# On Radical of Intuitionistic Fuzzy Primary Submodule

Authors: Abbas Taherpour; Shaban Ghalandarzadeh; Parastoo Malakooti Rad; Parvin Safari
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we further study the theory of Intuitionistic fuzzy submodules and we will define intuitionistic fuzzy primary submodule with the help of the definition of a radical submodule, and we also study the properties of these submodules. Furthermore, homomorphic image and preimage of intuitionistic fuzzy primary submodule are investigated. This manuscript is online on arXiv by the linkhttps://arxiv.org/abs/2207.09772.


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## Body

## 1. Introduction

Throughout this paper,R is a commutative ring with unity and M is an unitary R-module. A proper submodule P of M is called a prime submodule provided that for any r∈R and m∈M, rm∈P implies that m∈P or r∈P:M=r∈R|rM⊆P. The set of all prime submodules of M is called the prime spectrum of M and is denoted by Spec(M). The Weakly prime submodules have been introduced by Behboodi and Koohy in [1]. Also, the spectrum of weakly prime submodule has been investigated in [2]. Following this, the concepts of primal submodules over the noncommutative ring and nearly prime submodules have been presented in [3, 4].One of the structures that play an important role in mathematics is the structure of fuzzy algebra, which has been used extensively in many fields such as computer science, information technology, theoretical physics, and control engineering. Since the introduction of fuzzy sets in 1965 [5], researchers have done extensive research on various concepts of abstract algebra in the space of fuzzy sets, such as Rosenfeld [6], who in 1971 was the first to define the concept of fuzzy subgroups of a group. Many extensions of this concept have been proposed since then (especially in recent decades). In 1975, Relescu and Naegoita [7] applied the concept of fuzzy sets to the theory of modules. After presenting this definition, different types of fuzzy submodules were examined in the last two decades. In 1986, Atanassov [8] introduced intuitionistic fuzzy sets based on the degree of membership and degree of nonmembership, provided that the total membership and nonmembership should not exceed one another. In 1989, Biswas [9] applied the concept of intuitionistic fuzzy sets to group theory and studied the intuitionistic fuzzy subsets of a group. In the last few years a considerable number of papers have been done on fuzzy and intuitionistic fuzzy submodules in general, and fuzzy and intuitionistic fuzzy prime and primary submodules in particular. Hur et al. [10] introduced the notion of intuitionistic fuzzy prime ideals and intuitionistic fuzzy weakly completely prime ideals in a ring. The concept of the fuzzy prime submodule and fuzzy primary submodule was studied by Mashinchi and Zahedi in [11]. Also, intuitionistic fuzzy submodules and their properties were studied by many mathematicians [12–14]. In this paper, and in the first step we define the intuitionistic fuzzy radical of an intuitionistic fuzzy submodule of an R-module M and so we present some of its properties similar to the studies in the module theory. Also and related to this definition, we will define intuitionistic fuzzy primary submodule and investigate their properties. We show that for any intuitionistic fuzzy primary submodule of an R-module M, with sup property its radical will be an intuitionistic fuzzy prime ideal of R.


## 2. Definitions and Preliminary Results

In this paper, all rings are commutative with unity1≠0 and all R-modules are unitary. We use the symbol θ for the zero element of a R-module. Let X be a nonempty set. An intuitionistic fuzzy subset is a function A=μA,νA, which μAx denotes the degree of membership and νAx denotes the degree of nonmembership of x∈X to subset A, such that 0≤μAx+νAx≤1 for all x∈X, can be denoted by A=x,μAx,νAx. It is clear that when μAx+νAx=1, then A will be a fuzzy subset of X. The class of all intuitionistic fuzzy subsets of X is denoted by IFSX [8, 14].In this section, we give some basic concepts of intuitionistic fuzzy sets and intuitionistic fuzzy modules. For simplicity, we sometimes denote each intuitionistic fuzzy setA=x,μAx,νAx by A=μA,νA.Definition 1 (see [8]).
LetX be a nonempty set and A=μA,νA,B=μB,νB be intuitionistic fuzzy sets of X. Then,(1)A⊆B⇔μAx≤μBx,νAx≥νBx,for allx∈X,A∪B=x,μAx∨μBx,νAx∧νBx|x∈X,A∩B=x,μAx∧μBx,νAx∨νBx|x∈X.Definition 2 (see [15]).
LetR be a ring and A∈IFSR. Then, A is called an intuitionistic fuzzy ideal of R if for all x,y∈R, the followings hold:(1)
μAx−y≥μAx∧μAy,νAx−y≤νx∨νAy(2)
μAxy≥μAx∨μAy,νAxy≤νAx∧νAy
The class of intuitionistic fuzzy ideals ofR is denoted by IFI(R).Definition 3 (see [12]).
LetM be an R-module and A∈IFSM. Then A is called an intuitionistic fuzzy submodule if(1)
μAθ=1,νAθ=0(2)
μAx+y≥μAx∧μAy,νAx+y≤νAx∨νAy for all x,y∈M(3)
μArx≥μAx,νArx≤νAx for all x∈M and r∈R
We denote the class of all intuitionistic fuzzy submodule of an R-moduleM, by IFMM.
Notice that, whenR=M, then A∈IFMM if and only if μAθ=1,νAθ=0 and A∈IFIR.Now, for an intuitionistic fuzzy submodule, we define an intuitionistic fuzzy radical.Definition 4.
LetM be an R-module and A∈IFSM. The intuitionistic fuzzy subset A=μA,νA of R is called the intuitionistic fuzzy radical of A and is defined as follows:(2)μAr=∨n∈ℕ∧m∈MμArn.m,νAr=∧n∈ℕ∨m∈MνArn.m,for all r∈R, m∈M, n∈ℕ.It is clear that whenM=R, then the intuitionistic fuzzy radical of an intuitionistic fuzzy ideal of R will be defined by: A=μA,νA where μAr=∨n∈ℕμArn and νAr=∧n∈ℕνArn. Now we show that for any intuitionistic fuzzy submodule A of M, A is an intuitionistic fuzzy ideal of R.Proposition 1.
LetA=μA,νA be an intuitionistic fuzzy submodule of M then A=μA,νA is an intuitionistic fuzzy ideal of R.Proof.
Letr1,r2∈R then(3)μAr1+r2=∨n∈ℕ∧m∈MμAr1+r2n.m=∨n∈ℕ∧m∈MμAr1+r22n.m∨∨n∈ℕ∧m∈MμAr1+r22n+1.m.
Now, we have(4)∧m∈MμAr1+r22n.m=∧m∈MμA∑i=02nCi2nr12n−ir2i.m≥∧m∈MμAr12n.m∧μAr12n−1.r2.m∧⋯∧μAr1nr2n.m∧μAr1n−1r2n+1.m∧⋯∧μAr22n.m≥∧m∈MμAr12n.m∧μAr12n−1.m∧⋯∧μAr1n.m…∧μAr2n+1.m∧⋯∧μAr22n.m=∧m∈MμAr1n.m∧μAr2n+1.m≥∧m∈MμAr1n.m∧μAr2n.m=∧m∈MμAr1n.m∧∧m∈MμAr2n.m.
Also,(5)∧m∈MμAr1+r22n+1.m=∧m∈MμA∑i=02n+1Ci2n+1r12n+1−ir2i.m≥∧m∈MμAr12n+1.m∧μAr12n.r2.m∧⋯∧μAr1nr2n+1.m∧⋯∧μAr22n+1.m≥∧m∈MμAr12n+1.m∧μAr12n.m∧⋯∧μAr1n+1.m⋯∧μAr2n+1.m∧⋯∧μAr22n+1.m=∧m∈MμAr1n+1.m∧μAr2n+1.m≥∧m∈MμAr1n.m∧μAr2n.m=∧m∈MμAr1n.m∧∧m∈MμAr2n.m.
Therefore,(6)μAr1+r2=∨n∈ℕ∧m∈MμAr1+r22n.m∨∨n∈ℕ∧m∈MμAr1+r22n+1.m≥∨n∈ℕ∧m∈MμAr1n.m∧∧m∈MμAr2n.m∨∨n∈ℕ∧m∈MμAr1n.m∧∧m∈MμAr2n.m=∨n∈ℕ∧m∈MμAr1n.m∧∧m∈MμAr2n.m=∨n∈ℕ∧m∈MμAr1n.m∧∨n∈ℕ∧m∈MμAr2n.m=μAr1∧μAr2.
Similarly, we show thatνAr1+r2≤νAr1∨νAr2.(7)νAr1+r2=∧n∈ℕ∨m∈MνAr1+r2n.m=∧n∈ℕ∨m∈MνA∑i=0nCinr1n−ir2i.m≤∧n∈ℕ∨m∈MνAr1n.m∨νAr1n−1r2.m∨⋯∨νAr1r2n−1.m∨νAr2n.m≤∧n∈ℕ∨m∈MνAr1n.m∨νAr1n−1m∨⋯∨νAr2n−1.m∨νAr2n.msinceAis an intuitionistic fuzzy submodule ofM=∧n∈ℕ∨m∈MνAr1n.m∨νAr2n.m=∧n∈ℕ∨m∈MνAr1n.m∨∧n∈ℕ∨m∈MνAr2n.m=νAr1∨νAr2.
And,μAr1r2=∨n∈ℕ∧m∈MμAr1r2n.m=∨n∈ℕ∧m∈MμAr1nr2n.m≥∨n∈ℕ∧m∈MμAr1n.m=μAr1. Similarly, μAr1r2≥μAr2. Hence μAr1r2≥μAr1∨μAr2.
Also,νAr1r2=νAr1 and νAr1r2≤νAr2, hence νAr1r2≤νAr1∧νAr2. Next for r∈RμA−r=∨n∈ℕ∧m∈MμAr1μA−rn.m=∨n∈ℕ∧m∈MμArn.m=μAr;(8)νA−r=∧n∈ℕ∨m∈MνA−rn.m=∧n∈ℕ∨m∈MνArnm=νAr.
Therefore,A is an intuitionistic fuzzy ideal of R.In the following proposition, we will give some properties of intuitionistic fuzzy radicals.Proposition 2.
LetA, B∈IFMM then(1)
A=A(2)
A⊆B⟹A⊆B(3)
A∩B=A∩B(4)
A+B⊆A+BProof.
(1)
SinceA=μA,νA, so by Proposition 1A is an intuitionistic fuzzy ideal of R then μAr=∨n∈ℕμArn=∨n∈ℕ∨n′∈ℕ∧m∈MμArnn′.m=∨n∈ℕ∨n′∈ℕ∧m∈MμArnn′.m=∨n′′∈ℕ∧m∈MμArn′′.m=μAr. It is clear that νAr=νAr, since the proof is the dual of the proof of the first part. Hence A=μA,νA=A.(2)
By 3,A∩B=m,μAm∧μBm,νAm∨νBm:m∈M, hence(9)μA∧μBr=∨n∈ℕ∧m∈MμA∧μBrn.m=∨n∈ℕ∧m∈MμArn.m∧μBrn.m=∨n∈ℕ∧m∈MμArn.m∧∨n∈ℕ∧m∈MμBrn.m=μAr∧μAr.Similarly,(10)νA∨νBr=∧n∈ℕ∨m∈MνA∨νBrn.m=∧n∈ℕ∨m∈MνArn.m∨νBrn.m=∧n∈ℕ∨m∈MνArn.m∨∧n∈ℕ∨m∈MνBrn.m=νAr∨νAr.Therefore,A∩B=r,μAr∧μBr,νAr∨νBr=A∩B(3)
μA⊆μA+μB because μA+μBx=∨μAy∧μBz|y+z=x≥μAx∧μBθ;μBθ=1=μAx. And similarly, μB⊆μA+μB.
In the same way, we haveνA⊇νA+νB because νA+νBx=∧νAy∨νBz|y+z=x≤νAx∨νBθ;νBθ=0=νAx. Also νB⊇νA+νB.
Now, we haveμA⊆μA+μB, μB⊆μA+μB , hence. μA+μB⊆μA+μB⟹μA+μB⊆μA+μB=μA+μB.
And, alsoνA⊇νA+νB , νB⊇νA+νB hence νA+νB⊇νA+νB⟹νA+νB⊇νA+νB=νA+νB. Therefore A+B⊆A+B.Definition 5 (see [16]).
LetA be a submodule of M. Then intuitionistic fuzzy characteristic function χA=μχA,νχA is defined as follows:(11)μχAx=1,ifx∈A,0,ifx∉A,νχAx=0,ifx∈A,1,ifx∉A.Proposition 3.
LetA be a submodule of M, then χA=χA where χA is intuitionistic fuzzy characteristic function of A.Proof.
Letr∈R, then χAr=∨n∈ℕ∧m∈MμχArn.m,∧n∈ℕ∨m∈MνχArn.m. Now we consider two cases:Case 1:rn.m∈A then μχArn.m=1,νχArn.m=0 and hence χAr=1,0. By rn.m∈A it follows that r∈A and then χAr=1,0 this mean that χA=χA.Case 2:rn.m∉A then μχArn.m=0,νχArn.m=1 and χAr=0,1. Now since rn.m∉A for all n∈ℕ then r∉A hence χAr=0,1 and therefore for this case χA=χA.Definition 6 (see [17]).
IfA∈IFSR, then A is said to have the sup property if for every ∅≠Y⊆R, there is a y0∈Y such that supy∈YμAy=μAy0 , infy∈YνAy=νAy0.Definition 7.
Suppose thatA∈IFSX. Then the set XAα,β=x|x∈X such that μAx≥αandνAx≤β, where α,β∈0,1 with α+β≤1, is called α,β-cut set (crisp set).Proposition 4.
LetA∈IFMM and suppose that A has sup property, then MAα,β=MAα,β, where MAα,β=x∈M|μAx≥α,νAx≤β;α+β≤1;α,β∈ImμA×ImνA.Proof.
LetA be an intuitionistic fuzzy submodule of M with sup property, then(12)MAα,β=r∈R|μAr≥α,νAr≤β=r∈R|∨n∈ℕ∧m∈MμArn.m≥α,∧n∈ℕ∨m∈MνArn.m≤β=r∈R|∧m∈MμArn1.m≥α,∨m∈MνArn2.m≤βfor somen1,n2∈ℕSinceAhas sup property=r∈R|μArn1rn2.m≥α,νArn2rn1.m≤βfor allm∈MSinceμArn1rn2.m≥μArn1.m≥α,νArn1rn2.m≤νArn2.m≤β=r∈R|n″=n1+n2;μArn″.m≥α,νArn″.m≤β,for allm∈M=r∈R|rn′′.m∈MAα,β,for allm∈M,for somen″∈ℕ=r∈R|rn′′.M⊆MAα,βfor somen″∈ℕ=MAα,β.The above-given proposition is also true for strict level cuts, where the sup property is not required.


## 3. Intuitionistic Fuzzy Primary Submodules

In this section, we will give some characterization of an intuitionistic fuzzy primary submodule. We begin with a definition.Definition 8.
LetA=μA,νA be an intuitionistic fuzzy submodule of M. Then, the intuitionistic fuzzy subset A¯=μA¯,νA¯, is defined.(13)μA¯r=∧m∈MμAr.m,νA¯r=∨m∈MνArm,where A¯=A:M.Proposition 5.
LetA be an intuitionistic fuzzy submodule of M. Then, A¯ is an intuitionistic fuzzy ideal of R.Proof.
Letr1,r2∈R then(14)μA¯r1−r2=∧m∈MμAr1−r2.m=∧m∈MμAr1.m−r2.m≥∧m∈MμAr1.m∧μAr2.m=∧m∈MμAr1.m∧∧m∈MμAr2.m=μA¯r1∧μA¯r2.
It is clear thatνA¯r1−r2≤νA¯r1∨μA¯r2. Since the proof is the dual of the proof of the first part, hence we got that μA¯r1−r2≥μA¯r1∧μA¯r2, νA¯r1−r2≤νA¯r1∨μA¯r2. Now, suppose that r1,r2∈R. Then, we have μA¯r1r2=∧m∈MμAr1r2.m≥∧m∈MμAr1.m=μA¯r1, similarly νA¯r1r2≥νA¯r2. Now μA¯r1r2≥μA¯r1∨μA¯r2 and also νA¯r1r2≤νA¯r1∧νA¯r2. Therefore, A¯ is an intuitionistic fuzzy ideal of R.Remark 1.
IfA∈IFMM then its radical is equal to the radical of A¯, because(15)A¯=r,μA¯r,νA¯r=r,∨n∈ℕμA¯rn,∧n∈ℕνA¯rnSinceA¯is an intuitionistic fuzzy ideal ofR=r,∨n∈ℕ∧m∈MμArn.m,∨n∈ℕ∧m∈MνArn.m=r,μAr,νAr=A.Proposition 6.
LetM be an R-module and A∈IFMM. Then MA¯α,β=MAα,β¯, where MAα,β=x∈M:μAx≥αandνAx≤β.Proof.
(16)MAα,β¯=MAα,β:M=AnnMMAα,β=r∈R|r.m+MAα,β=MAα,β,for allm∈M=r∈R|r.m+MAα,β=MAα,β,for allm∈M=r∈R|r.m∈MAα,β,for allm∈M=r∈R|μAr.m≥α,νAr.m≤β,for allm∈M=r∈R|∧m∈MμAr.m≥α,∨m∈MνAr.m≤β=r∈R|μA¯r≥α,νA¯r≤β=MA¯α,β.Definition 9 (see [14]).
LetM and N be modules over the same ring R and let f be a mapping from M to N. Let A=μA,νA be an intuitionistic fuzzy submodule of M and B=μB,νB be an intuitionistic fuzzy submodule of N. Then, the image of the intuitionistic fuzzy submodule A of M, fA=fμA,νAf∈N can be defined for all y∈N as follows:(17)fμAy=∨μAx|x∈R,fx=y,iff−1y≠∅,0,otherwise,νAf=∧νAx|x∈R,fx=y,iff−1y≠∅,1,otherwise,and inverse image of an intuitionistic fuzzy submodule B of N can be defined as x∈M,f−1B=Bfx.Theorem 1.
Letf:M⟶M′ be an epimorphism of R-modules. If A∈IFMM then A¯⊆fA¯.Proof.
First, we show thatfA be an intuitionistic fuzzy submodule of M′. Let u,v∈R, then there exists x,y∈R such that fx=u,fy=v hence fx−y=u−v. Now, we have the following equations:(1)
fμAu−v=∨z∈f−1u−vμAz≥∨x−y∈f−1u−vμAx−y≥∨x∈f−1uy∈f−1vμAx∧μAy=∨x∈f−1uμAx∧∨y∈f−1uμAy=fμAu∧fμAv.(2)
fμAr.m=∨m′∈f−1r.mμAm′≥∨m′∈f−1mμAr.m′≥∨m′∈f−1mμAm′=fμAm.Similarly,νAfu−v=νAfu∧νAfvandνAfr.m=νAfm.(3)
fμA0M′=∨m∈f−10M′μAm=μA0M,νAf0M′=∧m∈f−10M′νAm=νA0M.
Thus, we prove thatfA∈IFMM′. Let r∈R, then(18)fμA¯r=fμA:fMr=∧m′∈fMfμAr.m′=∧m′∈fM∨m∈f−1r.m′μAm≥∧m′∈fM∨m∈f−1r.m′μAr.mSincem∈f−1m′thenr.m∈f−1r.m′≥∧m′∈fMm∈f−1m′μAr.m=∧m′∈fMm∈∪f−1m′μAr.m=∧m′∈fMm∈∪f−1m′μAr.m=∧m∈MμAr.m=μA¯r.
It is clear thatνAf¯r=νA¯r. Therefore A¯⊆fA¯=fμA¯,νAf¯.Theorem 2.
Letf:M⟶M′ be a homomorphism of R-modules. If B∈IFMM′, then f−1B¯⊇B¯. Equality holds if f is an epimorphism.Proof.
Letr∈R, then(19)f−1μB¯r=f−1μB:f−1M′r=∧m∈f−1M′f−1μBr.m=∧m∈f−1M′μBfr.m=∧m∈f−1M′μBrfm=∧fm∈M′μBr.fm≥∧m′∈M′μBr.m′=μB¯r.
Similar to the previous part, we can show thatf−1νB¯r≤νB¯r. Therefore, f−1B¯⊇B¯. Furthermore, if f is an epimorphism then f−1B¯=B¯.Proposition 7.
LetA,B∈IFMM. Then, the following hold:(1)
A¯¯=A¯(2)
A⊆B⟹A¯⊆B¯(3)
A∩B¯=A¯∩B¯(4)
A+B¯⊇A¯+B¯Proof.
Letr∈R, then(1)
Since we can considerA¯ as R-module, then μA¯¯r=∧r′∈RμA¯r.r′=μA¯r, and also νA¯¯r=∨r′∈RνA¯r.r′=νA¯r (Since 1∈R). Hence A¯¯=μA¯¯,νA¯¯=μA¯,μA¯=A¯(2)
SinceA⊆B then μA⊆μB and νA⊇νB, hence μA¯r=∧m∈MμAr.m≤∧m∈MμBr.m=μB¯r⟹μA¯⊆μB¯, and νA¯r=∨m∈MνAr.m≥∨m∈MνBr.m=νB¯⟹νA¯⊇νB¯r. Therefore, A¯⊆B¯.(3)
We know thatA∩B=μA∩B,νA∪B, so μA∩B¯r=∧m∈MμA∩Brm=∧m∈MμArm∧μBrm=∧m∈MμArm∧∧m∈MμBrm=μA¯r∧μB¯r=μA¯∩μB¯r. Similarly, νA∪B¯r=νA¯∪νBr and so we get A∩B¯=A¯∩B¯.(4)
It is clear thatμA⊆μA+μB, μB⊆μA+μB and νA⊇νA+νB, νB⊇νA+νB. So μA¯⊆μA+B¯,μB¯⊆μA+B¯. Hence μA¯+μB¯⊆μA+B¯ and νA¯⊇νA+B¯,νB¯⊇νA+B¯ and so νA¯+νB¯⊇νA+B¯. Therefore A+B¯⊇A¯+B¯.In this part of the paper, we define the intuitionistic fuzzy primary submodule.Definition 10.
LetA be an intuitionistic fuzzy submodule of R-module M, then A is called intuitionistic fuzzy primary submodule of M, if for r∈R,m∈M(20)μAr.m=μAm,νAr.m=νAmorμAr.m≤μAr,νAr.m≥νAr.Example 1.
Considerℤ4 as a ℤ−module. Define A=μA,νA by(21)μAz=1,ifz=0¯,0,ifz=1¯,1,2,ifz=2¯,0,ifz=3¯,νAz=0,ifz=0¯,1,ifz=1¯,12,ifz=2¯,1,ifz=3¯.
It is clear thatA is an intuitionistic fuzzy submodule of ℤ4. Since μAz=∨n∈ℕ∧m∈ℤ4μAznm and νAz=∧n∈ℕ∨m∈ℤ4νAznm, then(22)μAz=1,ifz=0,1,ifz∈2,0,ifz∉2,νAz=0,ifz=0,0,ifz∈2,1,ifz∉2,for all z∈ℤ. Thus, A=μA,νA is an intuitionistic fuzzy primary submodule of ℤ.Now, we would like to investigate the relationship between intuitionistic fuzzy primary and primary submodules of a module.Proposition 8.
LetA be a submodule of M. Then, its intuitionistic fuzzy characteristic function χA is an intuitionistic fuzzy primary submodule if and only if A is a primary submodule of M.Proof.
LetA be a primary submodule of M and r∈R,m∈M. By 5 we know that χA=μχA,νχA where(23)μχAx=1,ifx∈A,0,ifx∉A,νχAx=0,ifx∈A,1,ifx∉A.
IfχArm=1,0 then rm∈A. Since A is a primary submodule, then m∈A or r∈A. Now, we consider the following two cases:Case 1:m∈A then χAm=1,0 and hence χArm=χAmCase 2:r∈A then χA=1,0, but by Proposition 3, χA=χA hence χA=1,0, therefore χArm=χAr
IfχArm=0,1 then rm∉A and so m∉A, hence χAm=0,1. Thus, χArm=χAm or χArm≤χAr.
Conversely, letχA be an intuitionistic fuzzy primary submodule of M and suppose that rm∈A, r∈R,m∈M. Then, χArm=1,0. Now, since χA is an intuitionistic fuzzy primary submodule then(24)χArm=χAmorχArm≤χAr=χAr.
IfχArm=χAm then χAm=1,0 and so m∈A. But If χArm≤χAr, then(25)μχArm≤μχAr,νχArm≥νχAr⟹μχAr≥1,νχAr≤0⟹μχAr=1,νχAr=0⟹χA=1,0⟹r∈A.Proposition 9.
LetM be an R-module and A be an intuitionistic fuzzy primary submodule of M with sup property. Then, MAα,β is a primary submodule of M.Proof.
SinceA∈IFMM, so MAα,β is a submodule of M. Now let r∈R,m∈M and rm∈MAα,β, then μArm≥α,νArm≤β. Since A is an intuitionistic fuzzy primary submodule of R-module M then α≤μArm=μAm,β≥νArm=νAm or α≤μArm≤μAr,β≥νArm≥νAr. Hence μAm≥α,νAm≤β or μAr≥α,νAr≤β. This mean that m∈MAα,β or r∈MAα,β. But A has sup property and by Proposition 4 we have MAα,β=MAα,β, therefore m∈MAα,β or r∈MAα,β, and it follows that MAα,β is a primary submodule of M.
IfMAα,β is strict level cut or A is finite valued intuitionistic fuzzy submodule, then the above-given proposition holds without assumption of sup property. We know that A is finite valued intuitionistic fuzzy submodule of M, if Imμ=μx:x∈M, Imν=νx:x∈M be finite.Proposition 10.
LetA be an intuitionistic fuzzy submodule of M, such that every level cut of A is a primary submodule of M, then A is an intuitionistic fuzzy primary submodule of M.Proof.
Letr∈R,m∈M, and let μAr.m=α,νAr.m=β, then(26)r.m∈MAα,β⟹m∈MAα,βorr∈MAα,β⟹m∈MAα,βorrn,M⊆MAα,βfor somen∈ℕ⟹μAm≥α,νAm≤βorμArn.m≥α,νArn.m≤βfor allm∈M⟹α=μAr.m≥μAm≥α,β=νAr.m≤νAm≤βor∧m∈MμArn.m≥α,∨m∈MνArn.m≤β⟹μAr.m=μAm,νAr.m=νAmor∨n∈ℕ∧m∈MμArn.m≥α,∧n∈ℕ∨m∈MνArn.m≤β⟹μAr.m=μAm,νAr.m=νAmorμAr≥α,νAr≤β⟹μAr.m=μAm,νAr.m=νAmorμAr.m≤μAr,νAr.m≥νAr.
This means thatA is an intuitionistic fuzzy primary submodule of M.Definition 11.
LetA be an intuitionistic fuzzy submodule of the R-module M. The support of A is defined as A∗=x∈M|μAx>0,νAx<1.In the following, we investigated related between intuitionistic fuzzy submoduleA and support of A.Proposition 11.
LetA be an intuitionistic fuzzy primary submodule of the R-module M. Then A∗ is a primary submodule of M.Proof.
Suppose thatrm∈A∗ for r∈R,m∈M. Then, μAx>0,νAx<1. Since A is an intuitionistic fuzzy primary submodule, then
μArm=μAm or μArm≤μAr, νArm=νAm or νArm≥νAr, thus μAm>0 or μAr>0,νAm<1or νAr<1. Then m∈A∗ or r∈A∗. Therefore we prove that A∗ is a primary submodule of M.Remark 2.
The support of an intuitionistic fuzzy submodule may be a primary submodule, but it need not be an intuitionistic fuzzy primary submodule. So the converse of the above-given proposition need not be true. Note the following example.Example 2.
LetR=M=ℤ. Define an intuitionistic fuzzy subset A=μA,νA in ℤ by(27)μAx=1,ifx∈6ℤ,0.3,ifx∈2ℤ−6ℤ,0,ifx∈ℤ−2ℤ,νAx=0,ifx∈6ℤ,0.5,ifx∈2ℤ−6ℤ,1,ifx∈ℤ−2ℤ,
It is clear thatA is an intuitionistic fuzzy submodule of ℤ and A∗=2ℤ, which is a primary submodule of ℤ. We show that A is not an intuitionistic fuzzy primary submodule of ℤ. Consider r=2,m=3. Then, μArm=μA2.3=μA6=1≠μA3=0,νArm=νA2.3=νA6=0≠νA3=1 and μArm=μA2.3>μA2=∨n∈ℕ∧m∈ℤμA2n.m=0.3,νArm=νA2.3<νA2=∧n∈ℕ∨m∈ℤνA2n.m=0.5. Therefore, A is not an intuitionistic fuzzy primary submodule of ℤ.Definition 12 (see [18]).
An intuitionistic fuzzy idealA is said to be an intuitionistic fuzzy weakly primary ideal if for any x,y∈R(28)μAx.y=μAx,νAx.y=νAxorμAx.y≤μAyn,νAx.y≥νAyn,for some n∈ℕProposition 12.
LetA be an intuitionistic fuzzy primary submodule of M with sup property, then A¯ is an intuitionistic fuzzy weakly primary ideal of R and A is the intuitionistic fuzzy prime ideal of R.Proof.
Letr1,r2∈R, then
μA¯r1r2=∧m∈MμAr1r2.m=∧m∈MμAr1r2.m, νA¯r1r2=∨m∈MνAr1r2.m=∨m∈MνAr1r2.m. Now since A is an intuitionistic fuzzy primary submodule, then(29)νAr1r2.m=νAr1.morνAr1r2.m≥νAr2⟹⟹νA¯r1r2=∧m∈MνAr1.morνA¯r1r2≥∧m∈MνAr2=νAr2⟹νA¯r1r2=νA¯r1orνA¯r1r2≥νAr2=∨n∈ℕ∧m∈MνAr2n.m⟹νA¯r1r2=νA¯r1orνA¯r1r2≥∧m∈MνAr2n1.mfor somen1∈ℕBecauseAhas sup property⟹νA¯r1r2=νA¯r1orνA¯r1r2≥∧m∈MνAr2n1.m≥∧m∈MνAr2n1r2n2.m,and also(30)νAr1r2.m=νAr1.morνAr1r2.m≥νAr2⟹⟹νA¯r1r2=∨m∈MνAr1.morνA¯r1r2≥∨m∈MνAr2=νAr2⟹νA¯r1r2=νA¯r1orνA¯r1r2≥νAr2=∧n∈ℕ∨m∈MνAr2n.m⟹νA¯r1r2=νA¯r1orνA¯r1r2≥∨m∈MνAr2n1.mfor somen1∈ℕBecauseAhas sup property⟹νA¯r1r2=νA¯r1orνA¯r1r2≥∨m∈MνAr2n1.m≥∨m∈MνAr2n1r2n2.m,for some n1,n2∈ℕ. If we consider n′=n1+n2 then we have(31)μA¯r1r2=μA¯r1,νA¯r1r2=νA¯r1orμA¯r1r2≤μA¯r1n′,νA¯r1r2≥νA¯r1n′.
This proves thatA¯ is an intuitionistic fuzzy weakly primary ideal of R. Next, by Remark 1 we have A¯=A and since A¯ is intuitionistic fuzzy prime ideal then A is an intuitionistic fuzzy prime ideal of R.The opposite of the above proposition is true ifA is an ideal.Theorem 3.
LetA∈IFIR with sup property. Then, A is an intuitionistic fuzzy primary submodule if and only if A is an intuitionistic fuzzy weakly primary ideal of R.Proof.
Suppose thatA is an intuitionistic fuzzy primary submodule of R, then clearly A is an intuitionistic fuzzy primary ideal of R. Now since A is an intuitionistic fuzzy primary submodule, then for r1,r2∈R(32)μAr1r2=μAr1,νAr1r2=νAr1orμAr1r2≤μAr2,νAr1r2≥νAr2⟹⟹μAr1r2≤∨n∈ℕμAr2n,νAr1r2≥∧n∈ℕνAr2n⟹μAr1r2≤μAr2n1,νAr1r2≥νAr2n2forsomen1,n2∈ℕSinceAhas Sup property⟹μAr1r2≤μAr2n1≤μAr2n1r2n2,νAr1r2≥νAr2n2≥νAr2n1r2n2⟹μAr1r2≤μAr2n′,νAr1r2≥νAr2n′considern1+n2=n′.
Therefore, we obtain that(33)μAr1r2=μAr1,νAr1r2=νAr1orμAr1r2≤μAr2n′,νAr1r2≥νAr2n′.
HenceA is an intuitionistic fuzzy weakly primary ideal of R.
Conversely, letA be an intuitionistic fuzzy weakly primary ideal of R, then
μAr1r2=μAr1,νAr1r2=νAr1 or μAr1r2≤μAr2n,νAr1r2≥νAr2n for some n∈ℕ. Therefore, μAr1r2=μAr1,νAr1r2=νAr1 or μAr1r2≤∨n∈ℕμAr2n,νAr1r2≥∧n∈ℕνAr2n. Thus, μAr1r2=μAr1,νAr1r2=νAr1 or μAr1r2≤μAr2,νAr1r2≥νAr2 for all r1,r2∈R. Hence A is an intuitionistic fuzzy primary submodule of R.The above-given theorem shows that the concept of the intuitionistic fuzzy primary submodule is the generalization of the intuitionistic fuzzy primary ideal.Example 3.
Letℤ12 as ℤ−module. Define A=m,μAm,νAm such that m∈ℤ12. By(34)μAm=1,νAm=0,ifm∈0¯,4¯,8¯,μAm=12,νAm=12,ifm∈2¯,6¯,10¯,μAm=0,νAm=1,ifm∈1¯,3¯,5¯,7¯,9¯,11¯.
We get thatA is an intuitionistic fuzzy submodule of ℤ12. Now A¯=x,μA¯x,νA¯x for x∈ℤ defined by the following equation:(35)μA¯z=∧m∈ℤ12μAz.m=1,νA¯z=∨m∈ℤ12νAz.m=0ifz∈4¯,μA¯z=∧m∈ℤ12μAz.m=12,νA¯z=∨m∈ℤ12νAz.m=12,ifz∈2¯−4¯,μA¯z=∧m∈ℤ12μAz.m=0,νA¯z=∨m∈ℤ12νAz.m=1,otherwise.
ClearlyA¯ is an intuitionistic fuzzy ideal of ℤ. Now we know that A¯=A defined by A=μA,νA and(36)μAz=1,νAz=0,ifz∈2¯,μAz=0,νAz=1,otherwise.
It is easy to show thatAz.m=AmorAz.m≤Az for all m∈ℤ12,z∈ℤ. Hence A is an intuitionistic fuzzy primary submodule of ℤ12.Proposition 13.
Letf:M⟹M′ be an epimorphism of R-module M to R-module M′. If A∈IFMM, then A⊆fA. Equality holds if A is constant on kernel f.Proof.
By Theorem1 we have A¯⊆fA¯, therefore A¯⊆fA¯ and hence A⊆fA. Also A¯⊆fA¯ if A is constant on kernel f, hence A¯⊆fA¯ and therefore A=fA.Proposition 14.
Letf:M⟶M′ be a homomorphism of R-module M to R-module M′. If B∈IFMM′, then f−1B⊇B. Equality holds if f is an epimorphism.Proof.
By Theorem2, f−1B¯⊇B¯, hence f−1A′¯⊇B¯ and therefore f−1B⊇B. If f is an epimorphism, then f−1B¯=B¯ and hence f−1B=B.Theorem 4.
Letf:M⟶M′ be an epimorphism of R-module M to R-module M′, and let A be an intuitionistic fuzzy primary submodule of M, which is constant on kernel f, then the image f, fA is an intuitionistic fuzzy primary submodule of M′.Proof.
By Theorem1fA is an intuitionistic fuzzy submodule of M′. Let r∈R,m′∈M′, then(37)fμArm′=∨m∈f−1rm′μAm,fνArm′=∧m∈f−1rm′μAm.
SinceA constant on kernel f then(38)ifm∈f−1rm′⟹fm=rm′=rfm1Thereism1∈Msuchthatfm1=m′⟹fm−frm1=0⟹m−rm1∈kerf⟹μAm=μArm1,νAm=νArm1.
Therefore,fμArm′=∨m1∈f−1m′μArm1,fνArm′=∧m1∈f−1m′μArm1, but A is an intuitionistic fuzzy primary submodule of M and hence(39)μArm1=μAm1,νArm1=νAm1orμArm1≤μAr,νArm1≥νAr.
Now ifμArm1=μAm1,νArm1=νAm1 then(40)fμArm′=∨m1∈f−1m′μAm1=fμAm′;fνArm′=∧m1∈f−1m′μAm1=fνAm′.
And, ifμArm1≤μAr,νArm1≥νAr then(41)∨m1∈f−1m′μArm1≤∨m1∈f−1m′μAr⟹fμArm′≤∨m1∈f−1m′μAr=μAr≤fμAr,∧m1∈f−1m′νArm1≥∧m1∈f−1m′νAr⟹fνArm′≥∧m1∈f−1m′νAr=νAr≥fνAr.
Therefore, we got thatfμArm′=fμAm′,fνArm′=fνAm′ or fμArm′≤fμAr,fνArm′≥fνAr. This show that fA is an intuitionistic fuzzy primary submodule of M′.Theorem 5.
Letf:M⟶M′ be a homomorphism of R-module M to R-module M′. If B is an intuitionistic fuzzy primary submodule of M′, then f−1B is an intuitionistic fuzzy primary submodule of M.Proof.
Clearlyf−1B is an intuitionistic fuzzy submodule of M. Let r∈R,m∈M, then(42)f−1μBrm=μBfrm=μBrfm,f−1νBrm=νBfrm=νBrfm.
ButB is an intuitionistic fuzzy primary submodule, hence(43)μBrfm=μBfm=f−1μBm,νBrfm=νBfm=f−1νBmorμBrfm≤μBr≤f−1μBr,νBrfm≥νBr≥f−1νBr.
Thus, we show that(44)f−1μBrm=f−1μBm,f−1νBrm=f−1νBmorf−1μBrm≤f−1μBr,f−1νBrm≥f−1νBr.
This proves thatf−1B is an intuitionistic fuzzy primary submodule of M.


## 4. Concluding Remarks

In this paper, we defined the radical of an intuitionistic fuzzy submodule, and then we studied the intuitionistic fuzzy primary submodules with the help of this concept. We also presented some properties of homomorphic image and preimage of these submodules.



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