# A Generalized Definition of Fuzzy Subrings

Authors: Ying-Ying An; Fu-Gui Shi; Lan Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, under the condition thatL 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 L-fuzzy subring measures are presented. The properties of L-fuzzy subring measures under these two kinds of product operations are further studied. In addition, an L-fuzzy convexity is directly induced by L-fuzzy subring measure, and it is pointed out that ring homomorphism can be regarded as L-fuzzy convex preserving mapping and L-fuzzy convex-to-convex mapping. Next, we give the definition and related properties of the measure of L-fuzzy quotient ring and give a new characterization of L-fuzzy quotient ring when the measure of L-fuzzy quotient ring is 1.


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## Body

## 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 L-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 L-fuzzy subgroup measure. Li and Shi introduced the notion of L-fuzzy convex sublattice measure and induced an L-fuzzy convex structure [7]. Han and Shi [8] defined L-convex fuzzy ideal measure and further studied its induced L-fuzzy convex structure. The notions of M-fuzzifying convex structure and L,M-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 ofL-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 L-fuzzy subring measure can induce an L-fuzzy convex structure, and a ring homomorphism between two rings can exactly be regarded as an L-fuzzy convex-to-convex mapping and an L-fuzzy convex preserving mapping. And when the subring A of ring R is fuzzy and the ideal N is crisp, we can also obtain the concept of L-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 L-fuzzy quotient ring is 1. At the same time, we study the relationship between L-fuzzy subring measure and L-fuzzy quotient ring measure. It is proposed that the ring homomorphism from ring to quotient ring can be regarded as L-fuzzy convex-to-convex mapping and L-fuzzy convex preserving mapping.


## 2. Preliminaries

Throughout this study,L is a completely distributive lattice, and the smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively.An elementa in L is called a prime element if a⩾b∧c implies a⩾b or a⩾c. a in L is called co-prime if a⩽b∨c implies a⩽b or a⩽c [25]. The set of non-unit prime elements in L is denoted by PL. The set of non-zero co-prime elements in L is denoted by JL.The binary relation≺ in L is defined as follows: for a,b∈L, a≺b, if and only if, for every subset D⊆L, the relation b⩽supD always implies the existence of d∈D with a⩽d [26]. a∈L|a≺b is called the greatest minimal family of b in the sense of [27], denoted by βb; moreover, define the binary relation ∢ in L as follows: for a,b∈L, b∢a if and only if, for every subset E⊆L, the relation b⩾infE always implies the existence of e∈E with a⩾e [28]. a∈L|b∢a is called the greatest maximal family of a, denoted by αb.In a completely distributive latticeL, α is an ∧-∪ map, β is a union-preserving map, and there exist αb and βb for each b∈L such that b=∨βb=∧αb (see [29]).For a complete distributive latticeL, A∈LX and a∈L, we can define(1)Aa=x∈X|Ax⩾a,Aa=x∈X|Ax≰a,Aa=x∈X|a∈βAx,Aa=x∈X|a∉αAx.Theorem 1 (see [27, 30]).
For eachL-fuzzy set A∈LX, the following conditions are true:(1)
A=∨a∈La∧Aa=∨a∈MLa∧Aa=∨a∈La∧Aa=∨a∈MLa∧Aa(2)
A=∧a∈La∨Aa=∧a∈PLa∨Aa=∧a∈La∨Aa=∧a∈PLa∨Aa
Since a completely distributive lattice is a complete Heyting algebra, there exists a binary operation⟼ in L. Explicitly the implication is given by(2)a↦b=∨c∈L|a∧c⩽b.
We list some properties of the implication operation in the following lemma.Lemma 1 (see [6]).
LetL be a completely distributive lattice and let ⟼ be the implication operator corresponding to ∧. Then, for all a,b,c∈L, aii∈I⊆L, the following statements hold:(1)
⊤⟼a=a(2)
c⩽a⟼b⇔a∧c⩽b(3)
a⟼b=⊤⇔a⩽b(4)
a⟼∧i∈Iai=∧i∈Ia⟼ai, hence, a⟼b⩽a⟼c whenever b⩽c(5)
∨i∈Iai⟼b=∧i∈Iai⟼b, hence, b⟼c⩽a⟼c whenever a ⩽ b(6)
a⟼c∧c⟼b⩽a⟼b(7)
a⟼b∧c⟼d⩽a∧c⟼b∧dLemma 2 (see [6]).
Letf:R⟶R′ be a set mapping. Let μ and η be two L-fuzzy subsets in R and R′, respectively. Then, we have(i)
fL⟶fL←η=η if f is surjective(ii)
fL←fL⟶μ=μ if f is injective, where fL⟶:LR⟶LR′ and fL←:LR′⟶LR are defined by fL⟶μr′=∨μr|fr=r′,fL←η=η∘fDefinition 1.
(see [10]). A mapping C:LX⟶M is called an L,M-fuzzy convexity on X if it satisfies the following conditions:(LMC1)Cχ∅=CχX=⊤M(LMC2) IfAi|i∈Ω⊆LX is nonempty, then ∧i∈ΩCAi⩽C∧i∈ΩAi(LMC3) IfAi|i∈Ω⊆LX is nonempty and totally ordered, then ∧i∈ΩCAi⩽C∨i∈ΩAi
The pairX,C is called an L,M-fuzzy convex space. An L,L-fuzzy convex space is called an L-fuzzy convex space for short.Definition 2.
(see [10]). Let X,C and Y,D be L,M-fuzzy convexity spaces. A mapping f:X⟶Y is called(1)
AnL,M-fuzzy convex preserving mapping provided Dλ⩽CfL←λ for all λ∈LY(2)
AnL,M-fuzzy convex-to-convex mapping provided Cλ⩽DfL⟶λ for all λ∈LS(3)
A mappingf:X⟶Y is called an L,M-fuzzy isomorphism provided f is bijective, L,M-fuzzy convex preserving and L,M-fuzzy convex-to-convex
AnL,L-fuzzy convex preserving mapping is called an L-fuzzy convex preserving mapping, an L,L-fuzzy convex-to-convex mapping is called an L-fuzzy convex-to-convex mapping, and an L,L-fuzzy isomorphism is called an L-fuzzy isomorphism for short.
In [4], Malik and Mordeson introduced the following two operations.Definition 3.
(see [4]). Let A and B be two L-fuzzy subsets of a ring R. Define the L-fuzzy subset AB of R by ∀x∈R:(3)ABx=∨n∈ℕ,x=∑i=1nyizi∧i=1nAyi∧Bzi.Definition 4.
(see [4]). Let Ri|i∈Ω be a collection of rings, and let Ai be an L-fuzzy subset of Xi, ∀i∈Ω. Define the Cartesian product ∏i∈ΩAi of Ai|i∈Ω by ∀xi∈Ri:(4)∏i∈ΩAix=∧i∈ΩAixi.Theorem 2 (see [31]).
LetN be an ideal of ring R, A be an fuzzy subring of R, and A⊃N; then, A/N is an fuzzy subring of R/N.Definition 5.
(see [32]). LetA be an L-fuzzy subring of R and N be an subring of ring R, definition AZNx°N=∨n∈NAx°n. Then, AZN is an L-fuzzy subring of R/N, called L-fuzzy quotient ring of A/N.


## 3. A Generalized definition of Fuzzy Subrings

In this section, firstly, the definition ofL-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 L-fuzzy subgroup measure in [5]. It is natural to introduce the following concept.Definition 6.
Letμ be an L-fuzzy subset in a ring R. Then, the L-fuzzy subring measure ℛμ of μ is defined as(5)ℛμ=∧x,y∈Rμx∧μy⟼μxy∧μx−y.
We also say thatμ is an L-fuzzy subring of R with respect to measure ℛμ.Example 1.
LetZ3 be the remaining class ring of module 3 and specify the operations on Z3 such as regular addition and multiplication. Define A,B,C∈LZ3 by(6)Az=0.8,z=0,0.3,z=1,0.4,z=2,Bz=0,z=0,1,z=1,0.4,z=2,Cz=0,z=0,0,z=1,0,z=2.
We can getℛA=0.3,ℛB=0,andℛC=1.(1)
In fact,Ax∧Ay⟼Ax−y∧Axy=0.8⟼0.8=1,ifx=0,y=0,0.3⟼0.4=1,ifx=0,y=1,0.4⟼0.3=0.3,ifx=0,y=2,0.3⟼0.3=1,ifx=1,y=0,0.3⟼0.3=1,ifx=1,y=1,0.3⟼0.4=1,ifx=1,y=2,0.4⟼0.4=1,ifx=2,y=0,0.3⟼0.3=1,ifx=2,y=1,0.4⟼0.3=0.3,ifx=2,y=2.So, we can obtainℛA=1∧0.3=0.3.(2)
Bx∧By⟼Bx−y∧Bxy=0⟼0=1,ifx=0,y=0,0⟼0=1,ifx=0,y=1,0⟼0=1,ifx=0,y=2,0⟼0=1,ifx=1,y=0,1⟼0=0,ifx=1,y=1,0.4⟼0.4=1,ifx=1,y=2,0⟼0=1,ifx=2,y=0,0.4⟼0.4=1,ifx=2,y=1,0.4⟼0=0,ifx=2,y=2..So, we can obtainℛB=1∧0=0.(3)
Obviously,ℛC=1.Theorem 3.
Letμ be an L-fuzzy subset in a ring R. Then,(7)ℛμ=∧x,y∈Rμx∧μy⟼μxy∧∧x,y∈Rμx∧μy⟼μx+y∧∧y∈Rμy⟼μ−y.Proof.
From Definition6, we can obtain that ∀x,y∈R,(8)ℛμ∧μx∧μy⩽μxy,ℛμ∧μx∧μy⩽μx−y.
In particular, we have that(9)ℛμ∧μy⩽0,ℛμ∧μy⩽ℛμ∧μ0∧μy⩽μ−y,ℛμ∧μy=ℛμ∧μ−y,ℛμ∧μy∧μy=ℛμ∧μx∧μ−y.
This shows that(10)ℛμ⩽∧x,y∈Rμx∧μy⟼μxy,ℛμ⩽∧x,y∈Rμx∧μy⟼μx+y∧∧y∈Rμy⟼μ−y.
So,(11)ℛμ⩽∧x,y∈Rμx∧μy⟼μxy∧∧x,y∈Rμx∧μy⟼μx+y∧∧y∈Rμy⟼μ−y.
Analogously, we can prove(12)ℛμ⩾∧x,y∈Rμx∧μy⟼μxy∧∧x,y∈Rμx∧μy⟼μx+y∧∧y∈Rμy⟼μ−y.
The following lemma is obvious.Lemma 3.
Letμ be an L-fuzzy subset in a ring R. Then, ℛμ⩾a if and only if, for any x,y∈R,(13)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
The next theorem presents some equivalent descriptions ofL-fuzzy subring measure.Theorem 4.
Letμ be an L-fuzzy set in a ring R. Then,(1)
ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R(2)
ℛμ=∨a∈L|∀b⩽a,μbis a subring ofR(3)
ℛμ=∨a∈L|∀b∉αa,μbis a subring ofR(4)
ℛμ=∨a∈L|∀b∈PL,b≱a,μbis a subring ofR(5)
ℛμ=∨a∈L|∀b∈βa,μbis a subring ofR if βa∧b=βa∩βb for any a,b∈LProof.
1⇔2. Suppose that μx∧μy∧a⩽μxy and μx∧μy∧a⩽μx−y, for any x,y∈R. Then, for any b⩽a and for any x,y∈μb, we have(14)μxy⩾μx∧μy∧a⩾b,μx−y⩾μx∧μy∧a⩾b,this showsxy∈μb,x−y∈μb. Therefore, μb is a subring of R. Hence,(15)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩽∨a∈L|∀b⩽a,μbis a subring ofR.Conversely, assume thata∈L and ∀b⩽a, μb is a subring of R. For any x,y∈R, let b=μx∧μy∧a. Then, b⩽a and x,y∈μb; thus, xy, x−y∈μb, i.e.,(16)μxy⩾b=μx∧μy∧a,μx−y⩾b=μx∧μy∧a.This means that(17)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩾∨a∈L|∀b⩽a,μbis a subring ofR.So,1⇔2 is clearly established.1⇔3 Suppose that μx∧μy∧a⩽μxy and μx∧μy∧a⩽μx−y, for any x,y∈R. Then, for any b∉αa and x,y∈μb, we have(18)b∉αμx∪αμy∪αa=αμx∧μy∧μa.
By(19)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,we know(20)αμxy⊆αμx∧μy∧a,αμx−y⊆αμx∧μy∧a.
Hence,b∉αμxy and b∉αμx−y, i.e. xy,x−y∈μb.
This means thatμb is a subring of R and a∈a∈L|∀b∉αa,μbis a subring ofR.
This shows that(21)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩽∨a∈L|∀b∉αa,μbis a subring ofR.
Conversely, assume that(22)a∈a∈L|∀b∉αa,μbis a subring ofR.
Now, we prove, for anyx,y∈R,(23)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Suppose thatb∉αμx∧μy∧a. By(24)αμx∧μy∧a=αμx∪αμx∪αa,we know that b∉αa and x,y∈μb. Since μb is a subring of R, it holds that xy,x−y∈μb, i.e., b∉αμxy and b∉αμx−y.
This shows(25)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
It is proved that(26)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩾∨a∈L|∀b∉αa,μbis a subring ofR.
So,1⇔3 is clearly established;
1⇔4 Suppose that a∈L, and for any x,y∈R,(27)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Letb∈PL,b≱a, and x,y∈μb. Now, we prove xy,x−y∈μb. If xy,x−y∉μb, i.e., μxy⩽b and μx−y⩽b, then(28)μx∧μy∧a⩽μxy⩽b,μx∧μy∧a⩽μx−y⩽b.
Byb∈PL and x,y∈μb, we have a⩽b, which contradicts b≱a. Hence, xy,x−y∈μb. This shows that μb is a subring of R.
Therefore,(29)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩽∨a∈L|∀b∈PL,b≱a,μbis a subring ofR.
Conversely, assume that(30)a∈a∈L|∀b∈PL,b≱a,μbis a subring ofR.
Now, we prove that, for anyx,y∈R,(31)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Letb∈PL and μx∧μy∧a≰b. Then, μx≰b, μy≰b, and a≰b, i.e., x,y∈μb. Since μb is a subring of R, it holds that xy∈μb,x−y∈μb, i.e., μxy≰b,μx−y≰b.
This shows that(32)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Therefore,(33)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩾∨a∈L|∀b∈PL,b≱a,μbis a subring ofR.
So,1⇔4 is clearly established.
1⇔5 Suppose that(34)a∈a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R.
Then, for anyb∈βa and for any x,y∈μb, it holds that(35)b∈βμx∩βμy∩βa=βμx∧μy∧a⊆βμxy,b∈βμx∩βμy∩βa=βμx∧μy∧a⊆βμx−y,i.e., xy,x−y∈μb. This shows that μb is a subring of R. This means that(36)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩽∨a∈L|∀b∈βa,μbis a subring ofR.
Conversely, assume that(37)a∈a∈L|∀b∈βa,μbis a subring ofR.
Now, we prove that, for anyx,y∈R,(38)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Letb∈βμx∧μy∧a. By(39)βμx∧μy∧a=βμx∩βμy∩βa,we know that x,y∈μb and b∈βa. Since μb is a subring of R, it holds that xy,x−y∈μb, i.e., b∈βμxy and b∈βμx−y.
This shows that(40)μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y.
Therefore,(41)ℛμ=∨a∈L|μx∧μy∧a⩽μxy,μx∧μy∧a⩽μx−y,∀x,y∈R⩾a∈L|∀b∈βa,μbis a subring ofR.
So,1⇔5 is clearly established.
In conclusion,1⇔2⇔3⇔4⇔5.


## 4. The Relation betweenL-Fuzzy Subring Measure and L-Fuzzy Convexity

In this section, we will investigate the relation betweenL-fuzzy subring measure and L-fuzzy convexity. We will prove that a ring homomorphism is exactly an L-fuzzy convex preserving mapping and an L-fuzzy convex-to-convex mapping.For eachμ∈LR,ℛμ can be naturally considered as a mapping ℛ:LR⟶L defined by μ⟼ℛμ.The following theorem shows thatℛ is exactly an L-fuzzy convexity on R.Theorem 5.
LetR be a ring. Then, the mapping ℛ:LR⟶L defined by μ⟼ℛμ is an L-fuzzy convexity on R, which is called the L-fuzzy convexity induced by L-fuzzy subring measure on R.Proof.
(LMC1) It is straightforward that(42)ℛχ∅=ℛχR=⊤.(LMC2) Letμii∈Ω be a family of L-fuzzy subsets in a ring R. Now, we prove that(43)ℛ∧i∈Ωμi⩾∧i∈Ωℛμi.Suppose thata⩽∧i∈Ωℛμi, Then, ∀x,y∈R,(44)μix∧μiy∧a⩽μixy,μix∧μiy∧a⩽μix−y.Hence,(45)∧i∈Ωμix∧∧i∈Ωμiy∧a⩽∧i∈Ωμixy,∧i∈Ωμix∧∧i∈Ωμiy∧a⩽∧i∈Ωμix−y.This showsa⩽ℛ∧i∈Ωμi. So, we can obtain ℛ∧i∈Ωμi⩾∧i∈Ωℛμi.(LMC3) Letμi|i∈Ω⊆LR be nonempty and totally ordered. In order to prove ∧i∈Ωℛμi⩽ℛ∨i∈Ωμi, it needs to show that a⩽ℛ∨i∈Ωμi for any a⩽∧i∈Ωℛμi.
By Lemma3, for all i∈Ω, x,y,∈R, we have(46)μix∧μiy∧a⩽μixy,μix∧μiy∧a⩽μix−y.
Letb∈JL such that(47)b≺∨i∈Ωμix∧∨i∈Ωμiy∧a.
Then, we have(48)b≺∨i∈Ωμix,b≺∨i∈Ωμiy,b⩽a.
Hence, there exists somei,j∈Ω such that b⩽μix,b⩽μjy,b⩽a. Since μi|i∈Ω is totally ordered, we assume μj⩽μi; it follows that b⩽μix∧μiy∧a.
By(49)μix∧μiy∧a⩽μixy,μix∧μiy∧a⩽μix−y,we obtain b⩽μixy and b⩽μix−y. Hence, b⩽∨i∈Ωμixy and b⩽∨i∈Ωμx−y.
From the arbitrariness ofb, we have(50)∨i∈Ωμix∧∨i∈Ωμiy∧a⩽∨i∈Ωμixy,∨i∈Ωμix∧∨i∈Ωμiy∧a⩽∨i∈Ωμix−y,
Combining Lemma3, we have a⩽ℛ∨i∈Ωμi. By the arbitrariness of a, we obtain(51)∧i∈Ωℛμi⩽ℛ∨i∈Ωμi.
Therefore,ℛ is an L-fuzzy convexity on R.
Now, we consider theL-fuzzy subring measures of homomorphic image and preimage of L-fuzzy subsets.Theorem 6.
Letf:R⟶R′ be a ring homomorphism, and μ∈LRandη∈LR′. Then,(1)
ℛfL⟶μ⩾ℛμ, and if f is injective, then ℛfL⟶μ=ℛμ(2)
ℛfL←η⩾ℛη, and if f is surjective, then ℛfL←η=ℛηProof.
(1)
can be proved from Theorem4 and the following fact:(52)ℛfL⟶μ=∨a∈L|∀b∈PL,b≱a,fL⟶μbis a subring ofR=∨a∈L|∀b∈PL,b≱a,fL⟶μbis a subring ofR⩾∨a∈L|∀b∈PL,b≱a,μbis a subring ofR=ℛμ.Iff is injective, the above ≥ can be replaced by = . Hence, ℛμ=ℛfL⟶μ.(2)
can be similarly obtained by Theorem5:(53)ℛfL←η=∨a∈L|∀b⩽a,fL←ηbis a subring ofR=∨a∈L|∀b⩽a,fL←ηbis a subring ofR⩾∨a∈L|∀b⩽a,ηbis a subring ofR=ℛμ.
Iff is surjective, the above ≥ can be replaced by = . Thus, we can obtain that ℛη=ℛfL←η.
By Theorem6, we obtain the following theorem.Theorem 7.
Letf:R⟶R′ be ring homomorphism. Let ℛR and ℛR′ be the L-fuzzy convexities induced by L-fuzzy subring measures on R and R′, respectively. Then, f:R,ℛR⟶R′,ℛR′ is an L-fuzzy convex preserving mapping and an L-fuzzy convex-to-convex mapping.


## 5. The Operations ofL-Fuzzy Subrings

In this section, we shall discuss some operation properties ofL-fuzzy subring measures. In a ring R, given two L-fuzzy sets A, B, and AB is defined in Definition 3. Now, we present its representations by means of cut sets.Theorem 8.
LetR be a ring and A,B∈LR. Then, the following conditions are true.(1)
∀a∈L,ABa⊆AaBa⊆AaBa⊆ABa(2)
∀a∈L,ABa⊆AaBa⊆AaBa⊆ABa; in particular, if a∈PL, then ABa=AaBa(3)
AB=∨a∈La∧AaBa=∨a∈MLa∧AaBa(4)
AB=∨a∈La∧AaBa=∨a∈MLa∧AaBa(5)
AB=∧a∈La∨AaBa=∧a∈PLa∨AaBa(6)
AB=∧a∈La∨AaBa=∧a∈PLa∨AaBaProof.
(1)
∀a∈L; first, we prove that ABa⊆AaBa. By(54)x∈ABa⇒a∈βABx=β∨n∈ℕ,x=∑i=1nyizi∧i=1nAyi∧Bzi=∪n∈ℕ,x=∑i=1nyiziβ∧i=1nAyi∧Bzi⊆∪n∈ℕ,x=∑i=1nyizi∩i=1nβAyi∩βBzi.We know that there areyi,zi∈R and n∈ℕ such that x=∑i=1nyizi, a∈βAyi, and a∈βBzi, that is, yi∈Aa,zi∈Ba, so x=∑i=1nyizi∈AaBa. This shows ABa⊆AaBa.It is obvious thatAaBa⊆AaBa.Now, we proveAaBa⊆ABa. Suppose that x∈AaBa.Then, by(55)AaBa=x|∃n∈ℕ,yi∈Aaandzi∈Basuch thatx=∑i=1nyizi,we have(56)ABx=∨n∈ℕ,x=∑i=1nyizi∧i=1nAyi∧Bzi=∨n∈ℕ,x=∑i=1nyizi∧i=1na∧a⩾a.So, we obtainx∈ABa. It is proved that AaBa⊆ABa.(2)
ABa⊆AaBa can be proved from the following implications.(57)x∈ABa⇒ABx≰a⇒∨n∈ℕ,x=∑i=1nyizi∧i=1nAyi∧Bzi≰a⇒∃n∈ℕ,yi,zi∈Rsuch thatx=∑i=1nyizi,Ayi≰a,Bzi≰a⇒∃n∈ℕ,yi,zi∈Rsuch thatx=∑i=1nyizi,yi∈Aa,zi∈Ba⇒x∈AaBa.
In particular, ifa∈PL, then the inverse of the above implications are true. In this case, ABa=AaBa.
It is obvious thatAaBa⊆AaBa. Next, we prove that AaBa⊆ABa. Suppose that x∉ABa. Then, a∈αABx. By(58)a∈αABx=α∨n∈ℕ,x=∑i=1nyizi∧i=1nAyi∧Bzi⊆α∧i=1nAyi∧Bzi⊆∪i=1nαAyi∪αBzi⇒∃i∈ℕ,such that whenx=∑i=1nyizi,a∈αAyiora∈αBzi⇒∃i∈ℕ,such that whenx=∑i=1nyizi,yi∉Aaorzi∉Ba⇒x∉AaBa,we know that AaBa⊆AaBa.
From (1), (2), and Theorem1 we can obtain (3), (4), (5), and (6).Next, two theorems are generalizations of the results in [4].Theorem 9.
LetA,B be two L-fuzzy subsets of a ring R. Then, ℛAB≥ℛA∧ℛB.Proof.
By Theorem4 (4), we can obtain the following fact:(59)ℛAB=∨a∈L|∀b∈PL,b≱a,ABbis a subring ofR=∨a∈L|∀b∈PL,b≱a,AbBbis a subring ofRby Theorem82⩾∨a∈L|∀b∈PL,b≱a,Abis a subring ofR∧∨a∈L|∀b∈PL,b≱a,Bbis a subring ofR=ℛA∧ℛB.Theorem 10.
LetRi|i∈Ω be a collection of rings and Ai an L-fuzzy subset of Ri,∀i∈Ω. Then, ℛ∏i∈ΩAi⩾∧i∈ΩℛAi .Proof.
It is easy to check that∏i∈ΩAi=∧i∈ΩPi−1Ai, where Pi is the projection from ∏i∈ΩRi to Ri. From (LMC2) in the proof of Theorem 5, we can obtain(60)ℛ∏i∈ΩAi=ℛ∧i∈ΩPi−1Ai⩾∧i∈ΩℛPi−1Ai.
SincePi:∏i∈ΩRi⟶Ri is a ring homomorphism, by means of Theorem 5, we have ℛPi−1Ai⩾ℛAi. Thus, we obtain(61)ℛ∏i∈ΩAi⩾∧i∈ΩℛAi.From Theorem6, we can obtain the following corollary.Corollary 1.
LetRi|i∈Ω be a collection of rings, and let ℛΠ and ℛi are, respectively, L-fuzzy convexity induced by L-fuzzy subring measures of ∏i∈ΩRi, Ri. Then, Pi:∏i∈ΩRi,ℛΠ⟶Ri,ℛi is an L-fuzzy convex preserving mapping and an L-fuzzy convex-to-convex mapping.


## 6. A Generalized definition of Fuzzy Quotient Rings

In this section, firstly, we can get the definition ofL-fuzzy quotient ring measure from the given definition of L-fuzzy subring measure; Secondly, we study the relationship between L-fuzzy subring measure and L-fuzzy quotient ring measure. Finally, it is given that the ring homomorphism from ring to quotient ring is L-fuzzy convex-to-convex mapping and L-fuzzy convex preserving mapping. When the measure of L-fuzzy quotient ring is 1, a new characterization of L-fuzzy quotient ring is given by using cut set.Definition 7.
LetA be an L-fuzzy subset in a ring R, and quotient ring R/N=x+N|x∈R. Then, the L-fuzzy quotient ring measure ℛA/N of A is defined as(62)ℛA/N=∧x,y∈RA/Nx+N∧A/Ny+N⟼A/Nx+Ny+N∧A/Nx+N−y+N.
We also say thatA/N is an L-fuzzy quotient ring of R with respect to measure ℛA/N.
The following lemma is obvious.Lemma 4.
LetA be an L-fuzzy subset in a ring R. Then, ℛA/N⩾a if and only if, for any x,y∈R,(63)A/Nx+N∧A/Ny+N∧a⩽A/Nx+Ny+N,A/Nx+N∧A/Ny+N∧a⩽A/Nx+N−y+N.
Next, we study the relationship betweenL-fuzzy subring measure and L-fuzzy quotient ring measure.Corollary 2.
LetA be an L-fuzzy subset in a ring R, N be an ideal of R, and quotient ring R/N=x+N|x∈R. Then,(64)ℛA⩽ℛA/N.Proof.
Suppose thata⩽ℛA. Then, ∀x,y∈R:(65)Ax∧Ay∧a⩽Axy,Ax∧Ay∧a⩽Ax−y.
Hence,(66)A/Nx+N∧A/Ny+N∧a⩽A/Nx+Ny+N,A/Nx+N∧A/Ny+N∧a⩽A/Nx+N−y+N.
This showsa⩽ℛA/N. So, we can obtain(67)ℛA⩽ℛA/N.From the relationship betweenL-fuzzy subring measure and L-fuzzy convexity, we can get the following conclusion: R is a ring, the mapping ℛ defined by A/N⟼ℛA/N:LR⟶L is a L-fuzzy convexity on R, which is called L-fuzzy convexity of L-fuzzy quotient ring measure on R. The ring homomorphisms between the other two quotient rings are L-fuzzy convex preserving mapping and L-fuzzy convex–to-convex mapping.Finally, we show that the ring homomorphism from ringR to quotient ring R/N is L-fuzzy convex preserving mapping and L-fuzzy convex-to-convex mapping.Theorem 11.
LetR be a ring, R/N be a quotient ring, and A be a L-fuzzy subset of R. So, f:R⟶R/NA⟶A/N is a ring homomorphism, and this ring homomorphism is L-fuzzy convex preserving mapping and L-fuzzy convex-to-convex mapping.
Next, when the measure ofL-fuzzy quotient ring is 1, a new characterization of L-fuzzy quotient ring is given by using cut set.Theorem 12.
LetN be the ideal of R and A be the L-fuzzy subring of R. Then, the following conditions are true:(1)
∀a∈L,A/Na⊆Aa/N⊆Aa/N⊆A/Na(2)
A/Na=∩b∈βaA/Nb=∩b∈βaA/Nb(3)
A/Na=∪a∈βbA/Nb=∪a∈βbA/Nb(4)
A/N=∨a∈La∧Aa/N=∨a∈MLa∧Aa/N(5)
A/N=∨a∈La∧Aa/N=∨a∈MLa∧Aa/NProof.
(1)
∀a∈L; first, we prove that A/Na⊆Aa/N; let x+N∈A/Na. By(68)a∈βA/Nx+N=β∨n∈NAx+n=∪n∈NβAx+n.We know that there isn∈N, such that a∈βAx+n, so x+n∈Aa; let −n∈N,sox=x+n−n∈Aa+N, that is, x+N∈Aa/N. This shows A/Na⊆Aa/N.It is obvious thatAa/N⊆Aa/N.Now, we proveAa/N⊆A/Na; suppose that x+N∈Aa/N; then, by(69)Aa/N=x+N|x∈Aa,we haveA/Nx+N=∨n∈NAx+n⩾a. So, we obtain x+N∈A/Na. It is proved that Aa/N⊆A/Na.(2)
∀b∈βa; using the definition of cut set, we can easily obtain(70)A/Na⊆A/Nb⊆A/Nb⇒A/Na⊆∩b∈βaA/Nb⊆∩b∈βaA/Nb,in addition x+N∈∩b∈βaA/Nb⇒A/Nx+N⩾∨b∈L|b∈βa=a⇒x+N∈A/Na.
Therefore, (2) is proved. Similarly, (3) can be proved to be true.
From (1) and Theorem1, we can get that (4) and (5) hold.Similarly, using the other two cut sets, we can also give another form ofL-fuzzy quotient ring and the corresponding conclusions.Theorem 13.
LetN be the ideal of R and A be the L-fuzzy subring of R. Then, the following conditions are true:(1)
∀a∈L,A/Na=Aa/N⊆Aa/N⊆A/Na(2)
A/Na=∩a∈αbA/Nb=∩a∈αbA/Nb(3)
A/Na=∪b∈αaA/Nb=∪b∈αaA/Nb(4)
A/N=∧a∈La∨Aa/N=∧a∈PLa∨Aa/N(5)
A/N=∧a∈La∨Aa/N=∧a∈PLa∨Aa/NProof.
(1)
∀a∈L; first, we prove that A/Na=Aa/N.Suppose thatx+N∈A/Na; then, we have(71)x+N∈A/Na⇔A/Nx+N≰a⇔∨n∈NAx+n≰a⇔∀n∈N,Ax+n≰a⇔∀n∈N,x+n∈Aa⇔x+N∈Aa/N.So, we can obtainA/Na=Aa/N.Obviously,Aa/N⊆Aa/N.Now, we proveAa/N⊆A/Na. Suppose that x+N∉A/Na, so a∈αA/Nx+N. By(72)a∈αA/Nx+N=α∨n∈NAx+n⊆∩n∈NαAx+n⇒∀n∈N,a∈αAx+n⇒∀n∈N,x+n∉Ab⇒x+N∉Aa/N.So, we can obtainAa/N⊆A/Na.(2)
∀a∈αb;using the definition of cut set, we can easily obtain(73)A/Na⊆A/Nb⊆A/Nb⇒A/Na⊆∩a∈αbA/Nb⊆∩a∈αbA/Nb.
In addition,(74)x+N∈∩a∈αbA/Nb⇒a∉α∩a∈αbA/Nx+N⇒a∉αA/Nx+N⇒x+N∈A/Na.
Therefore, (2) is proved. Similarly, (3) can be proved to be true.
From (1) and Theorem1, we can get that (4) and (5) hold.


## 7. Conclusion

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



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