A Generalized Definition of Fuzzy Subrings

An, Ying-Ying; Shi, Fu-Gui; Wang, Lan

doi:https://doi.org/10.1155/2022/5341207

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5341207 | https://doi.org/10.1155/2022/5341207

A Generalized Definition of Fuzzy Subrings

Ying-Ying An,¹Fu-Gui Shi,^1,2and Lan Wang¹

Academic Editor: Jun Ye

Received27 Jan 2022

Revised04 Mar 2022

Accepted16 Mar 2022

Published30 Sept 2022

Abstract

In this study, under the condition that is a completely distributive lattice, a generalized definition of fuzzy subrings is introduced. By means of four kinds of cut sets of fuzzy subset, the equivalent characterization of -fuzzy subring measures are presented. The properties of -fuzzy subring measures under these two kinds of product operations are further studied. In addition, an -fuzzy convexity is directly induced by -fuzzy subring measure, and it is pointed out that ring homomorphism can be regarded as -fuzzy convex preserving mapping and -fuzzy convex-to-convex mapping. Next, we give the definition and related properties of the measure of -fuzzy quotient ring and give a new characterization of -fuzzy quotient ring when the measure of -fuzzy quotient ring is 1.

1. Introduction

Since Rosenfeld [1] introduced the concept of fuzzy subgroup, fuzzy algebra has developed rapidly. Liu et al. [2, 3] defined the concept of fuzzy subring. Malik and Mordeson discussed the direct sum operation properties of -fuzzy subrings [4]. Shi [5] proposed the concept of fuzzy subgroup measure for the first time. Shi and Xin [6] generalized fuzzy subgroup measure to -fuzzy subgroup measure. Li and Shi introduced the notion of -fuzzy convex sublattice measure and induced an -fuzzy convex structure [7]. Han and Shi [8] defined -convex fuzzy ideal measure and further studied its induced -fuzzy convex structure. The notions of -fuzzifying convex structure and -fuzzy convex structure are introduced by Shi and Xiu in [9, 10]. Actually, fuzzy convexity exists in many mathematical research areas, such as fuzzy vector spaces, fuzzy groups, fuzzy lattices, and fuzzy topologies (see [7, 8, 11–24]).

Inspired by this, the definition of -fuzzy subring measure was proposed in this study; its equivalent characterizations were given by using four kinds of cut sets, and the relevant properties were discussed. In addition, an -fuzzy subring measure can induce an -fuzzy convex structure, and a ring homomorphism between two rings can exactly be regarded as an -fuzzy convex-to-convex mapping and an -fuzzy convex preserving mapping. And when the subring of ring is fuzzy and the ideal is crisp, we can also obtain the concept of -fuzzy quotient ring measure. On this basis, we use the representation theorem to give the equivalent characterization of fuzzy quotient ring when the measure of -fuzzy quotient ring is 1. At the same time, we study the relationship between -fuzzy subring measure and -fuzzy quotient ring measure. It is proposed that the ring homomorphism from ring to quotient ring can be regarded as -fuzzy convex-to-convex mapping and -fuzzy convex preserving mapping.

2. Preliminaries

Throughout this study, is a completely distributive lattice, and the smallest element and the largest element in are denoted by and , respectively.

An element in is called a prime element if implies or . in is called co-prime if implies or [25]. The set of non-unit prime elements in is denoted by . The set of non-zero co-prime elements in is denoted by .

The binary relation in is defined as follows: for , , if and only if, for every subset , the relation always implies the existence of with [26]. is called the greatest minimal family of in the sense of [27], denoted by ; moreover, define the binary relation in as follows: for , if and only if, for every subset , the relation always implies the existence of with [28]. is called the greatest maximal family of , denoted by .

In a completely distributive lattice , is an - map, is a union-preserving map, and there exist and for each such that (see [29]).

For a complete distributive lattice , and , we can define

Theorem 1 (see [27, 30]). For each L-fuzzy set , the following conditions are true:(1)(2)Since a completely distributive lattice is a complete Heyting algebra, there exists a binary operation in . Explicitly the implication is given byWe list some properties of the implication operation in the following lemma.

Lemma 1 (see [6]). Let be a completely distributive lattice and let be the implication operator corresponding to . Then, for all , , the following statements hold:(1)(2)(3)(4), hence, whenever (5), hence, whenever a ⩽ b(6)(7)

Lemma 2 (see [6]). Let be a set mapping. Let and be two -fuzzy subsets in and , respectively. Then, we have(i) if is surjective(ii) if is injective, where and are defined by

Definition 1. (see [10]). A mapping is called an -fuzzy convexity on if it satisfies the following conditions: (LMC1) (LMC2) If is nonempty, then (LMC3) If is nonempty and totally ordered, then The pair is called an -fuzzy convex space. An -fuzzy convex space is called an -fuzzy convex space for short.

Definition 2. (see [10]). Let and be -fuzzy convexity spaces. A mapping is called(1)An -fuzzy convex preserving mapping provided for all (2)An -fuzzy convex-to-convex mapping provided for all (3)A mapping is called an -fuzzy isomorphism provided is bijective, -fuzzy convex preserving and -fuzzy convex-to-convexAn -fuzzy convex preserving mapping is called an -fuzzy convex preserving mapping, an -fuzzy convex-to-convex mapping is called an -fuzzy convex-to-convex mapping, and an -fuzzy isomorphism is called an -fuzzy isomorphism for short.
In [4], Malik and Mordeson introduced the following two operations.

Definition 3. (see [4]). Let and be two -fuzzy subsets of a ring . Define the -fuzzy subset of by :

Definition 4. (see [4]). Let be a collection of rings, and let be an -fuzzy subset of , . Define the Cartesian product of by :

Theorem 2 (see [31]). Let be an ideal of ring , be an fuzzy subring of , and ; then, A/N is an fuzzy subring of .

Definition 5. (see [32]). Let be an -fuzzy subring of and be an subring of ring , definition . Then, is an -fuzzy subring of , called -fuzzy quotient ring of .

3. A Generalized definition of Fuzzy Subrings

In this section, firstly, the definition of -fuzzy subring measure is given, and the equivalent characterizations are carried out with the help of four kinds of cut sets, and the related properties and their proofs are given, inspired by the concept of -fuzzy subgroup measure in [5]. It is natural to introduce the following concept.

Definition 6. Let be an -fuzzy subset in a ring . Then, the -fuzzy subring measure of is defined asWe also say that is an -fuzzy subring of with respect to measure .

Example 1. Let be the remaining class ring of module 3 and specify the operations on such as regular addition and multiplication. Define byWe can get .(1)In fact, So, we can obtain .(2). So, we can obtain .(3)Obviously, .

Theorem 3. Let be an -fuzzy subset in a ring . Then,

Proof. From Definition 6, we can obtain that ,In particular, we have thatThis shows thatSo,Analogously, we can proveThe following lemma is obvious.

Lemma 3. Let be an -fuzzy subset in a ring . Then, if and only if, for any ,The next theorem presents some equivalent descriptions of -fuzzy subring measure.

Theorem 4. Let be an -fuzzy set in a ring . Then,(1)(2)(3)(4)(5) if for any

Proof. . Suppose that and , for any . Then, for any and for any , we have this shows . Therefore, is a subring of . Hence, Conversely, assume that and , is a subring of . For any , let . Then, and ; thus, , , i.e., This means that So, is clearly established. Suppose that and , for any . Then, for any and , we haveBywe knowHence, and , i.e. .
This means that is a subring of and .
This shows thatConversely, assume thatNow, we prove, for any ,Suppose that . Bywe know that and . Since is a subring of , it holds that , i.e., and .
This showsIt is proved thatSo, is clearly established;
Suppose that , and for any ,Let , and . Now, we prove . If , i.e., and , thenBy and , we have , which contradicts . Hence, . This shows that is a subring of .
Therefore,Conversely, assume thatNow, we prove that, for any ,Let and . Then, , , and , i.e., . Since is a subring of , it holds that , i.e., .
This shows thatTherefore,So, is clearly established.
Suppose thatThen, for any and for any , it holds thati.e., . This shows that is a subring of . This means thatConversely, assume thatNow, we prove that, for any ,Let . Bywe know that and . Since is a subring of , it holds that , i.e., and .
This shows thatTherefore,So, is clearly established.
In conclusion, .

4. The Relation between L-Fuzzy Subring Measure and L-Fuzzy Convexity

In this section, we will investigate the relation between -fuzzy subring measure and -fuzzy convexity. We will prove that a ring homomorphism is exactly an -fuzzy convex preserving mapping and an -fuzzy convex-to-convex mapping.

For each can be naturally considered as a mapping defined by .

The following theorem shows that is exactly an -fuzzy convexity on .

Theorem 5. Let be a ring. Then, the mapping defined by is an -fuzzy convexity on , which is called the -fuzzy convexity induced by -fuzzy subring measure on .

Proof. (LMC1) It is straightforward that (LMC2) Let be a family of -fuzzy subsets in a ring . Now, we prove that Suppose that , Then, , Hence, This shows So, we can obtain . (LMC3) Let be nonempty and totally ordered. In order to prove , it needs to show that for any .By Lemma 3, for all , , we haveLet such thatThen, we haveHence, there exists some such that . Since is totally ordered, we assume ; it follows that .
Bywe obtain and . Hence, and .
From the arbitrariness of , we haveCombining Lemma 3, we have . By the arbitrariness of , we obtainTherefore, is an -fuzzy convexity on .
Now, we consider the -fuzzy subring measures of homomorphic image and preimage of -fuzzy subsets.

Theorem 6. Let be a ring homomorphism, and . Then,(1), and if is injective, then (2), and if is surjective, then

Proof. (1)can be proved from Theorem 4 and the following fact: If is injective, the above can be replaced by . Hence, .(2)can be similarly obtained by Theorem 5:If is surjective, the above can be replaced by . Thus, we can obtain that .
By Theorem 6, we obtain the following theorem.

Theorem 7. Let be ring homomorphism. Let and be the -fuzzy convexities induced by -fuzzy subring measures on and , respectively. Then, is an -fuzzy convex preserving mapping and an -fuzzy convex-to-convex mapping.

5. The Operations of -Fuzzy Subrings

In this section, we shall discuss some operation properties of -fuzzy subring measures. In a ring , given two -fuzzy sets , , and is defined in Definition 3. Now, we present its representations by means of cut sets.

Theorem 8. Let be a ring and . Then, the following conditions are true.(1)(2); in particular, if , then (3)(4)(5)(6)

Proof. (1); first, we prove that . By We know that there are and such that , , and , that is, , so . This shows . It is obvious that . Now, we prove . Suppose that . Then, by we have So, we obtain . It is proved that .(2) can be proved from the following implications.In particular, if , then the inverse of the above implications are true. In this case, .
It is obvious that . Next, we prove that . Suppose that . Then, . Bywe know that .
From (1), (2), and Theorem 1 we can obtain (3), (4), (5), and (6).

Next, two theorems are generalizations of the results in [4].

Theorem 9. Let be two -fuzzy subsets of a ring . Then, .

Proof. By Theorem 4 (4), we can obtain the following fact:

Theorem 10. Let be a collection of rings and an -fuzzy subset of . Then, .

Proof. It is easy to check that , where is the projection from to . From (LMC2) in the proof of Theorem 5, we can obtainSince is a ring homomorphism, by means of Theorem 5, we have . Thus, we obtain

From Theorem 6, we can obtain the following corollary.

Corollary 1. Let be a collection of rings, and let and are, respectively, -fuzzy convexity induced by -fuzzy subring measures of , . Then, is an -fuzzy convex preserving mapping and an -fuzzy convex-to-convex mapping.

6. A Generalized definition of Fuzzy Quotient Rings

In this section, firstly, we can get the definition of -fuzzy quotient ring measure from the given definition of -fuzzy subring measure; Secondly, we study the relationship between -fuzzy subring measure and -fuzzy quotient ring measure. Finally, it is given that the ring homomorphism from ring to quotient ring is -fuzzy convex-to-convex mapping and -fuzzy convex preserving mapping. When the measure of -fuzzy quotient ring is 1, a new characterization of -fuzzy quotient ring is given by using cut set.

Definition 7. Let be an -fuzzy subset in a ring , and quotient ring . Then, the -fuzzy quotient ring measure of is defined asWe also say that is an -fuzzy quotient ring of with respect to measure .
The following lemma is obvious.

Lemma 4. Let be an -fuzzy subset in a ring . Then, if and only if, for any ,Next, we study the relationship between -fuzzy subring measure and -fuzzy quotient ring measure.

Corollary 2. Let be an -fuzzy subset in a ring , be an ideal of , and quotient ring . Then,

Proof. Suppose that . Then, :Hence,This shows So, we can obtain

From the relationship between -fuzzy subring measure and -fuzzy convexity, we can get the following conclusion: is a ring, the mapping defined by is a -fuzzy convexity on , which is called -fuzzy convexity of -fuzzy quotient ring measure on . The ring homomorphisms between the other two quotient rings are -fuzzy convex preserving mapping and -fuzzy convex–to-convex mapping.

Finally, we show that the ring homomorphism from ring to quotient ring is -fuzzy convex preserving mapping and -fuzzy convex-to-convex mapping.

Theorem 11. Let be a ring, be a quotient ring, and be a -fuzzy subset of . So, is a ring homomorphism, and this ring homomorphism is -fuzzy convex preserving mapping and -fuzzy convex-to-convex mapping.
Next, when the measure of -fuzzy quotient ring is 1, a new characterization of -fuzzy quotient ring is given by using cut set.

Theorem 12. Let be the ideal of and be the -fuzzy subring of . Then, the following conditions are true:(1)(2)(3)(4)(5)

Proof. (1); first, we prove that ; let . By We know that there is , such that , so ; let , that is, . This shows . It is obvious that . Now, we prove ; suppose that ; then, by we have . So, we obtain . It is proved that .(2); using the definition of cut set, we can easily obtainin addition .Therefore, (2) is proved. Similarly, (3) can be proved to be true.
From (1) and Theorem 1, we can get that (4) and (5) hold.

Similarly, using the other two cut sets, we can also give another form of -fuzzy quotient ring and the corresponding conclusions.

Theorem 13. Let be the ideal of and be the -fuzzy subring of . Then, the following conditions are true:(1)(2)(3)(4)(5)

Proof. (1); first, we prove that . Suppose that ; then, we have So, we can obtain . Obviously, . Now, we prove . Suppose that , so . By So, we can obtain .(2)In addition,Therefore, (2) is proved. Similarly, (3) can be proved to be true.
From (1) and Theorem 1, we can get that (4) and (5) hold.

7. Conclusion

This study proposes the concept of -subset which is subring to some extent, and the -fuzzy subring measure is characterized by four kinds of cut sets. The properties of -fuzzy subring measures under these two operations are also studied. Furthermore, the -fuzzy convexity in a ring is directly induced by -fuzzy subring measure, and some properties of this fuzzy convexity are studied. Next, we give the definition and related properties of the measure of -fuzzy quotient ring and give a new characterization of -fuzzy quotient ring when the measure of -fuzzy quotient ring is 1.

Importantly, this idea can be applied to different algebraic systems, such as unitary rings, domains, and prime ideals. Therefore, fuzzy convexity can be derived from different algebraic systems. In addition, we can study the measurement of fuzzy relation and its application in information system, and the measurement of fuzzy filter is considered to provide a theoretical basis for image processing and fuzzy pattern recognition.

Data Availability

The data that support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the funding from Heilongjiang Education Department (nos. 1355ZD009 and 1354ZD009), the National Natural Science Foundation of China (11871097), the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Excellent Young Talents Project of Heilongjiang Province) (2020YQ07 and ZYQN2019071), and Student Science and Technology Innovation Program of Mudanjiang Normal University (kjcx2020-15mdjnu).

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Copyright © 2022 Ying-Ying An et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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