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Approximations of Maximal and Principal Ideal https://alqalam.utripoli.edu.ly/science/ eISSN 2707-7179 Review Article Approximations of Maximal and Principal Ideal 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 ring 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 ring theory. 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 Principal Ideal. 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 xU. 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 XU, 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)푓 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 commutative ring 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 + ..
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