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Research article

A nonstandard compact finite difference method for a truncated Bratu–Picard model

  • Received: 04 May 2024 Revised: 30 July 2024 Accepted: 09 August 2024 Published: 24 September 2024
  • MSC : 34B15, 26E35, 34A34, 65L12

  • In this paper, we used the nonstandard compact finite difference method to numerically solve one-dimensional truncated Bratu-Picard equations and discussed the convergence analysis of the proposed method. Depending on the parameters in the mentioned equation, it may have no solution, one solution, or two solutions; also, it may have infinitely many solutions. The numerical results show that our method covers all mentioned aspects depending on the parameters in the equation.

    Citation: Maryam Arabameri, Raziyeh Gharechahi, Taher A. Nofal, Hijaz Ahmad. A nonstandard compact finite difference method for a truncated Bratu–Picard model[J]. AIMS Mathematics, 2024, 9(10): 27557-27576. doi: 10.3934/math.20241338

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  • In this paper, we used the nonstandard compact finite difference method to numerically solve one-dimensional truncated Bratu-Picard equations and discussed the convergence analysis of the proposed method. Depending on the parameters in the mentioned equation, it may have no solution, one solution, or two solutions; also, it may have infinitely many solutions. The numerical results show that our method covers all mentioned aspects depending on the parameters in the equation.



    Zadeh [1] introduced the notion of fuzzy sets (FSs) which has numerous applications in different branches of science and technology. A fuzzy subset F on a universe K is denoted as {α,μF(α):αK}, where μF is a function from K to [0,1] and is called membership function. Undoubtedly fuzzy set is a generalization of conventional set. In a conventional set A, the membership function is the characteristic function χA. The utilization of fuzzy set theory can be observed in almost every scientific field, particularly those involving set theory and mathematical logic. After the invention of fuzzy sets, many theories were put forward to deal with uncertainty and imprecision. Some of those are extensions of fuzzy sets, while others strive to deal with uncertainty in another suitable way. Later, it has been found that only the membership function is not sufficient to describe certain types of information. In this way, Atanassov [2] extended fuzzy sets to intuitionistic fuzzy sets (IFSs) to give a proper illustration of the information and allow a greater degree of freedom and flexibility in representing uncertainty. An IFS F of a conventional set K is an object {(α,μF(α),νF(α)):αK}, where μF:K[0,1] and νF:K[0,1] are membership and non-membership functions respectively under the condition μR(α)+νR(α)1 for all αK. As compared to FS, the IFS handles uncertainty and vagueness in the field of decision-making [3,4] more effectively but even then, there is room for improvement. There exist many cases where IFSs unable to perform. For example: If a decision maker proposes μF(α)=0.7 and νF(α)=0.4 for some αK. Then μR(α)+νR(α)>1, therefore, such problems are beyond the limitations of IFS theory. To cope with these situations, Yager [5] generalized the notion of intuitionistic fuzzy sets by defining the Pythagorean fuzzy sets (PFSs). In [6], Zhang and Xu led the foundation of this novel concept. The Pythagorean fuzzy subset F of K is denoted by {(α,μF(α),νF(α)):αK}, where μF:K[0,1] and νF:K[0,1] such that (μF(α))2+(νF(α))21 for all αK. This idea is invented to transform vague and uncertain circumstances into mathematical form and to find an efficient solution [7,8].

    The theory of rings [9] is one of the important branches of mathematics. The idea of the ring was originally conceived to prove Fermat's Last Theorem, starting with Dedekind in the 1880s. After commitments from different fields (mainly number theory), the notion of the ring was summarized and firmly established in the decade of 1920 to 1930 by Noether and Krull. Present-day ring theory–an exceptionally dynamic numerical control–ponders rings in their very own right. To investigate rings, mathematicians have concocted different ideas to break rings into littler, better-reasonable pieces, for example, ideals, quotient rings and basic rings. In addition to these abstract attributes, ring theorists also make different qualifications between the theory of commutative rings and noncommutative rings. The commutative rings have a place in algebraic geometry and algebraic number theory. Noncommutative ring theory started with endeavors to extend complex numbers. The origins of commutative and non-commutative ring theories can be traced back to the early 19th century, while their maturity was achieved in the third decade of the 20th century. Over the past decade, ring theory has been applied to various branches of science, especially computer science, coding theory, and cryptography [10,11,12].

    In 1982, Liu [13] generalized the notion of the subring/ideal of a ring to the fuzzy subring/fuzzy ideal of a ring. In [14], different operations between fuzzy ideals of a ring have been defined. Mukherjee and Malik [15] published a fundamental paper on fuzzy ideals over Artinian rings. The author presented a characterization of Artinian rings with respect to fuzzy ideals. Hur et al. [16] defined the notion of intuitionistic fuzzy ideals of a ring. The notion of ω-fuzzy ideal of a ring is defined in [17]. Shabir et al. [18] investigated the approximation of bipolar fuzzy ideals of semirings. In [19], the notion of anti-fuzzy multi-ideals of the near ring is discussed. Madeline et al. [20] defined fuzzy multi-ideals of near rings and proved some fundamental theorems of this notion. A fuzzy version of Zorn's lemma is present in [21]. The author used it to demonstrate that every proper fuzzy ideal of a ring is contained in a maximal fuzzy ideal. Addis et al. [22] proved some important results regarding fuzzy homomorphism. The algebraic characteristics of (α,β)-Pythagorean fuzzy ideals of a ring are discussed in [23]. Hakim et al. presented a study [24] related to bipolar soft semiprime ideals over ordered semigroups. The concept of fuzzy ideals of LA rings is described in [25]. The authors presented a characterization of regular LA-rings with respect to fuzzy ideals.

    The above literature review highlights some research achievements in classical and intuitionistic fuzzy ring theory. Additionally, although certain results about (α,β)-Pythagorean fuzzy subrings of a ring and bipolar Pythagorean fuzzy subring of a ring have been demonstrated but some open questions remain to be answered.

    1) In classical ring theory, the interstation of two subrings/ideals of a ring K is a subring/ideal of K. Therefore, a question arises, whether the intersection of two Pythagorean fuzzy subrings/ideals is a Pythagorean fuzzy subring/ideal of K? Moreover, if S is a subring of a ring K and I is an ideal of K, then SI is an ideal of S. The analogous version of this theorem in the Pythagorean fuzzy framework needs to be studied.

    2) The characterization of classical/intuitionistic fuzzy subrings/ideals with respect to classical/intuitionistic fuzzy level sub-rings/ideals is given in the existing literature. Since a Pythagorean fuzzy ring theory is a generalization of intuitionistic fuzzy ring theory, therefore, it is important to understand the characterization of Pythagorean fuzzy subrings/ideals in terms of Pythagorean fuzzy level fuzzy subrings/ideals.

    3) In classical ring theory, cosets of subring/ideal of a ring K is an important notion because it gives rise to quotient rings. It is well known that the set of all cosets of an ideal of K forms ring under a certain binary operation. A natural question comes into mind, does the set of all Pythagorean fuzzy cosets of Pythagorean fuzzy ideal of K, forms a ring under certain binary operations?

    4) The fundamental theorem of ring homomorphism is one of the finest results in classical ring theory. Therefore, it is necessary to discuss this remarkable theorem in the context of Pythagorean fuzzy rings.

    5) In the literature, various algebraic properties of fuzzy semi-prime ideals have been discussed. Furthermore, the characterization of regular rings by virtue of fuzzy ideals is presented. In the context of Pythagorean fuzzy theory, these studies have yet to be examined from a broader perspective.

    Answering the above-mentioned open problems and bridging the knowledge gap in the existing literature is the ultimate aim of this research.

    The results proved in this paper are valid for Pythagorean fuzzy ideals. Since every IFS is a PFS, therefore, the same hold for intuitionistic fuzzy ideals as well. Moreover, every fuzzy set is an IFS, so the present study can also be applied to fuzzy ideals. However, we cannot apply these results directly to q-rung orthopair fuzzy ideals, picture fuzzy ideals, neutrosophic fuzzy ideals, fuzzy soft ideals and fuzzy hypersoft ideals. Therefore, separate studies are recommended for these generalized structures. This is the main limitation of our research.

    The rest of the paper is set up in this way: Section 2 contains basic definitions and concepts that are required to demonstrate our main results. In Section 3, the internal description of the Pythagorean fuzzy ideal along with its fundamental properties are discussed. The notions of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal is defined in Section 4. We prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Moreover, the Pythagorean fuzzy version of the fundamental theorem of ring homomorphisms has been proved. In Section 5, the concept of the Pythagorean fuzzy semi-prime ideal is defined. We investigate some important algebraic features of this newly defined notion. Furthermore, the characterization of regular rings by virtue of PFI is presented. The conclusion of this paper is presented in Section 6.

    This section contains some notions and concepts which are needed to prove our main theorems.

    Definition 2.1. [13] A FS F={(α,μF(α)):αK} of a ring K is called a fuzzy subring (FSR) of K if for all α1,α2K, the following properties are satisfied:

    ⅰ. μF(α1α2)min{μF(α1),μF(α2)},

    ⅱ. μF(α1α2)min{μF(α1),μF(α2)}.

    The fuzzy ideal (FI) of K has the same definition with the only difference that in condition (ⅱ) "min" is replaced by "max".

    Definition 2.2. [16] An IFS F={(α,μF(α),νF(α)):αK} of K is called an intuitionistic fuzzy subring (IFSR) of K if for all α1,α2K, the following requirements are fulfilled:

    ⅰ. μF(α1α2)min{μF(α1),μF(α2)} and νF(α1α2)max{νF(α1),νF(α2)},

    ⅱ. μF(α1α2)min{μF(α1),μF(α2)} and νF(α1α2)max{νF(α1),νF(α2)}.

    By interchanging "min" and "max" by "max" and "min" respectively, in condition (ⅱ), we obtain the definition of intuitionistic fuzzy ideal (IFI) of K.

    Next, we define Pythagorean fuzzy subrings (PFSRs) and Pythagorean fuzzy ideals (PFIs) of a ring K.

    Definition 2.3 A PFS F={(α,μF(α),νF(α)):αK} of K is called a PFSR of K if for all α1,α2K, the following requirements are fulfilled:

    ⅰ. (μF(α1α2))2min{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2max{(νF(α1))2,(νF(α2))2},

    ⅱ. (μF(α1α2))2min{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2max{(νF(α1))2,(νF(α2))2}.

    The PFS F is known to be PFI of K, if condition (ⅱ) is replaced with (μF(α1α2))2max{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2min{(νF(α1))2,(νF(α2))2}.

    It is evident from Definitions 2.2 and 2.3 that every IFR/IFI is a PFR/PFI. The following example demonstrates that the converse is not true.

    Example 2.1. We know Z4={0,1,2,3} is a ring with respect to addition and multiplication modulo 4. It is easy to verify that F={(0,0.9,0.3),(3,0.7,0.5),(2,0.8,0.4),(4,0.7,0.5)} is a PFSR/PFI of Z4={0,1,2,3}. Since the sum of membership and non-membership values is not less than or equal to one for all elements in Z4, therefore, F is not an IFS and hence not an IFR/IFI.

    Definition 2.4. Let F={(α,μF(α),νF(α):αK} be a PFS of K. Then the set F(σ,ρ)={αK:(μF(α))2σ,(νF(α))2ρ} is known as Pythagorean fuzzy level set (PFLS) of F.

    Definition 2.5. Let F and A be two PFS of K. The product FA of F and A is defined by

    FA={(α,μFA(α),νFA(α):αK},

    where (μFA(α))2=max{min((μF(α1))2,(μA(α2))2):α1,α2K,α1α2=α} and (νFA(α))2=min{max((νF(α1))2,(νA(α2))2):α1,α2K,α1α2=α}.

    This section contains some internal description of the Pythagorean fuzzy ideal and its fundamental properties.

    Remark. 3.1. Every PFI of K is a PFSR of K.

    In the following example, we see a PFSR of K which is not a PFI of K.

    Example. 3.1. Consider S={1,2,3}, then 2S is the power set of S, that is,

    2S={ϕ,S,{1},{2},{3},{1,2},{1,3},{2,3}}

    forms ring under symmetric difference Δ and intersection . Now, it is just a matter of simple calculation to conclude that

    F={(ϕ,0.90,0.20),(S,0.90,0.20),({1},0.60,0.70),({2},0.60,0.70),({3},0.90,0.20),({1,2},0.90,0.20),({1,3},0.60,0.70),({2,3},0.60,0.70)}

    is a PFSR of 2S. But

    (μF({1,2}{1}))2=(μF({1}))2=(0.60)2

    and

    max{(μF({1,2}))2,(μF({1}))2}=(0.90)2

    together imply that F is not a PFI of R.

    Theorem 3.1. Let F be a PFI of a ring K, then,

    ⅰ. (μF(0))2(μF(α))2 and (νF(0))2(νF(α))2 for all αK.

    ⅱ. μF(α1)μF(α2) and νF(α1)νF(α2) for some α1,α2K inplies that (μF(α1α2))2=min[(μF(α1))2,(μF(α2))2] and (νF(α1α2))2=max[(νF(α1))2,(νF(α2))2].

    Proof. ⅰ. Let αK, then (μF(0))2=(μF(αα))2min{(μF(α))2,(μF(α))2}=(μF(α))2. Similarly, we can prove (νF(0))2(νF(α))2 for all αK.

    ⅱ. Assume that α1, α2K such that μF(α1)>μF(α2), then obviously (μF(α1))2>(μF(α2))2.

    Consider

    (μF(α2))2=(μF(α1(α1α2)))2min[(μF(α1))2,(μF(α1α2))2]. (3.1)

    Since (μF(α1))2>(μF(α2))2, therefore, Eq (3.1) yields

    (μF(α2))2(μF(α1α2))2. (3.2)

    Furthermore,

    (μF(α1α2))2min{(μF(α1))2,(μF(α2))2}=(μF(α2))2. (3.3)

    The Eqs (3.2) and (3.3) together imply that

    (μF(α1α2))2=(μF(α2))2=min[(μF(α1))2,(μF(α2))2].

    In the identical way, it can be shown that

    (νF(α1α2))2=max[(νF(α1))2,(νF(α2))2].

    Theorem 3.2. The intersection of two PFIs of K is a PFI of K.

    Proof. Suppose that F1={α,μF1(α),νF1(α)} and F2={α,μF2(α),νF2(α)} are PFIs of K. Then for all α1,α2K, we have

    (μF1F2(α1α2))2=min[(μF1(α1α2))2,(μF2(α1α2))2]
    min[min((μF1(α1))2,(μF1(α2))2),min((μF2(α1))2,(μF2(α2))2)]
    =min[min((μF1(α1))2,(μF2(α1))2),min((μF1(α2))2,(μF2(α2))2)]
    =min[(μF1F2(α1))2,(μF1F2(α2))2].

    Therefore, (μF1F2(α1α2))2min[(μF1F2(α1))2,(μF1F2(α2))2]. Similarly, (νF1F2(α1α2))2max[(νF1F2(α1))2,(νF1F2(α2))2]. Next,

    (μF1F2(α1α2))2=min[(μF1(α1α2))2,(μF2(α1α2))2]
    min[max((μF1(α1))2,(μF1(α2))2),max((μF2(α1))2,(μF2(α2))2)]
    max[min((μF1(α1))2,(μF2(α1))2),min((μF1(α2))2,(μF2(α2))2)]
    =max[(μF1F2(α1))2,(μF1F2(α2))2].

    That is, (μF1F2(α1α2))2max[(μF1F2(α1))2,(μF1F2(α2))2]. The utilization of the same arguments gives (νF1F2(α1α2))2min[(νF1F2(α1))2,(νF1F2(α2))2]. Thus, F1F2 is a PFI of K.

    Theorem 3.3. Let F={α,μF(α),νF(α)} be a PFI of K. Then,

    ⅰ. F+={αK:(μF(α))2=(μF(0))2} is an ideal of K.

    ⅱ. F={αK:(νF(α))2=(νF(0))2} is an ideal of K.

    ⅲ. F={αK:(μF(α))2=(μF(0))2and(νF(α))2=(νF(0))2} is an ideal of K.

    Proof. ⅰ. By definition of F+, we have 0F+. Therefore, F+ is non-empty subset of K.

    Let α1,α2F+, then (μF(α1))2=(μF(0))2=(μF(α2))2. Consider

    (μF(α1α2))2min[(μF(α1))2,(μF(α2))2]=min[(μF(α1))2,(μF(α2))2]
    =min[(μF(0))2,(μF(0))2]=(μF(0))2.

    Moreover, from Theorem 3.1, it can be obtained (μF(0))2(μF(α1α2))2. Therefore, (μF(α1α2))2=(μF(0))2 implying that α1α2F+.

    Now, suppose that αF+ and βK. Then,

    (μF(αβ))2max[(μF(α))2,(μF(β))2]=(μF(0))2.

    In view of Theorem 3.1, (μF(0))2(μF(αβ))2. So, (μF(αβ))2=(μF(0))2αβF+.

    Similarly, it can be proved that βαF+. Thus, F+ is an ideal of K.

    ⅱ. The proof is similar to that of (ⅰ).

    ⅲ. The proof is straightforward by using (ⅰ) and (ⅱ).

    Theorem 3.4. The intersection of a PFSR F and a PFI A of a ring K is a PFI of F.

    Proof. Let α1,α2F, then,

    (μFA(α1α2))2=min[(μF(α1α2))2,(μA(α1α2))2]
    min[min((μF(α1))2,(μF(α2))2),min((μA(α1))2,(μA(α2))2)]
    =min[min((μF(α1))2,(μA(α1))2),min((μF(α2))2,(μA(α2))2)]
    =min[(μFA(α1))2,(μFA(α2))2].

    Therefore, (μFA(α1α2))2min[(μFA(α1))2,(μFA(α2))2]. Similarly, (νFA(α1α2))2max[(νFA(α1))2,(νFA(α2))2]. Next,

    (μFA(α1α2))2=min[(μF(α1α2))2,(μA(α1α2))2]
    min[min((μF(α1))2,(μF(α2))2),max((μA(α1))2,(μA(α2))2)]
    =min[max((μF(α1))2,(μF(α2))2),max((μA(α1))2,(μA(α2))2)],as(μF(α1))2=0=(μF(α2))2
    max[min((μF(α1))2,(μA(α1))2),min((μF(α2))2,(μA(α2))2)]
    =max[(μFA(α1))2,(μFA(α2))2].

    That is, (μFA(α1α2))2max[(μFA(α1))2,(μFA(α2))2]. By using the same arguments, it can be obtainable that (νFA(α1α2))2min[(νFA(α1))2,(νFA(α2))2]. Thus, FA is a PFI of F.

    We present the following example to explain Theorem 3.4.

    Example 3.2. Consider a PFSR F={(a,μF(a)={0.80,ifaZ,0.70,otherwise,νF(a)={0.30,ifaZ,0.40,otherwise):aR} and a PFI A={(a,μA(a)={0.90,ifa=0,0.75,otherwise,νA(a)={0.25,ifa=0,0.50,otherwise):aR} of the ring of real numbers R. Then, F=Z and FA={(a,μFA(a)={0.80,ifa=0,0.75,ifaZ{0}0.70,otherwise,νFA(a)={0.30,ifa=0,0.50,ifaZ{0}0.50,otherwise):aR}. It can be easily validated that FA is a PFI of F=Z.

    Theorem 3.5. A PFS F of a ring K is PFI of K if and only if F(σ,ρ) is an ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1].

    Proof. Suppose that F={(α,μF(α),νF(α):αK} is PFI of K. We want to prove that F(σ,ρ) is an ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1].

    For all such σ and ρ, clearly (μF(0))2σand(νF(0))2ρ. Therefore, 0F(σ,ρ) implying that F(σ,ρ) is a non-empty set.

    Suppose α1,α2F(σ,ρ), which means that (μF(α1))2,(μF(α2))2σ and (νF(α1))2,(νF(α2))2ρ. Since F is PFI of K, therefore,

    (μF(α1α2))2min{(μF(α1))2,(μF(α2))2}min{σ,σ}=σ

    and

    (νF(α1α2))2max{(νF(α1))2,(νF(α2))2}max{ρ,ρ}=ρ

    together imply that α1α2F(σ,ρ).

    Again, assume that αF(σ,ρ) and βK, then (μF(α))2σ and (νF(α))2ρ. Since F is PFI of K, therefore,

    (μF(αβ))2max{(μF(α))2,(μF(β))2}max{σ,(μF(β))2}σ

    and

    (νF(αβ))2max{(νF(α))2,(νF(β))2}min{ρ,(νF(β))2}ρ

    together imply that αβF(σ,ρ). In a similar way, we can prove that βαF(σ,ρ). Thus, F(σ,ρ) is an ideal of K.

    Conversely, let F(σ,ρ) be an ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1]. To show F is a PFI of K, firstly suppose that α1,α2K and let (μF(α1))2=σ1,(μF(α2))2=σ2,(νF(α1))2=ρ1 and (νF(α2))2=ρ2. Then,

    ⅰ. α1,α2F(min(σ1,σ2),max(ρ1,ρ2)), since F(min(σ1,σ2),max(ρ1,ρ2)) is an ideal of K, therefore α1α2F(min(σ1,σ2),max(ρ1,ρ2)), which yields (μF(α1α2))2min(σ1,σ2)=min{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2max(ρ1,ρ2)=max{(νF(α1))2,(νF(α2))2}.

    ⅱ. Either α1F(max(σ1,σ2),min(ρ1,ρ2)) or α2F(max(σ1,σ2),min(ρ1,ρ2)). In both the case, we yield α1α2F(max(σ1,σ2),min(ρ1,ρ2)), since F(max(σ1,σ2),min(ρ1,ρ2)) is an ideal of K, which further implies that (μF(α1α2))2max(σ1,σ2)=max{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2min(ρ1,ρ2)=min{(νF(α1))2,(νF(α2))2}.

    Thus, F is a PFSR of K.

    Theorem 3.6. Suppose that K is a division ring. Then, a PFS F is a PFI of K if and only if (μF(α))2=(μF(1))2(μF(0))2 and (νF(α))2=(νF(1))2(νF(0))2 for all αR{0}.

    Proof. Let F be a PFI of K. Then,

    (μF(α))2=(μF(α.1))2max{(μF(α))2,(μF(1))2}(μF(1))2=(μF(αα1))2
    max{(μF(α))2,(μF(α1))2}(μF(α))2,

    (μF(α))2=(μF(1))2. Similarly, we can obtain that (νF(α))2=(νF(1))2. Finally, the application of Theorem 3.1 (ⅰ) gives the desired result.

    Conversely, let (μF(α))2=(μF(1))2(μF(0))2 and (νF(α))2=(νF(1))2(νF(0))2 for all αR{0}:

    (ⅰ) For all α1,α2K, if α1α2, then, (μF(α1α2))2=(μF(1))2min{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2=(νF(1))2max{(νF(α1))2,(νF(α2))2}, and if α1=α2, then, (μF(α1α2))2=(μF(0))2min{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2=(νF(0))2max{(νF(α1))2,(νF(α2))2}.

    (ⅱ) For all α1,α2K, if α1=0 or α2=0, then (μF(α1α2))2max{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2min{(νF(α1))2,(νF(α2))2} is obvious, and if α10 and α20, then (μF(α1α2))2=(μF(1))2=max{(μF(α1))2,(μF(α2))2} and (νF(α1α2))2=(μF(1))2=min{(νF(α1))2,(νF(α2))2}.

    Thus, F is a PFI of K.

    In this section, we define the notion of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms ring under certain binary operations. Furthermore, we prove Pythagorean fuzzy version of fundamental theorem of ring homomorphism.

    We start this section with following theorem which provides basis to define Pythagorean fuzzy cosets (PFCs) of PFI in a ring K.

    Theorem 4.1. Assume that F={(α,μF(α),νF(α)):αK} is a PFI of a ring K and μΓF:KF[0,1] and νΓF: KF[0,1] are defined by μΓF(α+F)=μF(α) and μΓF(α+F)=νF(α) respectively. Then ΓF={(α+F,μΓF(α+F),νΓF(α+F)):α+FKF} is a PFI of KF.

    Proof. Since F is a PFI of K, therefore, F is an ideal of K. Firstly, we show that μΓF and νΓF, used to define PFS ΓF, are well-defined. For this, let

    α1+F=α2+F, where α1,α2K,

    α1α2Fα1,α2F(μF(α1))2=(μF(α2))2and(νF(α1))2=(νF(α2))2(μΓF(α1+F))2=(μΓF(α2+F))2and(νΓF(α1+F))2=(νΓF(α2+F))2.

    Next, we prove that ΓF is a PFI of KF, so let α1+F,α2+FKF. Then,

    (μΓF((α1+F)(α2+F)))2
    =(μΓF((α1α2)+F))2=(μΓF(α1α2))2
    min{(μΓF(α1))2,(μΓF(α2))2}
    =min{(μΓF(α1+F))2,(μΓF(α2+F))2}.

    Similarly,

    (νΓF((α1+F)(α2+F)))2max{(νΓF(α1+F))2,(νΓF(α2+F))2}.

    Moreover,

    (μΓF((α1+F)(α2+F)))2
    =(μΓF(α1α2+F))2=(μF(α1α2))2
    max{(μF(α1))2,(μF(α2))2}
    =max{(μΓF(α1+F))2,(μΓF(α2+F))2}.

    Similarly,

    (νΓF((α1+F)(α2+F)))2min{(νΓF(α1+F))2,(νΓF(α2+F))2}.

    Thus, we conclude that ΓF is a PFI of KF.

    The Example 4.1 describes the result proved in Theorem 4.1.

    Example 4.1. Consider a PFI F of Z6, a ring of integers modulo 6, that is,

    F={(0,0.95,0.15),(1,0.70,0.40),(2,0.95,0.15),(3,0.70,0.40),(4,0.95,0.15),(5,0.70,0.40)}.

    Then, we have F={0,2,4} and the quotient ring Z6F={{0,2,4},{1,3,5}}. Now, following the technique described in Theorem 4.1, we construct a PFS ΓF of Z6F as follows:

    ΓF={({0,2,4},0.95,0.15),({1,3,5},0.70,0.40)}.

    As can be seen, the constructed PFS ΓF is a PFI of Z6F.

    Theorem 4.2. Suppose that I and O are ideal and PFI of a ring K and KI respectively. If μO(α+I)=μO(I) and νO(α+I)=νO(I)αI, then there exists a PFI F of K such that F=I.

    Proof. Let us define a PFS F of K in the following way:

    μF(α)=μO(α+I) and νF(α)=νO(α+I), for all αK,

    (μF(α1α2))2=(μO((α1α2)+I))2
    =(μO((α1+I)(α1+I)))2
    min{(μO(α1+I))2,(μO(α2+I))2}
    =min{(μF(α1))2,(μF(α2))2}.

    Similarly,

    (μF(α1α2))2max{(νF(α1))2,(νF(α2))2}.

    Also,

    (μF(α1α2))2=(μO(α1α2+I))2=(μO((α1+I)(α1+I)))2
    max{(μO(α1+I))2,(μO(α2+I))2}
    =max{(μF(α1))2,(μF(α2))2}.

    Similarly,

    (μF(α1α2))2min{(νF(α1))2,(νF(α2))2}.

    This implies that F is a PFI of K.

    Next, αF(μF(α))2=(μF(0))2 and (νF(α))2=(νF(0))2(μO(α+I))2=(μF(μO(I)))2 and (νF(μO(α+I)))2=(νF(μO(I)))2αI. Thus, F=I.

    We verify the result proved in Theorem 4.2 in the following example.

    Example 4.2. Consider an ideal 4Z of Z. Then, Z4Z={4Z,1+4Z,2+4Z,3+4Z}. Keeping in mind the condition given in Theorem 4.2, we design a PFI O of Z4Z as follows:

    O={(4Z,0.90,0.20),(1+4Z,0.70,0.40),(2+4Z,0.80,0.35),(3+4Z,0.70,0.40)}.

    Again, we obtain a PFI F={n,μF(n)={0.90,ifn4Z,0.80,ifn2+4Z0.70,otherwise,,νF(n)={0.20,ifn4Z,0.35,ifn2+4Z,0.40,otherwise,} of Z by using the membership and non-membership functions defined in Theorem 4.2. It is easy to find F=4Z, which affirms Theorem 4.2.

    Definition 4.1. Let F be a PFI of a ring K and βK. Then, the PFS Fβ={(α,μFβ(α),νFβ(α):αK} of K, where (μFβ(α))2=(μF(αβ))2 and (νFβ(α))2=(νF(αβ))2, is called Pythagorean fuzzy coset (PFC) of PFI F in a ring K associated with β.

    The concept of PFC of a PFI F in a ring K is explained in the example below.

    Example 4.3. Consider a PFI F of Z6 as follows:

    F={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)}.

    Next, we find PFCs Fβ of F associated with all βZ6.

    (ⅰ) The PFC of 𝐽 with respect to 0 is

    F0={(0,μF(00),νF(00)),(1,μF(10),νF(10)),(2,μF(20),νF(20)),(3,μF(30),νF(30)),(4,μF(40),νF(40)),(5,μF(50),νF(50))}
    ={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)}.

    (ⅱ) The PFC of 𝐽 with respect to 1 is

    F1={(0,μF(01),νF(01)),(1,μF(11),νF(11)),(2,μF(21),νF(21)),(3,μF(31),νF(31)),(4,μF(41),νF(41)),(5,μF(51),νF(51))}
    ={(0,μF(5),νF(5)),(1,μF(0),νF(0)),(2,μF(1),νF(1)),(3,μF(2),νF(2)),(4,μF(3),νF(3)),(5,μF(4),νF(4))}
    ={(0,0.70,0.50),(1,0.80,0.40),(2,0.70,0.50),(3,0.70,0.50),(4,0.80,0.40),(5,0.70,0.50)}.

    (ⅲ) The PFC of 𝐽 with respect to 2 is

    F2={(0,μF(02),νF(02)),(1,μF(12),νF(12)),(2,μF(22),νF(22)),(3,μF(32),νF(32)),(4,μF(42),νF(42)),(5,μF(52),νF(52))}
    ={(0,μF(4),νF(4)),(1,μF(5),νF(5)),(2,μF(0),νF(0)),(3,μF(1),νF(1)),(4,μF(2),νF(2)),(5,μF(3),νF(3))}
    ={(0,0.70,0.50),(1,0.70,0.50),(2,0.80,0.40),(3,0.70,0.50),(4,0.70,0.50),(5,0.80,0.40)}.

    (iv) The PFC of 𝐽 with respect to 3 is

    F3={(0,μF(03),νF(03)),(1,μF(13),νF(13)),(2,μF(23),νF(23)),(3,μF(33),νF(33)),(4,μF(43),νF(43)),(5,μF(53),νF(53))}
    ={(0,μF(3),νF(3)),(1,μF(4),νF(4)),(2,μF(5),νF(5)),(3,μF(0),νF(0)),(4,μF(1),νF(1)),(5,μF(2),νF(2))}
    ={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)}.

    (v) The PFC of 𝐽 with respect to 4 is

    F4={(0,μF(04),νF(04)),(1,μF(14),νF(14)),(2,μF(24),νF(24)),(3,μF(34),νF(34)),(4,μF(44),νF(44)),(5,μF(54),νF(54))}
    ={(0,μF(2),νF(2)),(1,μF(3),νF(3)),(2,μF(4),νF(4)),(3,μF(5),νF(5)),(4,μF(0),νF(0)),(5,μF(1),νF(1))}
    ={(0,0.70,0.50),(1,0.80,0.40),(2,0.70,0.50),(3,0.70,0.50),(4,0.80,0.40),(5,0.70,0.50)}.

    (vi) The PFC of 𝐽 with respect to 5 is

    F5={(0,μF(05),νF(05)),(1,μF(15),νF(15)),(2,μF(25),νF(25)),(3,μF(35),νF(35)),(4,μF(45),νF(45)),(5,μF(55),νF(55))}
    ={(0,μF(1),νF(1)),(1,μF(2),νF(2)),(2,μF(3),νF(3)),(3,μF(4),νF(4)),(4,μF(5),νF(5)),(5,μF(0),νF(0))}
    ={(0,0.70,0.50),(1,0.70,0.50),(2,0.80,0.40),(3,0.70,0.50),(4,0.70,0.50),(5,0.80,0.40)}.

    Thus, there are three distinct PFCs of 𝐽 in terms of all elements of Z6, namely F0=F3, F1=F4 and F2=F5.

    Theorem 4.3. Let F be a PFI of K. Then RF, the set of all PFCs of F in K, forms ring with the following binary operations:

    Fβ+Fγ=Fβ+γandFβFγ=Fβγforallβ,γK.

    Proof. Firstly, we will prove that both the binary operations defined on RF are well-defined.

    Suppose that β,γ,ζ,ηK and Fβ=Fγ and Fζ=Fη. Then for all αK,

    μFβ(α)=μFγ(α)andνFβ(α)=νFγ(α), (4.1)
    μFζ(α)=μFη(α)andνFζ(α)=νFη(α). (4.2)

    So,

    μF(αβ)=μF(αγ)andνF(αβ)=νF(αγ), (4.3)
    μF(αζ)=μF(αη)andνF(αζ)=νF(αη). (4.4)

    Putting α=β+ζη in (4.3), α=ζ in (4.4) and α=β in (4.3), we have

    μF(ζη)=μF(β+ζηγ)andνF(ζη)=νF(β+ζηγ), (4.5)
    μF(0)=μF(ζη)andνF(0)=νF(ζη), (4.6)
    μF(0)=μF(βγ)andνF(0)=νF(βγ). (4.7)

    Now,

    (μFβ(α))2+(μFζ(α))2=(μFβ+ζ(α))2=(μF(αβζ))2
    =(μF((αγη)(βγ+ζη)))2
    min{(μF(α(γ+η)))2,(μF(βγ+ζη))2}
    min{(μF(α(γ+η)))2,(μF(0))2}(byusing(4.6)and(4.7))
    =(μF(α(γ+η)))2=(μFγ+η(α))2=(μFγ(α))2+(μFη(α))2.

    So,

    (μFβ(α))2+(μFζ(α))2(μFγ(α))2+(μFη(α))2. (4.8)

    Similarly, we can prove that

    (μFγ(α))2+(μFη(α))2(μFβ(α))2+(μFζ(α))2. (4.9)

    The inequalities (4.8) and (4.9) yields

    (μFβ(α))2+(μFζ(α))2=(μFγ(α))2+(μFη(α))2. (4.10)

    By using (4.1) and (4.2), we obtain

    μFβ(α)+μFζ(α)=μFγ(α)+μFη(α). (4.11)

    The similar reasoning leads us to

    νFβ(α)+νFζ(α)=νFγ(α)+νFη(α). (4.12)

    Again, the utilization of the same method gives us

    μFβ(α)μFζ(α)=μFγ(α)μFη(α), (4.13)

    and

    νFβ(α)νFζ(α)=νFγ(α)νFη(α). (4.14)

    The Eqs (4.11)–(4.14) yield that

    Fβ+Fζ=Fγ+FηandFβFζ=FγFη.

    Hence, both addition and multiplication defined on RF are well-defined. Next, it is easy to verify that F0=F serves as additive identity of RF, and for each FβRF there exists FβRF such that Fβ+Fβ=F0=Fβ+Fβ. The remaining properties are routine computations.

    The following example illustrates the fact mentioned in Theorem 4.3.

    Example 4.4. In Example 4.3, we find that the set of all PFCs of F={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)} in Z6 is RF={F0,F1,F2}. Consider the Cayley's tables (see Tables 1 and 2) of RF obtained by employing the operations defined in Theorem 4.3 as follows:

    Table 1.  Cayley's table of (RF,+).
    + F0 F1 F2
    F0 F0 F1 F2
    F1 F1 F2 F0
    F2 F2 F0 F1

     | Show Table
    DownLoad: CSV
    Table 2.  Cayley's table of (RF,.).
    . F0 F1 F2
    F0 F0 F0 F0
    F1 F0 F1 F2
    F2 F0 F2 F1

     | Show Table
    DownLoad: CSV

    From Tables 1 and 2, we see that RF is a ring under defined binary operations.

    Remark 4.1. If F={(α,μF(α),νF(α):αK} is a PFI of a ring K such that μF and νF are constant functions, then RF={F0}.

    Definition 4.2. Let F be a PFI of a ring K, then the PFI F' of RF defined by μF'(Fβ)=μF(β) and νF'(Fβ)=νF(β), for all βK, is called Pythagorean fuzzy quotient ideal (PFQI) associated with F.

    Theorem 4.4. If F={(α,μF(α),νF(α):αK} is a PFI of a ring K, then a mapping θ:KRF defined by θ(β)=Fβ for βK is a ring homomorphism with kernel F.

    Proof. Assume that β,γK, then,

    θ(β+γ)=Fβ+γ=Fβ+Fγ=θ(β)+θ(γ)

    and

    θ(βγ)=Fβγ=FβFγ=θ(β)θ(γ).

    First, we prove that if (μF(β))2=(μF(0))2 and (νF(β))2=(νF(0))2 if and only if Fβ=F0.

    Suppose that (μF(β))2=(μF(0))2 and (νF(β))2=(νF(0))2. Then for all αK, we have (μF(α))2(μF(β))2=(μF(0))2 and (νF(α))2(νF(β))2=(νF(0))2. If (μF(α))2<(μF(β))2(μF(αβ))2=(μF(α))2 by Theorem 3.1 (ⅱ). On the other hand, if (μF(α))2=(μF(β))2, then,

    α,β{γK:(μF(γ))2=(μF(0))2}(μF(αβ))2=(μF(α))2=(μF(0))2.

    So, in either case, we have

    (μF(αβ))2=(μF(α))2forallαK.

    In a similar way, we can prove

    (νF(αβ))2=(νF(α))2forallαK.

    Thus, Fβ=F0.

    Conversely, suppose that Fβ=F0, so for all αK, (μF(αβ))2=(μF(α0))2 and (νF(αβ))2=(νF(α0))2. Then,

    (μF(β))2=(μF(α(αβ)))2min{(μF(α))2,(μF(αβ))2}=(μF(α))2.

    Thus, for all αK, we obtain (μF(β))2(μF(α))2, hence, (μF(β))2=(μF(0))2.

    The similar reasoning yields that (νF(β))2=(νF(0))2. Now,

    Kerθ={βK:θ(β)=F0}={βK:Fβ=F0}={βK:(μF(β))2=(μF(0))2and(νF(β))2=(νF(0))2}=F.

    The following example explains the fact given in Theorem 4.4.

    Example 4.5. In Example 4.2, we computed RF={F0,F1,F2} for PFI F={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)} of Z6. By using the approach defined in Theorem 4.4, we define θ:Z6RF as follows:

    θ(0)=θ(3)=F0,θ(1)=θ(4)=F1andθ(2)=θ(5)=F2.

    Now, θ(1+2)=θ(3)=F0 and θ(1)+θ(2)=F1+F2=F3=F0 show that θ(1+2)=θ(1)+θ(2). Furthermore, θ((1)(2))=θ(2)=F2 and θ(1)θ(2)=F1F2=F2 together imply that θ((1)(2))=θ(1).θ(2). Similarly, it is easy to verify that θ(n+m)=θ(n)+θ(m) and ((n)(m))=θ(n).θ(m) for all n,mZ6. It means that θ is a ring homomorphism. Also, since F0 is zero of RF, therefore, Ker θ={0,3}=F satisfying Theorem 4.4.

    Next, we present an analogue of Fundamental theorem of homomorphism.

    Theorem 4.5. Let F be a PFI of a ring K, then every PFI of RF corresponds in a natural way to a PFI of K.

    Proof. Let F' be a PFI of RF. Define PFS P of K in the following way:

    (μP(β))2=(μF'(Fβ))2and(νP(β))2=(νF'(Fβ))2.

    It is simple to verify that P is a PFI of K.

    In this section, the concept of Pythagorean fuzzy semi-prime ideals is defined. We investigate some important algebraic features of this newly defined notion. Furthermore, the characterization of regular rings by virtue of PFI is presented.

    Definition 5.1. A PFI F of a ring K is called Pythagorean fuzzy semi-prime ideal (PFSPI) of K if for any PFI P of K, PnFPF for all nN.

    Theorem 5.1. A PFI F of a ring K is PFSPI if and only if F(σ,ρ) is a semi-prime ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1].

    Proof. Suppose that F is a PFSPI of K and βK such that βnF(σ,ρ). Let us define a PFI P of K in the following way:

    (μP(γ))2={σ,ifγβ,0,otherwise,and(νP(γ))2={ρ,ifγβ,1,otherwise.

    Suppose that (μnP(γ))20, then γ=γ1γ2,,γn such that (μP(γi))20, for all i=1,2,,n, because otherwise, there exists γj in each representation γ1γ2,,γn of γ such that (μP(γj))2=0, therefore,

    (μnP(γ))2=(μPμn1P(γjγ1γ2,,γj1γj+1,γj+2,,γn))2=0,

    which is a contradiction.

    Similarly, we can prove that (νnP(γ))21 implies γ=γ1γ2,,γn such that (νP(γi))21, for all i=1,2,,n. Thus,

    (μP(γi))2=σ=(μnP(γ))2and(νP(γi))2=ρ=(νnP(γ))2γiβ,bydefinitionofPγ=γ1,γ2,,γnβnF(σ,ρ),sinceβnF(σ,ρ)γF(σ,ρ)(μF(γ))2σand(νF(γ))2ρ(μF(γ))2(μnP(γ))2and(νF(γ))2(νnP(γ))2PnFPF,sinceFisaPFSPIofK(μF(β))2(μP(β))2=σand(νF(β))2(νP(β))2=ρβF(σ,ρ).

    Thus, F(σ,ρ) is a semi-prime ideal of K.

    Conversely, suppose that F(σ,ρ) is a semi-prime ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1]. Assume that F is not a PFSPI of K. It means that there exists a PFI P of K such that PnF but PF. Therefore, for some βK, we have

    (μP(β))2>(μF(β))2or(νP(β))2<(νF(β))2. (5.1)

    Let (μF(β))2=σ and (νF(β))2=ρ, then βF(σ,ρ), so βnF(σ,ρ). Moreover βF(σ',ρ') for all σ'>σ and ρ'<ρ. Since F(σ',ρ') is a semi-prime ideal of K, therefore, βnF(σ',ρ'). Thus, (μF(βn))2=σ and (νF(βn))2=ρ. Therefore,

    (μF(βn))2=(μF(β))2and(νF(βn))2=(νF(β))2. (5.2)

    Next, it can be easily verified that

    (μnP(β))2(μP(β))2and(νnP(β))2(νP(β))2. (5.3)

    Then, (5.1)–(5.3) imply that, either (μnP(β))2>(μF(βn))2 or (νnP(β))2<(νF(βn))2. This means that PnF, thus we reach at a contradiction.

    Example 5.1. Consider a PFI F={(0,0.90,0.35),(1,0.80,0.60),(2,0.80,0.60),(3,0.80,0.60),(4,0.90,0.35),(5,0.80,0.60),(6,0.80,0.60),(7,0.80,0.60),(8,0.90,0.35),(9,0.80,0.60),(10,0.80,0.60),(11,0.80,0.60)} of Z12. The Pythagorean fuzzy level ideal F(0.902,0.352)={0,4,8} is not a semi-prime ideal of Z12 since ({0,2,4,6,8,10})2={0,4,8}F(0.902,0.352) but {0,2,4,6,8,10}F(0.902,0.352).

    Also, F is not a PFSPI ideal of Z12 because there exists a PFI P={(0,0.90,0.35),(1,0.80,0.60),(2,0.90,0.35),(3,0.80,0.60),(4,0.90,0.35),(5,0.80,0.60),(6,0.90,0.35),(7,0.80,0.60),(8,0.90,0.35),(9,0.80,0.60),(10,0.90,0.35),(11,0.80,0.60)} of Z12 such that P2=PP=FF but PF. This satisfies the conditional statement "if F is PFSPI of K, then F(σ,ρ) is a semi-prime ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1]" expressed in Theorem 5.1.

    Example 5.2. Consider a PFI F={(0,0.90,0.30),(1,0.60,0.50),(2,0.60,0.50),(3,0.60,0.50),(4,0.80,0.40),(5,0.60,0.50),(6,0.60,0.50),(7,0.60,0.50)} of Z8. It is not a PFSPI ideal of Z8 because there exists a PFI P={(0,0.90,0.30),(1,0.60,0.50),(2,0.80,0.40),(3,0.60,0.50),(4,0.80,0.40),(5,0.60,0.50),(6,0.80,0.40),(7,0.60,0.50)} of Z8 such that P2=PP=FF but PF.

    Furthermore, for all σ[0,(μF(0))2] and ρ[(νF(0))2,1], we have the following three Pythagorean fuzzy level ideals F(σ,ρ):

    ⅰ. F(σ,ρ)={0}=I1 where 0.80<σ0.90 and 0.30ρ<0.40,

    ⅱ. F(σ,ρ)={0,4}=I2 where 0.60<σ0.80 and 0.40ρ<0.50,

    ⅲ. F(σ,ρ)=Z8=I3 where 0σ0.60 and 0.50ρ1.

    The Pythagorean fuzzy level ideal I2={0,4} is not a semi-prime ideal of Z8 as ({0,2,4,6})2={0,4}I2 but {0,2,4,6}I2. Thus, the conditional statement "if F(σ,ρ) is a semi-prime ideal of K for all σ[0,(μF(0))2] and ρ[(νF(0))2,1], then F is PFSPI of K" revealed in Theorem 5.1, is satisfied.

    Theorem 5.2. If F is a PFSPI of K, then RF, the set of all PFCs of F in K, has no non-zero nilpotent elements.

    Proof. Suppose that F is a PFSPI of a ring K, then by Theorem 5.1, it follows that F(σ,ρ) is a semi-prime ideal of K, where σ=(μF(0))2 and ρ=(νF(0))2. Moreover, in view of Theorem 4.3 it follows that RFKF(σ,ρ).

    Let α+F(σ,ρ) be non-zero nilpotent element of KF(σ,ρ). Therefore,

    (α+F(σ,ρ))n=F(σ,ρ)
    αn+F(σ,ρ)=F(σ,ρ)
    αnF(σ,ρ)αF(σ,ρ)
    α+F(σ,ρ)=F(σ,ρ).

    Thus, we have a contradiction, thus, KF(σ,ρ) has no non-zero nilpotent element. This together with RFKF(σ,ρ) leads to the desired result.

    Example 5.3. Consider a PFI F of Z6 as follows:

    F={(0,0.80,0.40),(1,0.70,0.50),(2,0.70,0.50),(3,0.80,0.40),(4,0.70,0.50),(5,0.70,0.50)}.

    Clearly, it has two non-empty Pythagorean fuzzy level subsets namely {0, 3} and Z6 which are semi-prime ideals of Z6. Therefore, by Theorem 5.1, F is PFSPI of Z6.

    Also, the set of all PFCs of F in Z6 is RF={F0,F1,F2}. One can see that both non-zero elements F1 and F2 are not nilpotent. Thus, Theorem 5.2 is satisfied.

    Definition 5.2. Let K be a ring and UK. Suppose that χU:K[0,1] and χUc:K[0,1] are defined by

    χU(α)={1,ifαU,0,ifαU,andχUc(α)={0,ifαU,1,ifαU.

    Then,

    Ψ(U)={(α,χU(α),χUc(α)):αK}

    is a PFS of K.

    Lemma 5.1. Ψ(U) is PFI of K if U is an ideal of K.

    The proof involves simple computation.

    Theorem 5.3. A ring K is regular if and only if FO=FO, where F and O are PFIs of K.

    Proof. Suppose that K is a regular ring. We want to show that FO=FO. From routine computations, we get FOFO. Let βK, the regularity of K ensures the existence of ζ and η in K such that β=ζη. Now,

    (μFO(β))2=max{min((μF(ζ))2,(μO(η))2)}.

    Since β=βγβ for some γK. Then,

    (μF(β))2=(μF(βγβ))2max{(μF(βγ))2,(μF(β))2}(μF(βγ))2
    max{(μF(β))2,(μF(γ))2}(μF(β))2.

    In short, (μF(β))2(μF(βγ))2(μF(β))2, therefore, (μF(βγ))2=(μF(β))2. Then,

    (μFO(β))2=max{min((μF(ζ))2,(μO(η))2)}min((μF(βγ))2,(μO(β))2),takingζ=βγandη=β
    =min((μF(β))2,(μO(β))2)=(μFO(β))2.

    Similarly, we obtain (ηFO(β))2(ηFO(β))2, which gives FOFO.

    Conversely, suppose that FO=FO. Let Y and Z be two ideals of K. In view of Lemma 5.1, Ψ(Y)={(α,χY(α),χYc(α)):αK} and Ψ(Z)={(α,χZ(α),χZc(α)):αK} are PFIs of K. Assume that βYZ, then (χY(β))2(χz(β))2=1 and (χYc(β))2(χZc(β))2=0. Since Ψ(Y)Ψ(Z)=Ψ(Z)Ψ(Z), therefore, (χY(β))2(χz(β))2=(χY(β))2(χz(β))2=1 and (χYc(β))2(χZc(β))2=(χYc(β))2(χZc(β))2=0, therefore,

    max{min((χY(β1))2,(χz(β2))2):β1β2=β,β1,β2K}=1andmin{max((χYc(β1))2,(χZc(β2))2):β1β2=β,β1,β2K}=0.

    It means that there exists γ1,γ2K such that

    (χY(γ1))2=1=(χZ(γ2))2and(χYc(γ1))2=0=(χZc(γ2))2,

    with β=γ1γ2. Thus, β=γ1γ2YZ, which gives YZY.Z. Furthermore, Y.ZYZ is obvious. So, YZ=Y.Z, then the regularity of K is directly followed by using the theorem on page 184 of [26].

    Theorem 5.4. A ring K is regular if and only if every PFI of K is idempotent.

    Proof. Let K be a regular ring and F be a PFI of K. Then, in view of Theorem 5.3, it is straightforward to show that F2=F.

    Conversely, let every PFI of K. Assume that F and O are PFIs of K. In view of Theorem 5.3, we require FO=FO to prove the regularity of K. For this, we proceed as follows: be idempotent

    (μFO(β))2=(μFO2(β))2=max{min((μFO(ζ))2,(μFO(η))2):β=ζη}
    max{min((μF(ζ))2,(μO(η))2):β=ζη}=(μFO(β))2.

    The same reasoning leads us to (νFO(β))2(νFO(β))2. So, FOFO. Furthermore FOFO is obvious. Thus, FO=FO.

    Lemma 5.2. (Ψ(β))2=Ψ(β2) for all βK.

    The proof is simple.

    Theorem 5.5. A commutative ring K is regular if and only if every PFI of K is PFSPI.

    Proof. Suppose that K is a regular ring and F is a PFI of K. Let P be any PFI of K such that PnF. Since K is regular, therefore by Theorem 5.4, we obtain Pn=P. Thus, PF. This shows that F is a PFSPI of K.

    Conversely, suppose that every PFI of K is PFSPI. In view of Lemma 5.2, we have (Ψ(β))2=Ψ(β2) for all βK. Since Ψ(β2) is PFSPI of K, therefore, Ψ(β)Ψ(β2). Also, Ψ(β2)Ψ(β) is obvious. Hence, Ψ(β)Ψ(β2). It means that βΨ(β2), therefore, β=αβ2=βαβ for some αK. Thus, K is a regular ring.

    Example 5.4. We know Z5Z is a regular ring. We design a PFI F of Z5Z as follows:

    F={(0+5Z,0.80,0.40),(1+5Z,0.70,0.50),(2+5Z,0.70,0.50),(3+5Z,0.70,0.50),(4+5Z,0.70,0.50)}.

    It is easy to find that F has two non-empty Pythagorean fuzzy level subsets namely {0+5Z} and Z5Z. Both of them are semi-prime ideals of Z5Z. Therefore, by Theorem 5.1, F is PFSPI of Z5Z satisfying the conditional statement "if a commutative ring K is regular then every PFI of K is PFSPI" expressed in Theorem 5.5.

    Example 5.5. Consider a PFI F={(0,0.90,0.35),(1,0.80,0.60),(2,0.80,0.60),(3,0.80,0.60),(4,0.90,0.35),(5,0.80,0.60),(6,0.80,0.60),(7,0.80,0.60),(8,0.90,0.35),(9,0.80,0.60),(10,0.80,0.60),(11,0.80,0.60)} of Z12. Since Z12 contains zero divisors, therefore it not a regular ring. Moreover, in Example 4.6, we see that F is not a PFSPI of Z12. Thus, the conditional statement "if every PFI of a commutative ring K is PFSPI then K is regular" revealed in Theorem 5.5 is verified.

    The basic purpose of this paper is to study the notion of the ideal of a classical ring under Pythagorean fuzzy environment. For this purpose, several notions of ring theory like cosets of an ideal, quotient ideal and semiprime ideal are converted into Pythagorean fuzzy format. We have proved that the intersection of two PFIs of a ring K is a PFI. We also show that the intersection of a PFSR F and PFI A of K is PFI of F. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present a Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. Next, we give the definition and related properties of Pythagorean fuzzy semi-prime ideals. Lastly, the characterization of regular rings by virtue of Pythagorean fuzzy ideals is presented.

    By using the outcomes of present study, our future intention is to investigate the algebraic properties of prime, maximal and irreducible ideals in Pythagorean fuzzy context. Moreover, in future work, we will extend the present concepts under different extensions of the fuzzy sets such as q-rung orthopair fuzzy sets, fuzzy soft sets and fuzzy hypersoft sets etc.

    The authors would like to dedicate this study to one of the leading fuzzy algebraists Professor Naseem Ajmal, Department of Mathematics, University of New Delhi, India.

    The authors declare no conflicts of interest.



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