https://alqalam.utripoli.edu.ly/science/ eISSN 2707-7179

Review Article

Approximations of Maximal and Principal

Faraj Abdunanbi

Department Mathematics, University of Ajdabiya, Libya.

ARTICLE INFO ABSTRACT https://doi.org/10.5281/zenodo.4 In this paper, we will be delving deeper into the connection between the 266815 rough theory and the theory precisely in the principle and maximal ideal. The rough set theory has shown by Pawlak as good formal tool for *Faraj Abdunanbi: Department modeling and processing incomplete information in information system. Mathematics, University of The rough theory is based on two concepts the upper approximation of a Ajdabiya, Libya, given set is the union of all the equivalence classes, which are subsets of the [email protected] set, and the lower approximation is the union of all the equivalence classes, Received: 25-10-2020 which are intersection with set non-empty. Many researchers develop this Accepted: 09-11-2020 theory and use it in many areas. Here, we will apply this theorem in the one Published: 10-11-2020 of the most important branches of mathematics that is . We will try to find the rough principal and maximal ideal as an extension of the Keywords: Lower Approximation, notion of a principal maximal ideal respectively in ring theory. In addition, Upper Approximation, Ring, Ideal. we study the properties of the upper and lower approximation of a

This work is licensed under the principal maximal ideal. However, some researchers use the rough theory Creative Commons Attribution in the group and ring theory. Our work, is Shaw there are rough maximal International License (CC BY and principle ideal as an extension of the maximal and principle ideal 4.0). respectively. Our result will introduce the rough maximal ideal as an extended notion of a classic maximal ideal and we study some properties

of the lower and the upper approximations a maximal ideal.

Cite this article: Abdunanbi F. Approximations of Maximal and . Alq J Med App Sci. 2021;4(1):7-11.

INTRODUCTION rough theory in the group and ring theory. For example, the notation of rough subring with respect The rough set theory has shown by Pawlak [1] in ideal has presented by B. Davvaz [2]. Algebraic 1982. It is a good formal tool for modeling and properties of rough sets have been studied by processing incomplete information in information Bonikowaski [3]. John N. Mordeson [4], has used system. In recently 40 years, some researchers develop covers of the universal set to defined approximation this theory and use it in many areas. The upper operators on the power set of the given set. Some approximation of a given set is the union of all the concept lattice in Rough set theory has studied by Y.Y. equivalence classes, which are subsets of the set, and Yao [5]. Ronnason Chinram, [6] studies rough prime the lower approximation is the union of all the ideas and Rough fuzzy prime ideals in gamma- equivalence classes, which are intersection with set semigroups. Some other substitute an algebraic non-empty. Many researchers develop and use the

Faraj Abdunanbi. Alq J Med App Sci. 2021;4(1):7-11 7

https://alqalam.utripoli.edu.ly/science/ eISSN 2707-7179 structure instead of the universe set. Like Biswas and objects which can be with certainty classified as Nanda [7], they make some notions of rough members of X with respect to R is called the R-lower subgroups. Kuroki in [8], introduced the notion of a approximation of a set X with respect to R. rough ideal in a semi group. Some properties of the Also, Kuroki and Mordeson in [9] studied the Definition 2.2: For an approximation space (U, R) ,we define the boundary region by 퐵푋 = ̅푹̅̅푋̅ − 푹푋. structure of rough sets and rough groups. S. B 푅 i.e . 퐵푋 is the set of all objects which can be decisively Hosseinin [10], he introduced and discussed the 푅 classified neither as members of X nor as the members concept of T-rough (prime, primary) ideal and T- c of X with respect to R. If 퐵푋푅 = ∅ , we say X is exact rough fuzzy (prime, primary) ideal in a commutative (crisp) set . But if 퐵푋푅 ≠ ∅ , we say X Rough set ring. In addition, B. Davvaz [11] applied the concept ( inexact). We have Properties of approximations as: of approximation spaces in the theory of algebraic 1) 푅푋 ⊆ 푋 ⊆ 푅푋̅̅̅̅̅ hyperstructures, and in investigated the similarity 2) 푅∅ = 푅̅̅̅∅̅̅ , 푅푈 = 푅푈̅̅̅̅̅ , between rough membership functions and 3) 푅(푋 ∪ 푌) ⊇ 푅(푋) ∪ 푅(푌), conditional probability. In this paper, we shall 4) 푅(푋 ∩ 푌) = 푅(푋) ∩ 푅(푌), introduce the maximal ideal. Our result will introduce 5) ̅푅̅̅(̅푋̅̅̅∪̅̅̅푌̅̅)̅ = 푅̅̅̅(̅푋̅̅)̅ ∪ 푅̅̅̅(̅푌̅̅)̅ the rough maximal ideal as an extended notion of a 6) ̅푅̅̅(̅푋̅̅̅∩̅̅̅푌̅̅)̅ ⊆ 푅̅̅̅(̅푋̅̅)̅ ∩ 푅̅̅̅(̅푌̅̅)̅ classic maximal ideal and we study some properties of 7) 푅̅̅푋̅̅̅퐶̅ = ( 푅푋)푐 the lower and the upper approximations a maximal 8) 푅푋퐶 = (푅푋̅̅̅̅̅ )푐 ideal. 9) 푅( 푅푋) = 푅̅̅̅(̅푅푋̅̅̅̅̅) = 푅푋 ̅̅̅̅̅̅̅̅̅̅̅ 10) ( 푅( 푅푋) = 푅( 푅푋) = 푅푋̅̅̅̅̅ PRELIMINARIES

Suppose that U (universe) be a nonempty finite set. Example 2.1: Let us consider set of objects U ={ x1, x2, Let R an equivalence relation (reflexive, symmetric, and x3, x4, x5, x6, x7 }, and the Equivalence relation R = {{ x1}, transitive) on an U. Some authors say R is { x2}, { x3, x4 }, { x5, x7 }, { x6 }}, and let X= { x1, x4, x6 }. Now, ̅̅̅̅ indiscernibility relation. The discernibility relation the upper approximations of X: 푹푋 = {푥 ∈ 푈: [푥]푅 ∩ represents our lack of knowledge about elements of U. 푋 ≠ ∅}. So ̅푹̅̅푋̅ = { x1,x3,x4,x6} and the lower For simplicity, they assume R an equivalence relation. approximation of X :푅푋 = {푥 ∈ 푈: [푥]푅 ⊆ 푋 }.So 푅푋 = ̅̅̅̅ We use U/R to denote the family of all equivalent { x1, x6 }. The boundary region by 퐵푋푅 = 푅푋 − 푅푋 .So, classes of R (or classifications of U), and we use [x]R to 퐵푋푅= { x3, x4 }. Then 퐵푋푅 ≠ ∅, so X is Rough set . denote an equivalence class in R containing an element xU. The pair (U, R) is called an Now, we define the ideal and maximal idea. Then approximation space. The empty set  and the we will study the upper and lower approximations elements of U/R are call elementary sets. For any XU, ideal. We suppose we have a ring ℛ and I be an Ideal we write Xc to denote the complementation of X in U. of a ring ℛ , and X be a non-empty subset of ℛ.

Definition 2.1: For an approximation space (U, R), we Definition 2.3: Let I be an Ideal ofℛ; For a,b  ℛ we define the upper approximation of X by ̅푹̅̅푋̅ = {푥 ∈ say a is congruent of b mod I, we express this fact in 푈: [푥]푅 ∩ 푋 ≠ ∅};i.e. ̅푹̅̅푋̅ the set of all objects which can symbols as 푎 ≡ 푏(푚표푑 퐼)𝑖푓 푎 − 푏 ∈ 퐼 …….……….(1) be only classified as possible members of X with respect to R is called the R-upper approximation of a set X with Not that, it easy to see the relation (1) is an respect to R. And the lower approximation of X equivalents relation. Therefore, when we let U= ℛ and by 푹푋 = {푥 ∈ 푈: [푥]푅 ⊆ 푋 }. i.e 푹푋 is the set of all we suppose a relation R is the equivalents relation (1),

Faraj Abdunanbi. Alq J Med App Sci. 2021;4(1):7-11 8

https://alqalam.utripoli.edu.ly/science/ eISSN 2707-7179 so we can defined the upper and lower In this section, we introduce the maximal ideal and approximation of X with respect of I as:̅퐼̅(̅푋̅̅̅) =∪ { 푥 ∈ we study some properties of upper and lower ℛ ∶ (푥 + 퐼) ∩ 푋 ≠ ∅} , 퐼(푋) =∪ { 푥 ∈ ℛ ∶ 푥 + 퐼 ⊆ 푋} , maximal ideal. Let consider the example2-4: respectively. Moreover, the boundary of X with respect of I is BX =̅퐼̅(̅푋̅̅̅) − 퐼(푋). If BX=∅ we say X is Example 3.1. Let consider the example2-4 ℛ = ℤ12 , M Rough set with respect I. For any approximation space = {0, 3, 6, 9}. X={1,2,6,7,9} For ∈ ℛ ∶ 푥 + 푀 , we get (U,R) by rough approximation on (U,R), we mean a {0,3,6,9}, {1,4,7,10},{2,5,8,11}. Now, the upper ̅̅̅̅̅̅̅ mapping Apr(X): 푝(푈) → 푃(푈) × 푃(푈) defined by for approximations of X with respect of M: 푀(푋) =∪ { 푥 ∈ ℛ ∶ (푥 + 푀) ∩ 푋 ≠ ∅} {0,1,2,3,4,5,6,7,8,9,10,11} all 푥 ∈ 푃(푈), Apr(X)= (̅퐼̅(̅푋̅̅̅), 퐼(푋)), where ̅퐼̅(̅푋̅̅̅) = { 푥 ∈ , = and lower approximation of X with respect of M= 푀(푋) = ℛ ∶ (푥 + 퐼) ∩ 푋 ≠ ∅}, 퐼(푋) = { 푥 ∈ ℛ ∶ 푥 + 퐼 ⊆ 푋}. ∪ { 푥 ∈ ℛ ∶ 푥 + 푀 ⊆ 푋} , So, 퐼(푋) = ∅ because no

푀(푋) Definition2.4: An ideal M in a ring ℛ we called element satisfy the definition of . Moreover, BX ̅̅̅̅̅̅ maximal if M ≠ ℛ and the only ideal strictly containing = 퐼(푋) − 퐼(푋) = {0,1,2,3,4,5,6,7,8,9,10,11}. Then, X is M is ℛ. rough set with respect M.

Definitions 2.5: Let ℛ be a with Example 3.2. Let us consider the ring ℛ = ℤ6 . identity. Let S be a subset of ℛ . The ideal generated Suppose let maximal ideals is M={0,2,4} and by S is the subset < S > = {r1s1 + r2s2 + ... X={1,2,3,4,5}. For ∈ ℛ ∶ 푥 + 푀, we get {0,2,4}, {1,3,5}. + rksk ∈ ℛ | r1 , r2 , ... ∈ ℛ , s1 , s2 , ... ∈ S, k ∈ N}.In The upper approximations of X with respect of M: {0, particular, if S has a single element s this is called 2, 4} ∪{1, 3, 5}.푠표, ̅푀̅̅(̅푋̅̅̅) = {0,1,2,3,4,5} and the lower the principal ideal generated by s. That is, < s > = approximation of X with respect of M :푀(푋) = {1,3,5}. {rs | r ∈ ℛ }. BX =푀̅̅̅(̅푋̅̅̅) − 푀(푋) = {0,2,4}. Then X is rough set with respect maximal ideal M. Examples 2.2: The ideal 2 ℤ of ℤ is the principal ideal <2>. We can study the properties of maximal ideal in next proposition: Examples 2.3: In ℤ, the ideal < 5 > is maximal. For suppose that I is an ideal of ℤ properly containing <5>. Proposition3-1: For every approximation (ℛ,M) and Then there exists some m ∈ I with m  < 5 >, i.e. 5 does Every subset A,B  ℛ we have: not divide m. Then gcd(5, m) = 1 since 5 is prime, and 1) 푀(퐴) ⊆ 퐴 ⊆ 푀̅̅̅(̅̅퐴̅̅); we can write 1 = 5x + my for integers x and y. Since 5x∈ 2) 푀(∅) = ∅ = 푀̅̅̅(̅̅∅̅̅); I and my ∈ I, this means 1 ∈ I. Then I = ℤ, and < 5 >, is 3) 푀(ℛ) = ℛ = 퐼̅̅(̅ℛ̅̅̅); a maximal ideal in ℤ. Note that the maximal ideals in ℤ are precisely the ideals of the form < p >, where p is 4) 푀(퐴 ∩ 퐵) = 푀(퐴) ∩ 푀(퐵); prime. 5) 푀̅̅̅(̅퐴̅̅̅∪̅̅̅퐵̅̅)̅ = ̅푀̅̅(̅̅퐴̅̅)̅ ∪ ̅푀̅̅(̅̅퐵̅̅)̅ 6) If A B , then 푀(퐴) 푀(퐵), 푎푛푑 ̅푀̅̅(̅̅퐴̅̅) ⊆ 푀̅̅̅(̅퐵̅̅̅); Proposition 1.4 Let ℛ be a commutative ring with 7) 푀(퐴 ∪ 퐵) ⊇ 푀(퐴) ∪ 푀(퐵); identity. Then every maximal ideal of ℛ is prime. 8) 푀̅̅̅(̅퐴̅̅̅∩̅̅̅퐵̅̅)̅ ⊆ ̅푀̅̅(̅̅퐴̅̅)̅ ∩ ̅푀̅̅(̅̅퐵̅̅)̅ 9) 푀̅̅̅(̅̅A̅̅)̅ = ( 푀(퐴퐶)푐 Example 2.4. For ℛ = ℤ12, two maximal ideals are M1 = 10) 푀(퐴) = (푀̅̅̅̅(̅퐴̅̅퐶̅)푐 {0, 2, 4, 6, 8, 10} and M2 = {0, 3, 6, 9}. Two other ideals, 11) 푀( 퐼(퐴) = 푀̅̅̅(̅푀̅̅̅(̅퐴̅̅)̅̅) = 푀(퐴) which are not maximal are {0, 4, 8} and {0, 6}. ̅̅̅̅̅̅̅̅̅̅̅̅̅ 12) ( 푀(푀(퐴) = ( 푀(퐴) = ̅푀̅̅(̅̅퐴̅̅)̅ UPPER AND LOWER MAXIMAL IDEALS

Faraj Abdunanbi. Alq J Med App Sci. 2021;4(1):7-11 9

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Proof: 8) 퐼̅̅(̅퐴̅̅̅∩̅̅̅퐵̅̅)̅ ⊆ 퐼̅̅(̅퐴̅̅̅)̅ ∩ 퐼̅̅(̅퐵̅̅̅)̅ 1) If x 푀(퐴) , then x 푀(퐴) = { 푥 ∈ ℛ ∶ 푥 + 푀 ⊆ 9) ̅퐼̅(̅A̅̅)̅ = ( 퐼(퐴퐶)푐 퐴, then x A,Hence 푀(퐴) ⊆ 퐴, next if x  A, 10) 퐼(퐴) = (퐼̅̅(̅퐴̅̅퐶̅)푐 ̅푀̅̅(̅̅퐴̅̅) = { 푥 ∈ ℛ ∶ (푥 + 푀) ∩ 퐴 ≠ ∅} ,, then x 11) 퐼( 퐼(퐴) = 퐼̅̅(̅퐼̅̅(̅퐴̅̅)̅̅) = 퐼(퐴) 푀̅̅̅(̅̅퐴̅̅) ⊆ 푀̅̅̅(̅̅퐴̅̅) then A . 2) And 3) it easy to see that. ̅̅̅̅̅̅̅̅̅̅ 12) ( 퐼(퐼(퐴) = 퐼( 퐼(퐴) = 퐼̅̅(̅퐴̅̅̅)̅ 4) If x 푀(퐴 ∩ 퐵) , then x 푀(퐴 ∩ 퐵) = { 푥 ∈ ℛ ∶ 13) 퐼(푥 + 퐼) = 퐼̅̅(̅푥̅̅+̅̅̅퐼̅̅) 푓표푟 푎푙푙 푥 ∈ ℛ. 푥 + 푀 ⊆ 퐴 ∩ 퐵} , then x  ℛ ∶ 푥 + 푀 ⊆ 퐴 푎푛푑 푥 + 푀퐵 , then then x 푀(퐴) ∩ 푀(퐵). Definition 4.1. If A and B are anon-empty subset ofℛ, 5) It says way in 4) we denote AB for the set of all finite sums { a1 b1 + a2 6) Since A B, then A∩B=A, by 4) then 푀(퐴) ∩ 푛 b2 ,…, an bn :n ℕ, ai A, bi B}.i.e: AB=∑푖=1(푎푖 . 푏푖), ai 푀(퐵); It implies̅푀̅̅(̅̅퐴̅̅) ⊆ 푀̅̅̅(̅퐵̅̅̅), also, by 5) we A, bi B. ̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅ get 푀(퐴) ⊆ 푀(퐵). Proposition 4-2: Let I be maximal or principal Ideal 7) Since A  A∪ B, B  A∪ B , by 6) we get ofℛ, and A, B are non-empty subset of the ringℛ, then ̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅̅̅ 푀(퐴) 푀(퐴 ∪ 퐵), 푎푛푑 푀(퐵) ⊆ 푀(퐴 ∪ 퐵) ; 1) 퐼̅̅(̅퐴̅̅̅) + 퐼̅̅(̅퐵̅̅̅) = 퐼̅̅(̅퐴̅̅̅+̅̅퐵̅̅̅) 푤𝑖푐ℎ 푦𝑖푒푙푑푠 푀(퐴 ∪ 퐵) ⊇ 푀(퐴) ∪ 푀(퐵); 2) ̅퐼̅(̅퐴̅̅.̅퐵̅̅̅) = 퐼̅̅(̅퐴̅̅̅). 퐼̅̅(̅퐵̅̅̅) 8) It says way in 7) 3) 퐼(퐴) + 퐼(퐵) ⊆ 퐼(퐴 + 퐵) 9) -13) it is easy to see that by using of definition 4) 퐼(퐴. 퐵) ⊇ 퐼(퐴). 퐼(퐵) of upper and lower approximations of A with Proof respect M. ̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ 1) We need to proof퐼(퐴 + 퐵) ⊆ 퐼(퐴) + 퐼(퐵) and ̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅ UPPER AND LOWER PRINCIPAL IDEAL: 퐼(퐴 + 퐵) ⊇ 퐼(퐴) + 퐼(퐵)So, suppose x퐼(퐴 + 퐵) , by definition of upper approximation of A+B with respect I, (x+I) ⋂ (A+B) ≠ ∅ . Hence there exists Example 4.1. Let us consider the ring ℛ = ℤ6 , 푛 J={0,2,4} and X={1,2,3,4,5}. For x ∈ ℛ ∶ 푥 + 퐽 , we get s(x+I) and yA+B, also, 푠 = ∑푖=1 푎푖 + 푏푖 for some 푛 {0,2,4}, {1,3,5}. The upper approximations of X with ai A, bi B. we have, 푥 ∈ 푠 + 퐼 = ∑푖=1(푎푖 + 푏푖) + 푛 respect of J: {0, 2, 4} ∪{1, 3, 5}. 푠표, 퐽̅̅(̅푋̅̅̅) = {0,1,2,3,4,5}. 퐼 = ∑푖=1(푎푖 + 퐼) + (푏푖 + 퐼). Then there exist 푛 And the lower approximation of X with respect of 푥푖(푎푖+I), and 푠푖(푏푖+I) such that 푥 = ∑푖=1 푥푖 + 푠푖 ̅̅̅̅̅̅ ̅̅̅̅̅̅ J:퐽(푋) = {1,3,5} ,BX =퐽̅̅(̅푋̅̅̅) − 퐽(푋) = {0,2,4}. Then X is So, xi퐼(퐴) And si퐼(퐵). Because, 푎푖(푥푖+I)A and 푏 ( 푠 +I)B, Then, 퐼̅̅(̅퐴̅̅̅+̅̅퐵̅̅̅) ⊆ 퐼̅̅(̅퐴̅̅̅) + rough set with respect J. 푖 푖 퐼̅̅(̅퐵̅̅̅) .

The other side, we suppose x퐼̅̅(̅퐴̅̅̅) + 퐼̅̅(̅퐵̅̅)̅, Proposition4-1: suppose ℛ commtive ring and I Then principle ideal. For every approximation (ℛ,I) w and 푛 ̅̅̅̅̅̅ ̅̅̅̅̅̅ Every subset A,B  ℛ we have: 푥 = ∑ 푎푖 + 푏푖for some 푥푖퐼(퐴) And 푠푖퐼(퐵). 푖=1 1) 퐼(퐴) ⊆ 퐴 ⊆ 퐼̅̅(̅퐴̅̅̅); Hence, (푎푖+I)A≠ ∅ and 푠푖 ∈(푏푖+I)B≠ ∅ for 1 ≤ 2) 퐼(∅) = ∅ = 퐼̅̅(̅∅̅̅̅); 𝑖 ≤ 푛. 푛 푛 3) 퐼(ℛ) = ℛ = 퐼̅̅(̅ℛ̅̅̅); Also, ∑푖=1 푥푖 + 푠푖 ∈ 퐴 + 퐵 푎푛푑 ∑푖=1 푥푖 + 푠푖 ∈ ∑푛 (푎 + 푏 ) + 퐼. 푡ℎ푢푠푒 (푥 + 퐼) ∩ (퐴 + 퐵) ≠ 4) 퐼(퐴 ∩ 퐵) = 퐼(퐴) ∩ 퐼(퐵); 푖=1 푖 푖 ∅, 푡ℎ푎푡 푚푒푎푛 푥 ∈ ̅퐼̅(̅퐴̅̅̅+̅̅̅퐵̅̅)̅, so, 퐼̅̅(̅퐴̅̅̅+̅̅퐵̅̅̅) ⊇ 퐼̅̅(̅퐴̅̅̅) + 5) 퐼̅̅(̅퐴̅̅̅∪̅̅̅퐵̅̅)̅ = 퐼̅̅(̅퐴̅̅̅)̅ ∪ 퐼̅̅(̅퐵̅̅̅)̅ 퐼̅̅(̅퐵̅̅̅). 6) If A B , then 퐼(퐴) 퐼(퐵), 푎푛푑 퐼̅̅(̅퐴̅̅̅) ⊆ 퐼̅̅(̅퐵̅̅̅); 7) 퐼(퐴 ∪ 퐵) ⊇ 퐼(퐴) ∪ 퐼(퐵); 2) Similar 1) by using definition of AB.

Faraj Abdunanbi. Alq J Med App Sci. 2021;4(1):7-11 10

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3) &4) Similar way in (1)&(2) by using [5] Yao, Y.Y., Perspectives of Granular Computing definition of lower approximation. Proceedings of IEEE International Conference on Granular Computing. 2005:1;85-90. [6] Ronnason Chinram., Rough prime ideas and Example 4.3. Let consider the ring ℛ = ℤ6 , I={0,2,4} Rough fuzzy prime ideals in Gamma-semigroups, 푛 and A={1,2,3,4,5 }, B={0,1,2,4},then AB=∑푖=1(푎푖 . 푏푖), ai Commun. Korean Math. Soc. 2009:24(3):341-351. A, bi B.AB= {0,1,2,3,4,5}. [7] Biswas, S. Nanda. Rough groups and rough We 푔푒푡 퐼̅̅(̅퐴̅̅̅) = {0,1,2,3,4,5} . In addition, 퐼̅̅(̅퐵̅̅̅) = subgroups, Bull. Polish Acad. Sci. Math.1994: {0,1,2,3,4,5}. 42 :251–254.

Then̅퐼̅(̅퐴̅̅.̅퐵̅̅̅) = {0,1,2,3,4,5}. [8] N. Kuroki.Rough ideals in semigroups, Inform. Sci. 1997: 100 139–163. However,퐼̅̅(̅퐴̅̅̅). 퐼̅̅(̅퐵̅̅̅).= {0,1,2,3,4, ,5}. [9] N. Kuroki, J.N. Mordeson. Structure of rough sets Then ̅퐼̅(̅퐴̅̅.̅퐵̅̅̅) = 퐼̅̅(̅퐴̅̅̅) . 퐼̅̅(̅퐵̅̅)̅ . 퐼(퐴) = {1,3,5} , 퐼(퐵) = and rough groups, J. Fuzzy Math. 1997:5(1):183–191. {0,2,4} , 퐼(퐴. 퐵) = {0,1,2,3,4,5}.퐼(퐴). 퐼(퐵) = {0,2,4}. [10] S.B. Hosseini.T- Rough (Prime, Primary) ideal and So, 퐼(퐴. 퐵) ⊇ 퐼(퐴). 퐼(퐵). t- Rough fuzzy Prime, Primary) ideal on commutative rings, int.J, contemp. Math science. 2012;7(4):337-350. CONCLUSION [11] B. Davvaz. Rough sets in a fundamental ring, Bull. In this paper, we find there exists rough maximal Iranian Math. Soc.1998: 24 :49. ideal and principle that will be as an extension of the notion of a maximal ideal and principle ideal respectively

Acknowledgments

I would like to thank Tracy Tian Editorial Assistant of OJDM suggestion on this paper.

Conflict of Interest

There are no financial, personal, or professional conflicts of interest to declare.

REFERENCE

[1] Pawlak Z. Rough sets, Int J Inf Comp Sci. 1982;11:341–356. [2] B. Davvaz Roughness in Rings, Inform. Sci. 2004: 164 :147-163. [3] Z. Bonikowaski. Algebraic structures of rough sets, in: W.P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin. 1995: 242–247. [4] John N. Mordeson, Elsevier. Rough set theory applied to (fuzzy) , Fuzzy Sets and Systems.2001:121:315–324.

Faraj Abdunanbi. Alq J Med App Sci. 2021;4(1):7-11 11