𝑒𝑛 , 𝑋∗|𝐴∗ 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐴∗ 𝑖𝑠 semimaxmal submodule of 𝑋∗ Proposition 3. 3 Let A be F- submodule of F- module X which satisfies X/A ∗ 𝑋∗ 𝐴∗⁄ . If 𝐴∗ is a semimaximal submodule of 𝑋∗ , then A is a F- semimaximal submodule of X . Proof: Since 𝐴∗ is a semimaxmal submodule of 𝑋∗ ℎ𝑒𝑛𝑐𝑒 , 𝑋∗ 𝐴∗⁄ 𝑖𝑠 semisimple Hence , 𝑋 𝐴⁄ is semisimple F module . Therefore 𝐴 is a F semimaximal submodule of X . Remarks and Examples 3.4 1) Every F- maximal submodule of fuzzy module is a F- semimaxmal submodule.   142  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Proof: A is a F- maximal submodule of F- module X .Then X /A is simple , and so X /A is a semisimple .Hence , A is semimaximal. The converse is not true in general see the following example : Example: Let 𝑀 𝑍 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 . 𝐷𝑒𝑓𝑖𝑛𝑒 𝑋 𝑥 1, ∀ 𝑥 ∈ 𝑀 𝐿𝑒𝑡 𝐴 𝑋 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴 𝑥 1 𝑖𝑓 𝑥 ∈ 6𝑍, 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Thus 𝑋∗ 𝐴∗⁄ 𝑀 𝐴∗⁄ 𝑍 6𝑍⁄ 𝑍 𝑖𝑠 semisemple and A is semimaximal. But 𝐴∗ is not a maximal 𝑖𝑛 𝑋∗ . Thus A is not maximal F- submodule by Prop 3.3.Hence X/𝐴 ∗ M / 𝐴∗ 𝑍 is semisemple , and s𝑜 𝑏𝑦 𝑃𝑟𝑜𝑝 2.8 , X/𝐴 is semisemple ; that is 𝐴 is a F semimaxmal submodule 2) A F- submodule of semimaximal F-module need not to be semimaximal . For example : Let A , B ≤ X , such that 𝐴 𝑥 1 𝑖𝑓 𝑥 ∈ 2 1/2 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝐵 𝑥 1 𝑖𝑓 𝑥 ∈ 8 1 3⁄ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , So 𝑋∗ / 𝐴∗ 𝑍 / 2 𝑍 ∗ 𝑖𝑠 semisimple . So 𝐴∗ is semimaximal .But 𝑋∗ 𝐵∗ 𝑍 / 8 𝑍 ⁄ , 𝑖𝑠 𝑛𝑜𝑡 semisimple. Hence, 𝐵∗ is not semimaximal submodule . Thus B is not semimaximal F- submodule by Proposition 3.3. 3) If A and B are F- submodules of F- module X such that A 𝐵 𝑋 𝑎𝑛𝑑 𝐴 is semimaximal in X .Then B is semimaximal in X . Proof: Since A is a fuzzy semimaximal submodule in X , 𝑋 𝐴⁄ is semisimple . Hence , 𝑋 𝐴⁄ 𝑋 𝐵⁄⁄ is semisimple image of semisimple Proposition 2.12 By the second isomorphism theorem[14] , 𝑋 𝐴⁄ 𝑋 𝐵⁄⁄ 𝑋 𝐵⁄ is semisimple . Thus , B is a fuzzy semimaximal submodule . 4- Let A and B be two F- submodules of F- module X . If is A a semimaximal F -submodule of X , then 𝐴 𝐵 is also semimaximal F- submodule of X. Proof : Clearly 𝐴 𝐴 𝐵 and , hence the result follows directly. 5) Let {𝐴 1,2,3, … 𝑛} be a finite collection of semimaximal F- submodules of F- module X. Then 𝐴 , 𝑖 1,2,3 … 𝑛 is a semimaximal F- submodule. Proof: It is clear by 4 6) Let A and B be two F- submodules of F- module X such that A 𝐵 . If A is semimaximal in B and B is semimaximal in X , then A is not necessary semimaximal of X ,as the following example shows : Example: Take 𝑀 𝑍 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 ,let 𝑋 ∶ 𝑀 → 0,1 , 𝑑𝑒𝑓𝑖𝑛𝑒 𝑋 𝑥 1, ∀𝑥 ∈ 𝑀, 𝐴 𝑥 1 𝑖𝑓 𝑥 ∈ 9 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒   143  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 𝐵 𝑥 1 𝑖𝑓 𝑥 ∈ 3 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑋 𝐵⁄ ∶ 𝑍 3⁄ ≅ 𝑍 → 0,1 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑋 𝐵⁄ 𝑥 1, ∀𝑥 ∈ 𝑍 𝑋 𝐴⁄ ∶ 𝑍 9 ⁄ ≅ 𝑍 → 0,1 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑋 𝐴⁄ 𝑥 1, ∀𝑥 ∈ 𝑍 𝑁𝑜𝑤 , 𝑋 /𝐵 ∗ 𝑍 𝑖𝑠 𝑠𝑖𝑚𝑝𝑙𝑒 , 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑋∗ 𝐵∗⁄ 𝑍 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝐵 𝑖𝑠 𝑎 𝐹 𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 , 𝑏𝑦 𝑃𝑟𝑜𝑝 3.3 𝐵𝑢𝑡 𝑋∗ 𝐴∗ 𝑍 𝑖𝑠 𝑛𝑜𝑡 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 . 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐴∗ 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 𝑖𝑛 𝑋∗ 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑖𝑛 𝑋. 𝑆𝑖𝑛𝑐𝑒 𝐴 𝐵 𝑎𝑛𝑑 𝐵 𝐴: 𝑍 / 9 → 0,1 ⁄ ∶ 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 ∶ B A x⁄ 1 if x ∈ 3 / 9 ≅ Z 0 otherwise Now , 𝐵 /A ∗ 3 / 9 ≅ 𝑍 .But 𝐵∗ 𝐴∗ 3 / 9 ≅ 𝑍⁄ 𝑖𝑠 𝑠𝑖𝑚𝑝𝑙𝑒 . Then 𝐴∗ is semimaximl in 𝐵∗ . Proposition 3.5 Let 𝐴 be a semimaximal F- submodule of F- module 𝑋 , 𝑖 1,2,3, … … 𝑛 Then 𝐴 is a semimaximal F- submodule of F- module  𝑋 , 𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 X/A ∗ 𝑋∗ 𝐴∗⁄ . 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝐹 𝑚𝑜𝑑𝑢𝑙𝑒 𝑋 𝑎𝑛𝑑 𝐴 𝑋, ∀ 𝑖 1,2,3 … . 𝑛. Proof: Since 𝐴 is a semimaximal F- submodule of F- module 𝑋 , ∀𝑖 1,2,3, … . 𝑛. 𝑡ℎ𝑒𝑛 𝑋 𝐴⁄ , is F- 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒 𝑚𝑜𝑑𝑢𝑙𝑒 , 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖 1,2,3 … . 𝑛. Hence  𝑋 /𝐴 isF-semisimple module and so by Remarks and Examples 3.4(5)  ∗ is a semisimple . B ut  𝑋 𝐴 ∗  𝑋 𝐴 ∗ by 2  𝑋 ∗ 𝐴 ∗  𝑋 ∗  𝐴 ∗  𝑋 ∗ 𝐴 ∗ Hence  𝐴 ∗ 𝑖𝑠 𝑎 semimaximal submodule in  𝑋 ∗ . By hypothesis   ∗  ∗  ∗ Hence  𝐴 is a F- semimaximal submodule in  𝑋 . Remark 3.6 If X is a F- module of an R- module M ,then is not necessary that X has semimaximal F- submodule , for example . Example : Let 𝑀 𝑍 𝑝 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 , let 𝑋 ∶ 𝑀 → 0,1 , 𝑑𝑒𝑓𝑖𝑛𝑒 𝑏𝑦 𝑋 𝑥 1 , ∀ 𝑥 ∈ 𝑀 . Assume X has a semimaximal submodule say A . Let 𝐴∗ N is a a semimaximal submodule in 𝑋∗ 𝑍 𝑏𝑦 𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 3.2 ; 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑋∗/𝑁 𝑍 /𝑁 is semisimple . But ≅ 𝑍 , 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑍 𝑖𝑠 semisimple, which is a contradiction .   144  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Thus A is not semimaximal F- submodule of X . The following proposition give sufficient condition but not necessary condition for F- submodule to be semimaximal. Proposition 3.7 Let A be F- submodule of fuzzy module X such that A is intersection of a finite number of F- maximal submodules of X . Then , A is a F- semimaximal submodule . Proof : Let 𝐴 𝐴 ∩ 𝐴 ∩ … … … ∩ 𝐴 , 𝐴 𝑖𝑠 𝑎 𝐹 submodule of X , ∀i 1,2, … … … . n. Hence,byProposition1.18, 𝑋 𝐴⁄ ≅ 𝐹 𝑠𝑢𝑏𝑚𝑑𝑢𝑙𝑒 𝑜𝑓 𝑋|𝐴  𝑋|𝐴 … … … … . .  𝑋|𝐴 But is simple for each 𝑋/ 𝐴 , so 𝑋/ 𝐴  𝑋 / 𝐴 … … … … . . 𝑋| /𝐴 is semisimple and since a F- submodule of fuzzy semisimple is F- semisimple, Therefore 𝑋 𝐴⁄ is a F- semisimple module Thus A is F- semimaxmal submodule . Remark 3.8 The converse of Proposition 3.7 is not true in general . We can give the following example : Let 𝑀  𝑍 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 , 𝑝 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 Define 𝑋 ∶ 𝑀 → 0,1 and 𝐴: 𝑀 → 0,1 by 𝑋 𝑥 1 , ∀𝑥 ∈ 𝑀 𝑎𝑛𝑑 𝐴 𝑥 1, ∀x ∈ 𝑍 But 𝑋 𝐴⁄ 𝑥 𝐴∗ 1 𝑖𝑓 𝑥 ∈ 𝐴∗ sup 𝑋 𝑥 𝑏 𝑖𝑓 𝑏 ∈ 𝐴∗, 𝑥 𝐴∗ Thus 𝑋 𝐴⁄ 𝑥 𝐴∗ 1, ∀ x 𝐴∗ ∈ 𝑀|𝐴∗  𝑍 , 𝑝 2, is semisimple . Thus 𝐴 is 𝑎 F semimaxmal submodule . But A is not a finite intersection of F maximal submodule . Proposition 3.9 If X is a F- module of an R- module M and R is a semisimple ring ,then every F- submodule A of X is semimaximal .Provided 𝑋𝑙𝐴 ∗ 𝑋∗|𝐴∗ Proof : Let R be semisimple ring . Then M is a semisimple R-module .Since X is a F- module over M , then 𝑋∗ 𝑀 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝑋∗ is semisimple which implies that 𝑋∗ / 𝐴∗ 𝑖𝑠 semisimple. 𝑇ℎ𝑢𝑠 𝐴∗ 𝑖𝑠 𝑎 semimaximal submodule of 𝑋∗. Then by Proposition 3.4 𝐴 is a F semimaximal submodule of X . Proposition 3.10 The intersection of two semimaximal F- submodules is aslo semimaximal. Proof : If A , B are two semimaximal F- submodules of module X .Then 𝑋 𝐴 𝑎𝑛𝑑 𝑋 𝐵⁄⁄ are semisimple . Therefore , 𝑋 𝐴 ⨁ 𝑋 𝐵⁄⁄ is semisimple . As 𝑋 𝐴 ∩ 𝐵⁄ ≅ F subodule of 𝑋 𝐴 ⨁ 𝑋 𝐵⁄⁄ . But any F submodule of 𝑋 𝐴 ⨁ 𝑋 𝐵⁄⁄ is semisimple , hence 𝑋 𝐴⋂𝐵⁄ is semisimple Therefore 𝐴 ∩ 𝐵 is a F -semimaximal submodule of X .   145  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Proposition 3.11 If A , B are two F- submodules of F- module X such that A is semimaximal in X and B contains A . Then B is semimaximal in X . Proof: 𝐵 𝐴⁄ is F- submodule of 𝑋 𝐴⁄ since A B and 𝑋 𝐴⁄ ≅ 𝑋 𝐵⁄ , by third isomorphism theorem. But X /A is semisimple , imples 𝑋 /𝐴 / 𝐵 /𝐴 is semisimple . That is 𝑋 / 𝐵 is semisimple . Therefore B is semimaximal in X . The following Corollary immediately consequence of Proposition 3.11 Corollary 3.12 If Å is a semimaximal F- submodule of F- module X and K be a F- ideal of a ring R . Then Å: 𝐾 is a F- submodule semimaximal. Proof: Since Å: 𝐾 is a fuzzy submodule of X containing Å , then result follows by Proposition 3.11 However the converse of Corollary 3.12. is not true in general ,for example Example 3.13 Consider 𝑀 𝑍 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 𝑎𝑛𝑑 𝑙𝑒𝑡 𝑋 ∶ 𝑀 → 0,1 , 𝐴 ∶ 𝑀 → 0,1 , 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦: 𝑤ℎ𝑒𝑟𝑒 𝑋 𝑥 1 , ∀𝑥 ∈ 𝑀 , 𝐴 𝑥 1 𝑖𝑓 𝑥 ∈ 0 , , 4 , 8 , 1 2⁄ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 A is a F- submodule of X . Let 𝐾: 𝑍 → 0,1 defined by: 𝐾 𝑥 1 𝑖𝑓 𝑥 ∈ 2𝑍 1|3 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 K is a F- ideal of Z , then 𝑋 𝐴 𝑥⁄ 1 𝑖𝑓 𝑥 ∈ 𝑍 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Note that ∗ ∗ 𝑍 / 4 𝑍 is not semisimple . Hence 𝐴∗ is not semimaximal . Thus A is is not semimaximal Proposirion 3.4. However ,[ 𝐴∗: 𝐾∗ 4 : 2𝑍 2 is a semimaximal submodule of 𝑍 . But A ∶ K ∗ 𝐴∗ ∶ 𝐾∗ 𝑏𝑦 13 . Therefore 𝐴: 𝐾 is a semimaximal F- submodule of X . Proposition 3.14 If A and B are two F- submodules of F- module X such that 𝐵 𝐴 Then , 𝐴 is semimaximal in 𝑋 if and only if 𝐴 𝐵⁄ is a semimaximal F- submodule of 𝑋 𝐵⁄ . Proof : If A is semimaximal in 𝑋 .Then 𝑋 𝐴⁄ is semisimple ,which implies is semisimple , since 𝑋 𝐴⁄ 𝑋 /𝐵 / 𝐴 / 𝐵 .by [second isomorphism theorem [14] Hence 𝐴 / 𝐵 is a semimaximal F- submodule of 𝑋 /𝐵 .The converse is smilarly.   146  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Proposition 3.15 If K is semimaximal F- ideal of a ring R and X is be a F- module of R- module M , then KX is a semimaximal F- submodule , provided ∗ X∗ / K∗ X∗ Poorf: Suppose K is semimaximal F- ideal a ring R , then 𝐾∗ is a semimaximal ideal by [4] . Hence by[1] K∗ X∗ is a semimaximal submodule and so KX ∗ 𝑖𝑠 𝑎 semimaximal submodule of 𝑋∗. Thus KX is a F-semimaximal submodule. Remark 3.16 The following example shows that the converse of Proposition 3.15 is not true in general, Consider 𝑀 𝑍 𝑎𝑠 𝑍 𝑚𝑜𝑑𝑢𝑙𝑒 , 𝑑𝑒𝑓𝑖𝑛𝑒 𝑋: 𝑀 → 0,1 𝑏𝑦 𝑋 𝑥 1 , ∀𝑥 ∈ 𝑀 𝐷𝑒𝑓𝑖𝑛𝑒 𝐾: 𝑍 → 0,1 , 𝑏𝑦: 𝐾 𝑥 1 𝑖𝑓 𝑥 ∈ 4𝑍 , 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 X is a F- module and K is F- ideal of Z (it is easy to prove ) . Note that X∗ 𝑍 and K∗ 4𝑍 which is not is semimaximal ideal . Since , 𝐾𝑋 ∗ 𝐾∗ 𝑋∗ .Then 𝐾𝑋 ∗ 4𝑍 𝑍 0′, 2′, 4′ 2 and so . 𝑋/𝐾𝑋 ∗ 𝑍 / 2 is simple Thus, KX ∗ is a maximal submodule in X∗. On the other hand , by Proposition 1.17 𝑋/𝐾𝑋 ∗ 𝑋∗ 𝐾𝑋 ∗⁄ Thus 𝐾𝑋 is a semimaximal F submodule of X . Corollary 3.17 X is be a multiplication F- module of an R module - M . If every F- ideal K in a ring R is semimaximal , then every F- submodule of X is semimaximal. Provided that 𝑋 /𝐴 ∗ 𝑋∗ /𝐴∗ , ∀ 𝐴 𝑋 poorf: It is directly from Proposition 3.15 . Proposition 3.18 Every epimorphic image of semimaximal F- submodule is a semimaximal F- submodule . Proof: Let 𝑔 ∶ 𝑋 𝐴⁄ → 𝑌 / 𝑓 𝐴 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑔 𝑥 𝐴 𝑦 𝑓 𝐴 , ∀𝑦 ∈ 𝑌. 𝑔 is well-defined and 𝑔 is an epimorphism . Since 𝑋 𝐴⁄ is semisimple . Hence , 𝑔 𝑋 𝐴 𝑌 𝑋 𝐴⁄ 𝑌⁄ is semisimple . Thus , 𝑓 𝐴 is a semimaximal fuzzy submodule. Proposition 3.19 Let X be a finitely generated F- R- module M such that 𝐹 𝑎𝑛𝑛 𝐴 is a semimaximal F- ideal , . ∀ 𝐴 𝑋 . If X /A ∗ ∗ ∗ , ∀ A 𝑋 , Then every F- submodule of X is semimaximal. Proof: Since X is a F- finitely generated R- module , then 𝑋∗ is finitely generated R- module [15]. Since F annA ∗ 𝑎𝑛𝑛 𝐴∗ 𝑎𝑛𝑑 F 𝑎𝑛𝑛𝐴 𝑖𝑠 semimaximal F-ideal ,implies F 𝑎𝑛𝑛𝐴 ∗ is semimaximal 𝑏𝑦 4 , 𝑠𝑜 𝑎𝑛𝑛 𝐴∗ 𝑖𝑠 semimaximal ideal . Then by 1 , every submodule of 𝑋∗ 𝑖𝑠 semimaximal.   147  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Hence , 𝐴∗ 𝑖𝑠 semimaximal . But this implies A is semimaximal , since X /A ∗ ∗ ∗ References 1. Hatem, Y. K., semimaximal Modules, ph. D. Thesis, University of Baghdad,2005. 2. kasch , F. modules and Rings ,Academic press, 1982. 3. Maysoun A. 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