37 This work is licensed under a Creative Commons Attribution 4.0 International License. Study of Fuzzy 𝛔– 𝐑𝐒𝐧𝐠 and Some Related Concepts Abstract This paper introduces the concept of fuzzy σ– ring as a generalization of fuzzy Οƒ βˆ’algebra and basic properties; examples of this concept have been given. As the first result, it has been proved that every fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— and construct their converse by example. Furthermore, the fuzzy ring concept has been studied to generalize fuzzy algebra and its relation. Investigating that the concept of fuzzy σ– ring is a stronger form of a fuzzy ring that is every fuzzy σ– ring over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ— and construct their converse by example. In addition, the idea of the smallest, as an important property in the study of real analysis, is studied as well. Finally, the main goal of this paper is to study these concepts and give basic properties, examples, characterizations and relationships between them. Keywords: Οƒ βˆ’algebra, σ– ring, fuzzy Οƒ βˆ’algebra, fuzzy algebra, measure. 1. Introduction The generalized measure theory, which is the subject of this thesis emerged from the well- established classical measure theory by the process of generalization. As is well known, classical measures are nonnegative real-valued set functions, each defined on a specific class of subsets of a given universal set, that satisfies certain axiomatic requirements. One of these requirements, crucial to classical measures, is known as the requirement of additively. Measure theory plays a vital role in mathematics, particularly in probability theory's foundation. The theory of measure has been extensively studied and is used in modeling the physical world. The notion of σ–field is essential in measure theory and probability theory. In 2019 Ahmed and Ebrahim [1] studied the concept of σ–field and discussed many details about some generalizations of this concept. They proved some important results in measure theory. Many authors were interested in studying σ–field and σ– ring, Ibn Al Haitham Journal for Pure and Applied Sciences Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/35.2.2767 Article history: Received 5, January, 2022, Accepted,25, January, 2022, Published in April 2022. Ibrahim S. Ahmed Mathematical Department /College of Computer Science and Mathematics /Tikrit University ibrahim1992@tu.edu.iq Hassan H. Ebrahim Mathematical Department /College of Computer Science and Mathematics /Tikrit University hassan1962pl@tu.edu.iq Ali Al-Fayadh Department of Mathematics and Computer Applications / College of Science / Al – Nahrain University aalfayadh@yahoo.com https://creativecommons.org/licenses/by/4.0/ mailto:ibrahim1992@tu.edu.iq mailto:hassan1962pl@tu.edu.iq mailto:aalfayadh@yahoo.com Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 38 for example see [2-6]. Zadeh in 1965 [7] first introduced the concept of the fuzzy set where 𝒳 is a nonempty set, then a fuzzy set F in 𝒳 was defined as a set of ordered pairs {(Ο‰, 𝓋F(πœ”)) : πœ” ∈ 𝒳 } where 𝓋F : 𝒳 β†’ [0 , 1] was a function such that for every πœ” ∈ 𝒳, 𝓋F(πœ”) represented the degree of membership of πœ” in F. Brown [8] and Wang [9] studied some types of fuzzy sets such as fuzzy power set, empty fuzzy set, universal fuzzy set, the complement of a fuzzy set, the union of two fuzzy sets and the intersection of two fuzzy sets. Ahmed et al [10] first introduced the concept of fuzzy Οƒ βˆ’algebra and fuzzy algebra, where 𝒳 β‰  βˆ…. A nonempty class β„‹ βˆ— βŠ† π’«βˆ—(𝒳) is called a fuzzy σ–algebra over a fuzzy set 𝒳 βˆ—, if 1. βˆ…βˆ— ∈ β„‹ βˆ—, where βˆ…βˆ— ={(Ο‰, 0) : βˆ€πœ” ∈ 𝒳 }. 2.If E ∈ β„‹ βˆ—, then E𝑐 ∈ β„‹ βˆ—. 3.If E1, E2, … ∈ β„‹ βˆ—, then β‹‚ Ek ∞ π‘˜=1 ∈ β„‹ βˆ—. If condition 3 is satisfied only for finite sets, then β„‹ βˆ— is said to be a fuzzy algebra over a fuzzy set 𝒳 βˆ—. In this work we introduce the concept of fuzzy σ– ring and fuzzy ring which are generalizations of fuzzy Οƒ βˆ’algebra. The main goal of this paper is to study these concepts, give basic properties, examples, characterizations and studied relationships between them. 2. Preliminaries This section is going to review some well-known definitions in measure theory. Definition 2.1 [3] Let 𝒳 β‰  βˆ…. A collection β„‹ is called σ– ring iff : 1.If F, E ∈ β„‹ , then F βˆ– E ∈ β„‹ . 2.If E1, E2, … ∈ β„‹ , then ⋃ Ek ∞ π‘˜=1 ∈ β„‹ . Definition 2.2 [1] Let 𝒳 β‰  βˆ…. A collection β„‹ is called σ– field iff : 1.𝒳 ∈ β„‹ . 2.If F ∈ β„‹ , then F𝑐 ∈ β„‹ . 3.If E1, E2, … ∈ β„‹ , then ⋃ Ek ∞ π‘˜=1 ∈ β„‹ . Proposition 2.3 [5] Every σ–field is a σ– ring. Definition 2.4 [8 , 9] Let 𝒳 be a nonempty set. Then: 1.The collection of all fuzzy sets in 𝒳 is called a fuzzy power set and is denoted by π’«βˆ—(𝒳), In symbols: π’«βˆ—(𝒳) = { F ∢ F is a fuzzy set in 𝒳 }. 2.The empty fuzzy set in 𝒳 is denoted by βˆ…βˆ— and defined as: βˆ…βˆ— ={(Ο‰, 0) : βˆ€πœ” ∈ 𝒳 }. 3.The fuzzy set 𝒳 βˆ— in 𝒳 is defined as: 𝒳 βˆ— ={(Ο‰, 1) : βˆ€πœ” ∈ 𝒳 }. Definition 2.5 [7] Let 𝒳 be a nonempty set. Then the union of the two fuzzy sets F and E in 𝒳 with respective membership functions 𝓋F(πœ”) and 𝓋E(πœ”) is a fuzzy set G in 𝒳 whose membership function is related to those of F and E by 𝓋G(πœ”) = max πœ”βˆˆπ’³ {𝓋F(πœ”) , 𝓋E(πœ”)}, In symbols: Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 39 G = F⋃E ⇔ G = {(Ο‰, max πœ”βˆˆπ’³ {𝓋F(πœ”) , 𝓋E(πœ”)}) ∢ πœ” ∈ 𝒳 }. Definition 2.6 [8] Let 𝒳 be a nonempty set. Then the intersection of two fuzzy sets F and E in 𝒳 with respective membership functions 𝓋F(πœ”) and 𝓋E(πœ”) is a fuzzy set G in 𝒳 whose membership function is related to those of F and E by 𝓋G(πœ”) = min πœ”βˆˆπ’³ {𝓋F(πœ”) , 𝓋E(πœ”)}, In symbols: G = Fβ‹‚E ⇔ G = {(Ο‰, min πœ”βˆˆπ’³ {𝓋F(πœ”) , 𝓋E(πœ”)}) ∢ πœ” ∈ 𝒳 }. Definition 2.7 [9] Let 𝒳 be a nonempty set and F is a fuzzy set in 𝒳. Then the complement of a fuzzy set F is denoted by Fc and defined as: Fc ={(Ο‰, 1 βˆ’ 𝓋F(πœ”)) : πœ” ∈ 𝒳 }. Proposition 2.8 [10] Every fuzzy σ–algebra is a fuzzy algebra. 3. The main results: In this section, the basic definitions and facts related to this work are recalled, which starts with the following definition. Definition 3.1 Let 𝒳 β‰  βˆ…. A collection β„‹ βˆ— βŠ† π’«βˆ—(𝒳) is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—, iff 4. βˆ…βˆ— ∈ β„‹ βˆ—. 5.If F, E ∈ β„‹ βˆ—, then F βˆ– E ∈ β„‹ βˆ—. 6.If E1, E2, … ∈ β„‹ βˆ—, then ⋃ Ek ∞ π‘˜=1 ∈ β„‹ βˆ—. Definition 3.2 A fuzzy measurable space relatively to a fuzzy σ– ring is an ordered pair ( 𝒳 βˆ—, β„‹ βˆ— ), where 𝒳 β‰  βˆ…and β„‹ βˆ— βŠ† π’«βˆ—(𝒳) be a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— and an element of β„‹ βˆ— is called a measurable set relatively to fuzzy σ– ring. Example 3.1 Let 𝒳 β‰  βˆ…. Then each of βˆ…βˆ— and π’«βˆ—(𝒳) is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— . Example 3.2 Assume 𝒳 ={ π‘Ž ,b} and β„‹ βˆ— = { Ek , Ek c βŠ‚ 𝒳 βˆ— : Ek βŠƒ Ek+1 , for every k = 1,2, …}. Then βˆ…βˆ—, 𝒳 βˆ— ∈ β„‹ βˆ—. If F, E ∈ β„‹ βˆ—, then F c, E c ∈ β„‹ βˆ— and either E βŠ‚ F c or E βŠƒ F c. If E βŠ‚ F c, then F βŠ‚ E c, hence Fβ‹‚E c ∈ β„‹ βˆ—, that is F\E ∈ β„‹ βˆ—. If E βŠƒ F c, then E c βŠ‚ F, hence Fβ‹‚E c ∈ β„‹ βˆ—, that is F\E ∈ β„‹ βˆ—. Now, If E1, E2, … ∈ β„‹ βˆ—, then Ek βŠƒ Ek+1 for every (k = 1,2, … ) and hence 𝓋Ek (Ο‰) > 𝓋Ek+1 (Ο‰) for all Ο‰ ∈ 𝒳 and hence ⋃ Ek ∞ π‘˜=1 = {(Ο‰, sup{ 𝓋E1 (Ο‰), 𝓋E2 (Ο‰), 𝓋E3 (Ο‰), … } : βˆ€Ο‰ ∈ 𝒳)} = {(Ο‰, 𝓋E1 (Ο‰)) ∢ βˆ€Ο‰ ∈ 𝒳} = E1 Thus, ⋃ Ek ∞ π‘˜=1 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— and hence ( 𝒳 βˆ—, β„‹ βˆ— ) is a fuzzy measurable space relatively to the fuzzy σ– ring. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 40 Example 3.3 Let 𝒳 ={ π‘Ž ,b ,c} and β„‹ βˆ— ={ βˆ…βˆ—,{( π‘Ž,0.3),( 𝑏,0.3),(c,0.4)},{( π‘Ž,0.4),(𝑏,0.4),(c,0.2)}}. Then β„‹ βˆ— is not a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—, because {( π‘Ž,0.3),( 𝑏,0.3),(c,0.4)}, {( π‘Ž,0.4),(𝑏,0.4),(c,0.2)} ∈ β„‹ βˆ—, but {( π‘Ž, 0.3), ( 𝑏, 0.3), (𝑐, 0.4)} ⋃ {( π‘Ž, 0.4), (𝑏, 0.4), (𝑐, 0.2)} ={( π‘Ž,0.4),(𝑏,0.4),(c,0.4)}βˆ‰ β„‹ βˆ—. Lemma 3.1 Let {β„‹i βˆ—}iβˆˆΞ™ be a nonempty collection of fuzzy σ– ring over a fuzzy set 𝒳 βˆ— . Then β‹‚ β„‹i βˆ— iβˆˆΞ™ is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Proof Since β„‹i βˆ— is a nonempty collection of a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— βˆ€i ∈ Ξ™, then there is βˆ…βˆ— β‰  E ∈ β„‹i βˆ— βˆ€i ∈ Ξ™, which implies that E ∈ β‹‚ β„‹i βˆ— iϡΙ and hence β‹‚ β„‹i βˆ— iϡΙ β‰  βˆ… βˆ—. It is clear that βˆ…βˆ— ∈ β‹‚ β„‹i βˆ— iϡΙ . Let F, E ∈ β‹‚ β„‹i βˆ— iϡΙ . Then F, E ∈ β„‹i βˆ— βˆ€i ∈ Ξ™, hence F\E ∈ β„‹i βˆ— βˆ€i ∈ Ξ™. Therefore F\E ∈ β‹‚ β„‹i βˆ— iϡΙ . Let E1, E2, … ∈ β‹‚ β„‹i βˆ— iϡΙ , then E1, E2, … ∈ β„‹i βˆ— βˆ€i ∈ Ξ™ and hence ⋃ Ek ∞ k=1 ∈ β„‹i βˆ— βˆ€i ∈ Ξ™, thus ⋃ Ek ∞ k=1 ∈ β‹‚ β„‹i βˆ— iϡΙ . Therefore, β‹‚ β„‹i βˆ— iβˆˆΞ™ is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. In the following example shows the union for two fuzzy σ– ring over a fuzzy set 𝒳 βˆ— needs not be a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Example 3.4 Let 𝒳 ={ πœ”1 , πœ”2} and β„‹1 βˆ— ={βˆ…βˆ—,{( πœ”1,0.1),( πœ”2,0.5)} , {( πœ”1,0.9),( πœ”2 ,0.5)} , 𝒳 βˆ—}, β„‹2 βˆ— ={βˆ…βˆ—,{( πœ”1,0.2),(πœ”2,0.4)} , {( πœ”1,0.8),( πœ”2,0.6)} , 𝒳 βˆ—}. Then β„‹1 βˆ— and β„‹2 βˆ— are fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Now, β„‹1 βˆ— ⋃ β„‹2 βˆ— = { βˆ…βˆ—, {( πœ”1, 0.1), ( πœ”2, 0.5)}, {( πœ”1, 0.9), ( πœ”2 ,0.5)}, {( πœ”1, 0.2), (πœ”2, 0.4)} , {( πœ”1, 0.8), ( πœ”2, 0.6)} , 𝒳 βˆ—}. Put,E1 = {( πœ”1, 0.1), ( πœ”2, 0.5)} , E2 = {( πœ”1, 0.9), ( πœ”2 ,0.5)} , E3 = {( πœ”1, 0.2), (πœ”2, 0.4)} , E4 = {( πœ”1, 0.8), ( πœ”2, 0.6)} .Then Ek ∈ β„‹1 βˆ— ⋃ β„‹2 βˆ— for all k=1,2,…,4. So, we have ⋃ Ek 4 k=1 = { (πœ”1, π‘€π‘Žπ‘₯{0.1,0.9,0.2,0.8}), (πœ”2, π‘€π‘Žπ‘₯{0.5,0.5,0.4,0.6}), } ={( πœ”1,0.9),( πœ”2,0.6)} βˆ‰ β„‹1 βˆ— ⋃ β„‹2 βˆ— . Thus, β„‹1 βˆ— ⋃ β„‹2 βˆ— is not fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Definition 3.3 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳), then the intersection of all fuzzy σ– ring over a fuzzy set 𝒳 βˆ—, which includes π”—βˆ— is said to be the fuzzy σ– ring over a fuzzy set 𝒳 βˆ— generated by π”—βˆ— and denoted by Οƒπ‘Ÿ (π”—βˆ—), that is: Οƒπ‘Ÿ (π”—βˆ—) = β‹‚{β„‹i βˆ—: β„‹i βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—and β„‹i βˆ— βŠ‡ π”—βˆ—, βˆ€i ∈ Ξ™}. Proposition 3.1 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then Οƒπ‘Ÿ (π”—βˆ—) is the smallest fuzzy σ– ring over a fuzzy set 𝒳 βˆ— that includes π”—βˆ—. Proof The result is directed by the definition of Οƒπ‘Ÿ (π”—βˆ—) and Lemma 3.1. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 41 Example 3.5 Let 𝒳 ={ πœ”1 , πœ”2} and 𝔗 βˆ— = { {( πœ”1, 0.1), ( πœ”2, 0.5)}, {( πœ”1, 0.9), ( πœ”2 ,0.5)}}. Then, Οƒπ‘Ÿ (π”—βˆ—) ={βˆ…βˆ—,{( πœ”1,0.1),( πœ”2,0.5)},{( πœ”1,0.9),( πœ”2,0.5)},𝒳 βˆ—} is the smallest fuzzy σ– ring over a fuzzy set 𝒳 βˆ— that include π”—βˆ—. Proposition 3.2 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳), then Οƒπ‘Ÿ (π”—βˆ—) = π”—βˆ— if and only if π”—βˆ— is fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Proof The result direct by definition of Οƒπ‘Ÿ (π”—βˆ—) and Proposition 3.1. Proposition 3.3 Every fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Proof Let β„‹ βˆ— be a fuzzy σ– algebra over a fuzzy set 𝒳 βˆ—. Then from the definition of fuzzy σ– algebra we get, βˆ…βˆ— ∈ β„‹ βˆ—. Let F, E ∈ β„‹ βˆ—. Then E𝑐 ∈ β„‹ βˆ—, hence Fβ‹‚E𝑐 ∈ β„‹ βˆ—, but Fβ‹‚E𝑐 = F\E implies that F\E ∈ β„‹ βˆ—. Let E1, E2, … ∈ β„‹ βˆ—. Then by definition of fuzzy σ– algebra we have, Ek 𝑐 (for all k = 1,2, … ) and β‹‚ Ek π‘βˆž k=1 ∈ β„‹ βˆ— and (β‹‚ Ek π‘βˆž k=1 ) 𝑐 ∈ β„‹ βˆ—. By De-Morgan law, we get (β‹‚ Ek π‘βˆž k=1 ) 𝑐 = ⋃ Ek ∞ k=1 , hence ⋃ Ek ∞ k=1 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— be a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. While the converse is not true as shown below: Example 3.6 Let 𝒳 ={ πœ”1 , πœ”2} and β„‹ βˆ— = { βˆ…βˆ—, {( πœ”1, 0), (πœ”2, 0.5)}, {( πœ”1, 0.6), ( πœ”2, 0.5)}, {( πœ”1, 0.4), ( πœ”2, 0.5)} }. Put F={( πœ”1,0), (πœ”2,0.5)}, then F \βˆ… βˆ— = {(πœ”1, 𝑀𝑖𝑛{0,1 βˆ’ 0}), (πœ”2, 𝑀𝑖𝑛{0.5,1 βˆ’ 0})} = {( πœ”1, 0),( πœ”2, 0.5)}= F In the same way, we get βˆ…βˆ—\ F = βˆ…βˆ—. Since Fc = {( πœ”1,1), (πœ”2,0.5)}, then 𝓋𝐹 (πœ”) ≀ 𝓋Fc (πœ”), hence F βŠ† Fc and F\ F = F. If E = {( πœ”1,0.6),( πœ”2,0.5)}, then E\βˆ… βˆ— = E and βˆ…βˆ—\ E = βˆ…βˆ— and If F = {( πœ”1,0), (πœ”2,0.5)} and E = {( πœ”1,0.6),( πœ”2,0.5)}, then E c ={( πœ”1,0.4),( πœ”2,0.5)} and E \E = {(πœ”1, 𝑀𝑖𝑛{0.6,0.4}), (πœ”2, 𝑀𝑖𝑛{0.5,0.5})} = {(πœ”1, 0.4), (πœ”2, 0.5)} =E c F\ E = {(πœ”1, 𝑀𝑖𝑛{0,1 βˆ’ 0.6}), (πœ”2, 𝑀𝑖𝑛{0.5,1 βˆ’ 0.5})} = {( πœ”1, 0),( πœ”2, 0.5)}= F Similarly, E\ F = E and Ec\F = Ec. . Now, F⋃ E⋃Ec = {(πœ”1, 𝑆𝑒𝑝{0,0,0.6,0.4}), (πœ”2, 𝑆𝑒𝑝{0,0.5,0.5,0.5})} = {( πœ”1, 0.6),( πœ”2, 0.5)}= E. Therefore, β„‹ βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. In contrast, β„‹ βˆ— is not fuzzy σ– algebra over a fuzzy set 𝒳 βˆ—, because{( πœ”1,0),(πœ”2,0.5)}∈ β„‹ βˆ—, but {( πœ”1, 0), (πœ”2, 0.5)} c = {( πœ”1, 1), (πœ”2, 0.5)} βˆ‰ β„‹ βˆ—. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 42 Theorem 3.1 Assume 𝒳 β‰  βˆ… and β„‹ βˆ— βŠ† π’«βˆ—(𝒳) with 𝒳 βˆ— ∈ β„‹ βˆ—. Then β„‹ βˆ— is fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— if and only if β„‹ βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Proof Assume that β„‹ βˆ— is fuzzy σ– algebra over a fuzzy set 𝒳 βˆ—, then by Proposition 3.3 we get β„‹ βˆ— is fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Conversely: Suppose that β„‹ βˆ— is a fuzzy σ– ring over a fuzzy set 𝒳 βˆ— such that 𝒳 βˆ— ∈ β„‹ βˆ—, then βˆ…βˆ— ∈ β„‹ βˆ—. Let E ∈ β„‹ βˆ—. Then 𝒳 βˆ—\E ∈ β„‹ βˆ—, but 𝒳 βˆ—\E = 𝒳 βˆ—β‹‚E𝑐 = {(Ο‰, Min {𝓋𝒳 βˆ— , 𝓋E𝑐 (Ο‰)}) ∢ βˆ€Ο‰ ∈ 𝒳} = {(Ο‰, Min {1,1 βˆ’ 𝓋E (Ο‰)}) ∢ βˆ€Ο‰ ∈ 𝒳} = {(Ο‰, 1 βˆ’ 𝓋E (Ο‰)) ∢ βˆ€Ο‰ ∈ 𝒳} = E 𝑐 This implies that, 𝒳 βˆ—\E = E𝑐 ∈ β„‹ βˆ—. Let E1, E2, … ∈ β„‹ βˆ—. Then as shown above we have, Ek 𝑐 (for all k = 1,2, … ). Hence by definition of a fuzzy σ– ring we get ⋃ Ek π‘βˆž k=1 ∈ β„‹ βˆ—, thus (⋃ Ek π‘βˆž k=1 ) 𝑐 ∈ β„‹ βˆ—. By De-Morgan law, we get (⋃ Ek π‘βˆž k=1 ) 𝑐 = β‹‚ Ek ∞ k=1 hence β‹‚ Ek ∞ k=1 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— be a fuzzy σ– algebra over a fuzzy set 𝒳 βˆ—. Proposition 3.4 Let 𝒳 be a nonempty set and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then Οƒπ‘Ÿ (π”—βˆ—) βŠ† Οƒ(π”—βˆ—), where Οƒ(π”—βˆ—) is the smallest fuzzy σ–algebra over a fuzzy set 𝒳 βˆ— that includes π”—βˆ—. Proof Clearly. Proposition 3.5 Let 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). If 𝒳 βˆ— ∈ Οƒπ‘Ÿ (π”—βˆ—), then Οƒπ‘Ÿ (π”—βˆ—) = Οƒ(π”—βˆ—). Proof The proof follows from Theorem 3.1. Definition 3.4 Let assume 𝒳 β‰  βˆ… . A class β„‹ βˆ— βŠ† π’«βˆ—(𝒳) is said to be a fuzzy ring over a fuzzy set 𝒳 βˆ—, if 1. βˆ…βˆ— ∈ β„‹ βˆ—. 2.If F, E ∈ β„‹ βˆ—, then F βˆ– E ∈ β„‹ βˆ—. 3.If E1, E2, , En ∈ β„‹ βˆ—, then ⋃ Ek 𝑛 π‘˜=1 ∈ β„‹ βˆ—. Definition 3.5 A fuzzy measurable space relatively to fuzzy ring is an ordered pair ( 𝒳 βˆ—, β„‹ βˆ— ), where 𝒳 is a nonempty set and β„‹ βˆ— βŠ† π’«βˆ—(𝒳) be a fuzzy ring over a fuzzy set 𝒳 βˆ— and an element of β„‹ βˆ— is called a measurable set relatively to fuzzy ring. Example 3.7 Suppose 𝒳 β‰  βˆ…. Then each of βˆ…βˆ— and π’«βˆ—(𝒳) is a fuzzy ring over a fuzzy set 𝒳 βˆ— . Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 43 Example 3.8 Let 𝒳 ={ πœ”1 , πœ”2} and β„‹ βˆ— ={βˆ…βˆ—,{( πœ”1,0.4),( πœ”2,0.3)} , {( πœ”1,0.6),( πœ”2,0.7)}}. Then β„‹ βˆ— and is a fuzzy ring over a fuzzy set 𝒳 βˆ—. Example 3.9 Let 𝒳 ={ πœ”1 , πœ”2} and β„‹ βˆ— ={βˆ…βˆ—,{( πœ”1,0.3),(πœ”2,0.7)},{( πœ”1,0.2),( πœ”2,0.8)}, 𝒳 βˆ—}. Then β„‹ βˆ— is not a fuzzy ring over a fuzzy set 𝒳 βˆ—, because {( πœ”1,0.3),(πœ”2,0.7)} and {( πœ”1,0.2),( πœ”2,0.8)} ∈ β„‹ βˆ—, but {( πœ”1, 0.3), (πœ”2, 0.7)}⋃{( πœ”1, 0.2), ( πœ”2, 0.8)} = {( πœ”1, 0.3), ( πœ”2, 0.8)} βˆ‰ β„‹ βˆ—. Lemma 3.2 Let {β„‹i βˆ—}iβˆˆΞ™ be a nonempty collection of a fuzzy ring over a fuzzy set 𝒳 βˆ— . Then β‹‚ β„‹i βˆ— iβˆˆΞ™ is a fuzzy ring over a fuzzy set 𝒳 βˆ—. Proof Direct. Definition 3.6 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then the intersection of all fuzzy rings over a fuzzy set 𝒳 βˆ—, which includes π”—βˆ— is said to be the fuzzy ring over a fuzzy set 𝒳 βˆ— generated by π”—βˆ— and denoted by R(π”—βˆ—), that is, R(π”—βˆ—)= β‹‚{β„‹i βˆ—: β„‹i βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—and β„‹i βˆ— βŠ‡ π”—βˆ—, βˆ€i ∈ Ξ™}. Proposition 3.6 If 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then R(π”—βˆ—) is the smallest fuzzy ring over a fuzzy set 𝒳 βˆ— that includes π”—βˆ—. Proof The result is directed by the definition of R(π”—βˆ—) and Lemma 3.2. Proposition 3.7 Every fuzzy σ– ring over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. Proof Let β„‹ βˆ— be a fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. Then by definition of fuzzy σ– ring we have, βˆ…βˆ— ∈ β„‹ βˆ—. Let F, E ∈ β„‹ βˆ—. Then F\E ∈ β„‹ βˆ—. Let E1, E2, En ∈ β„‹ βˆ—. Consider, Em = βˆ… βˆ— for all m > n, then we get E1, E2, E3, … ∈ β„‹ βˆ— and hence from the definition of fuzzy σ– ring, we have ⋃ Ek ∞ k=1 ∈ β„‹ βˆ— , but ⋃ Ek ∞ k=1 = ⋃ Ek n k=1 ⋃En+1⋃En+2⋃…= ⋃ Ek n k=1 β‹ƒβˆ… βˆ—β‹ƒβˆ…βˆ—β‹ƒβ€¦= ⋃ Ek n k=1 . Thus E1⋃E2 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. In general, the converse of the above proposition is not true as shown in the following example: Example 3.10 Let 𝒳 = ℝ and π’₯ = finite disjoint union of right – semi-closed intervals. Assume that β„‹ βˆ— = { all ( π’₯, 𝓋π’₯ ) }. Then β„‹ βˆ— is a fuzzy ring over a fuzzy set β„βˆ—, but β„‹ βˆ— is not fuzzy σ– ring over a Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 44 fuzzy set β„βˆ—. Because if we take Ek = {((0, 1- (1βˆ•k)], 𝓋Ek ) }, k=1,2,…, then Ek ∈ β„‹ βˆ— βˆ€n, but ⋃ Ek ∞ k=1 = {((0,1), 𝓋Ek )} βˆ‰ β„‹ βˆ—. Proposition 3.8 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then R(π”—βˆ—) βŠ† Οƒπ‘Ÿ (π”—βˆ—). Proof The proof is directed by proposition 3.7 and proposition 3.6. Proposition 3.9 Every fuzzy algebra over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. Proof Let β„‹ βˆ— be a fuzzy algebra over a fuzzy set 𝒳 βˆ—. Then by the definition of fuzzy algebra we have, 𝒳 βˆ— ∈ β„‹ βˆ—, hence βˆ…βˆ— = 𝒳 βˆ— 𝑐 ∈ β„‹ βˆ— Let F, E ∈ β„‹ βˆ—. Then E𝑐 ∈ β„‹ βˆ—, hence Fβ‹‚E𝑐 ∈ β„‹ βˆ—, but Fβ‹‚E𝑐 = F\E implies that F\E ∈ β„‹ βˆ—. Let E1, E2 ∈ β„‹ βˆ—. Then by definition of fuzzy algebra implies that E1⋃E2 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— be a fuzzy ring over a fuzzy set 𝒳 βˆ—. while the converse is not true as shown in the next example: Example 3.11 Let 𝒳 ={ π‘Ž , 𝑏} and β„‹ βˆ— ={βˆ…βˆ—,{( π‘Ž,0), (𝑏,0.5)},{( π‘Ž,0.6),( 𝑏,0.5)},{( π‘Ž,0.4),( 𝑏,0.5)}}. Then, β„‹ βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. In contrast, β„‹ βˆ— is a fuzzy algebra over a fuzzy set 𝒳 βˆ—, because {( π‘Ž,0), (𝑏,0.5)}∈ β„‹ βˆ—, but {( π‘Ž, 0), (𝑏, 0.5)}c = {( π‘Ž, 1), (𝑏, 0.5)} βˆ‰ β„‹ βˆ—. Proposition 3.10 Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳), then R(π”—βˆ—) βŠ† AL(π”—βˆ—) where AL(π”—βˆ—) is the smallest fuzzy algebra over a fuzzy set 𝒳 βˆ— that include π”—βˆ—. Proof The proof is followed by proposition 3.9 with proposition 3.6. Proposition 3.11 Assume 𝒳 to be a non-empty set and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then R(π”—βˆ—) βŠ† Οƒ(π”—βˆ—). Proof Observe. Theorem 3.2 Assume 𝒳 β‰  βˆ… and β„‹ βˆ— βŠ† π’«βˆ—(𝒳) such that 𝒳 βˆ— ∈ β„‹ βˆ—. Then β„‹ βˆ— is a fuzzy algebra over a fuzzy set 𝒳 βˆ— if and only if β„‹ βˆ— is fuzzy ring over a fuzzy set 𝒳 βˆ—. Proof Assume that β„‹ βˆ— is fuzzy algebra over a fuzzy set 𝒳 βˆ—, then by Proposition 3.9, we get β„‹ βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 45 Conversely: Suppose that β„‹ βˆ— is fuzzy ring over a fuzzy set 𝒳 βˆ— such that 𝒳 βˆ— ∈ β„‹ βˆ—. Let E ∈ β„‹ βˆ—. Then 𝒳 βˆ—\E ∈ β„‹ βˆ—, but 𝒳 βˆ—\E = 𝒳 βˆ—β‹‚E𝑐 = {(Ο‰, Min {𝓋𝒳 βˆ— , 𝓋E𝑐 (Ο‰)}) ∢ βˆ€Ο‰ ∈ 𝒳} = {(Ο‰, Min {1,1 βˆ’ 𝓋E (Ο‰)}) ∢ βˆ€Ο‰ ∈ 𝒳} = {(Ο‰, 1 βˆ’ 𝓋E (Ο‰)) ∢ βˆ€Ο‰ ∈ 𝒳} = E 𝑐 Which implies that, 𝒳 βˆ—\E = E𝑐 ∈ β„‹ βˆ—. Let E1, E2, … , En ∈ β„‹ βˆ—. Then ⋃ Ek n k=1 ∈ β„‹ βˆ—. Therefore, β„‹ βˆ— be a fuzzy algebra over a fuzzy set 𝒳 βˆ—. Proposition 3.12 Suppose 𝒳 be a non-empty set and π”—βˆ— βŠ† π’«βˆ—(𝒳). If 𝒳 βˆ— ∈ R(π”—βˆ—), then R(π”—βˆ—) = AL(π”—βˆ—). Proof The proof is followed by Theorem 3.2. Proposition 3.13 Every fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. 4. Conclusions We will try to generalize the concept of fuzzy σ– ring to some other concepts in future works. We define the concept of measure on fuzzy σ– ring and discuss many properties of this concept. In this study, the concepts of fuzzy σ– ring and fuzzy ring over a fuzzy set 𝒳 βˆ— weakly are introduced as a generalization of fuzzy σ– algebra and fuzzy algebra over the same fuzzy set 𝒳 βˆ—. Furthermore, some properties of these concepts are investigated such as: 1. every fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— is a fuzzy σ– ring over fuzzy set 𝒳 βˆ—. 2.Assume 𝒳 β‰  βˆ… and π”—βˆ— βŠ† π’«βˆ—(𝒳). Then Οƒπ‘Ÿ (π”—βˆ—) is the smallest fuzzy σ– ring over a fuzzy set 𝒳 βˆ— that include π”—βˆ—. 3.Assume 𝒳 β‰  βˆ… and β„‹ βˆ— βŠ† π’«βˆ—(𝒳) such that 𝒳 βˆ— ∈ β„‹ βˆ— . Then β„‹ βˆ— is a fuzzy σ– algebra over a fuzzy set 𝒳 βˆ— if and only if β„‹ βˆ— is fuzzy σ– ring over a fuzzy set 𝒳 βˆ—. 4.Every fuzzy σ– ring over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. 5.Every fuzzy algebra over a fuzzy set 𝒳 βˆ— is a fuzzy ring over a fuzzy set 𝒳 βˆ—. References 1.Ahmed, I.S. ; Ebrahim, H.H. Generalizations of Οƒ-field and new collections of sets noted by Ξ΄- field, AIP Conf Proc. 2019, 2096, (020019-1 )-(020019-6). 2.Ahmed, I.S. ; Asaad, S.H. ; Ebrahim, H.H. Some new properties of an outer measure on a σ– field, Journal of Interdisciplinary Mathematics. 2021, 24 (4), 947–952. 3.Endou, N. ; Nakasho, K. ; Shidama, Y. Οƒ-ring and Οƒ-algebra of Sets, Formaliz. Math. 2015, 23 (1), 51–57. 4.Ahmed, I.S. ; Ebrahim, H.H. On Ξ±-field and Ξ²-field, J. Phys.: Conf. Ser. 2019, 1294, 1-8. 5.Ebrahim, H.H.; Ahmed, I.S. 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