# On 2-Absorbing Primary Fuzzy Ideals of Commutative Rings

Authors: Deniz Sönmez; Gürsel Yeşilot; Serkan Onar; Bayram Ali Ersoy; Bijan Davvaz
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this work, we give a characterization of generalizations of prime and primary fuzzy ideals by introducing 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals and establish relations between 2-absorbing (primary) fuzzy ideals and 2-absorbing (primary) ideals. Furthermore, we give some fundamental results concerning these notions.


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## Body

## 1. Introduction

The fundamental concept of fuzzy set was introduced by Zadeh [1] in 1965. In 1982, Liu introduced the notion of fuzzy ideal of a ring [2]. Mukherjee and Sen have continued the study of fuzzy ideals by introducing the notion of prime fuzzy ideals [3]. To the present day, fuzzy algebraic structures have been developed and many interesting results were obtained.Prime ideals and primary ideals play a significant role in commutative ring theory. Because of this importance, the concept of 2-absorbing ideals, which is a generalization of prime ideals [4], and the concept of 2-absorbing primary ideals, which is a generalization of primary ideals [5], were introduced. While the prime fuzzy ideals and primary fuzzy ideals have been investigated [3, 6], the concepts of 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals have not been studied yet. In this paper, we introduce the 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals and some generalizations of 2-absorbing primary fuzzy ideals and describe some properties of 2-absorbing primary fuzzy ideals.LetR be a commutative ring with identity. Recall that a proper ideal I of R is called a 2-absorbing ideal if whenever a,b,c∈R and abc∈I, then either ab∈I or ac∈I or bc∈I [4] and a proper ideal I of R is called a 2-absorbing primary ideal if whenever a,b,c∈R and abc∈I, then either ab∈I or ac∈I or bc∈I [5]. Based on these definitions, a nonconstant fuzzy ideal μ of R is called a2-absorbing fuzzy ideal of R if for any fuzzy points xr,ys,zt of R, xryszt∈μ implies that either xrys∈μ or xrzt∈μ or yszt∈μ and a nonconstant fuzzy ideal μ of R is said to be a2-absorbing primary fuzzy ideal of R if for any fuzzy points xr,ys,zt of R, xryszt∈μ implies that either xrys∈μ or xrzt∈μ or yszt∈μ. It is shown that if μ is a 2-absorbing primary fuzzy ideal of R, then μ is 2-absorbing fuzzy ideal. We introduce the notions ofweakly completely 2-absorbing fuzzy ideal andweakly completely 2-absorbing primary fuzzy ideal, which is a weakened status of the 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideal, respectively. Then relationship between the 2-absorbing primary fuzzy ideals and weakly completely 2-absorbing primary fuzzy ideals is analyzed. Based on the definition of the level set, the transition of 2-absorbing primary ideals of R and 2-absorbing primary fuzzy ideals of R is examined. By a 2-absorbing primary ideal of R, a 2-absorbing primary fuzzy ideal is established (Proposition 20). For a ring homomorphism f:R→S, it is shown that f-1(ξ) is a 2-absorbing primary fuzzy ideal of R, where ξ is 2-absorbing primary fuzzy ideal of S. If μ is 2-absorbing primary fuzzy ideal of R, which is constant on Kerf, then it is proved that f(μ) is a 2-absorbing primary fuzzy ideal of S. It is shown under what condition the intersection of the collection of 2-absorbing primary fuzzy ideals is 2-absorbing primary fuzzy ideal. It is shown that the intersection of two 2-absorbing primary fuzzy ideals need not be a 2-absorbing primary fuzzy ideal if this condition is not satisfied (Example 27). Also, it is proved that union of a directed collection of 2-absorbing primary fuzzy ideals of R is 2-absorbing primary fuzzy ideal.


## 2. Preliminaries

We assume throughout that all rings are commutative with1≠0. Unless stated otherwise L=[0,1] stands for a complete lattice. Z denotes the ring of integers, L(R) denotes the set of fuzzy sets of R, and LI(R) denotes the set of fuzzy ideals of R. For μ,ξ∈L(R), we say μ⊆ξ if and only if μ(x)≤ξ(x) for all x∈R. When r∈L, x,y∈R we define xr∈L(R) as follows: (1)xry=rx=y,0otherweise,and xr is referred to as fuzzy point of R.Also, forμ∈L(R) and t∈L, define μt as follows: (2)μt=x∈R:μx≥t.Definition 1 (see [2]).
A fuzzy subsetμ of a ring R is called a fuzzy ideal of R if for all x,y∈R the following conditions are satisfied:(i)
μ(x-y)≥μ(x)∧μ(y),∀x,y∈R.(ii)
μ(xy)≥μ(x)∨μ(y),∀x,y∈R.
Letμ be any fuzzy ideal of R; x,y∈R, and let 0 be the additive identity of R. Then it is easy to verify the following:(i)
μ(0)≥μ(x), μ(x)=μ(-x) and μt⊂μs, where s,t∈Im(μ) and t>s.(ii)
Ifμ(0)=μ(x-y), then μ(x)=μ(y), μ(x)=s iff x∈μs, and x∉μt,∀t>s.Definition 2 (see [7]).
Letμ be any fuzzy ideal of R. The ideals μt, (μ(0)≥t) are called level ideals of μ.Definition 3 (see [3]).
A fuzzy idealμ of R is called prime fuzzy ideal if for any two fuzzy points xr,ys of R, xrys∈μ implies either xr∈μ or ys∈μ.Definition 4 (see [6]).
Letμ be a fuzzy ideal of R. Then μ, called the radical of μ, is defined by μ(x)=⋁n≥1μ(xn).Definition 5 (see [6]).
A fuzzy idealμ of R is called primary fuzzy ideal if for x,y∈R, μ(xy)>μ(x) implies μ(xy)≤μ(yn) for some positive integer n.Theorem 6 (see [6]).
Letμ be fuzzy ideal of a ring R. Then μ is a fuzzy ideal of R.Definition 7 (see [3]).
LetR be a ring. Then a nonconstant fuzzy ideal μ is said to be weakly completely prime fuzzy ideal iff for x,y∈R, μ(xy)=max⁡{μ(x),μ(y)}.Theorem 8 (see [8]).
Ifμ and ξ are two fuzzy ideals of R, then μ∩ξ=μ∩ξ.Theorem 9 (see [8]).
Letf:R→S be a ring homomorphism and let μ be a fuzzy ideal of R such that μ is constant on Kerf and let ξ be a fuzzy ideal of S. Then, (i)
f(μ)=f(μ),(ii)
f-1(ξ)=f-1(ξ).Definition 10 (see [4]).
A nonzero proper idealI of a commutative ring R with 1≠0 is called a 2-absorbing ideal if whenever a,b,c∈R with abc∈I, then either ab∈I or ac∈I or bc∈I.Definition 11 (see [5]).
A proper idealI of R is called a 2-absorbing primary ideal of R if whenever a,b,c∈R with abc∈I, then either ab∈I or ac∈I or bc∈I.Theorem 12 (see [5]).
IfI is a 2-absorbing primary ideal of R, then I is a 2-absorbing ideal of R.Definition 13 (see [9]).
An element1>α∈L is called a 2-absorbing element if for any x,y,z∈L, x∧y∧z<α implies either x∧y<α or x∧z<α or y∧z<α.


## 3. 2-Absorbing Primary Fuzzy Ideals

Definition 14.
Letμ be a nonconstant fuzzy ideal of R. Then μ is called a 2-absorbing fuzzy ideal of R if for any fuzzy points xr,ys,zt of R, xryszt∈μ implies that either xrys∈μ or xrzt∈μ or yszt∈μ.Theorem 15.
Every prime fuzzy ideal ofR is a 2-absorbing fuzzy ideal.Proof.
The proof is straightforward.Lemma 16.
Letμ be a fuzzy ideal and t∈[0,μ(0)]. If μ is a 2-absorbing fuzzy ideal, then μt is a 2-absorbing ideal of R.Proof.
Letμ be a 2-absorbing fuzzy ideal and t∈[0,μ(0)]. If xyz∈μt for any x,y,z∈R, then μ(xyz)≥t. Thus, (xyz)t(xyz)=t≤μ(xyz) and (xyz)t=xtytzt∈μ. Since μ is a 2-absorbing fuzzy ideal, we have xtyt=(xy)t∈μ or xtzt=(xz)t∈μ or ytzt=(yz)t∈μ. Hence, xy∈μt or xz∈μt or yz∈μt. Therefore, μt is a 2-absorbing ideal of R.The following example shows that the converse of the lemma need not be true.Example 17.
LetR=Z, the ring of integers. Define the fuzzy ideal μ of Z by (3)μx=1,x=0,14,x∈4Z-0,0,x∈Z-4Z.
Thenμt is (0), 4Z, Z in case t≥1, t≥1/4, t≥0, respectively. Thus, it is seen that μt is 2-absorbing ideal for all t∈Imμ. Since 21/221/211/4(4)=1/4≤μ(4)=1/4 so 21/221/211/4∈μ but 21/221/2(4)=1/2>μ(4)=1/4 and 21/211/4(2)=1/4>μ(2)=0. Hence, μ is not 2-absorbing fuzzy ideal.Definition 18.
Letμ be a nonconstant fuzzy ideal of R. Then μ is said to be a 2-absorbing primary fuzzy ideal of R if xryszt∈μ implies that either xrys∈μ or xrzt∈μ or yszt∈μ for any fuzzy points xr,ys,zt.Theorem 19.
Every primary fuzzy ideal ofR is a 2-absorbing primary fuzzy ideal of R.Proof.
It is clear from the definition of primary fuzzy ideal.Proposition 20.
LetI be a 2-absorbing primary ideal of R and 1≠α∈L a 2-absorbing element. If μ is the fuzzy ideal of R defined by(4)μx=1,x∈I,α,x∉I,for all x∈R, then μ is a 2-absorbing primary fuzzy ideal of R.Proof.
Assume thatxryszt∈μ but xrys∉μ and xrzt∉μ and yszt∉μ for any x,y,z∈R. Then μ(xy)<r∧s and μ((yz)n)≤μ(yz)<s∧t and μ((xz)n)≤μ(xz)<r∧t for all n≥1. In this case, μ(xy)=α and xy∉I, μ((yz)n)=α and (yz)n∉I so yz∉I, μ((xz)n)=α, and (xz)n∉I so xz∉I. Since I is a 2-absorbing primary ideal of R, then we get xyz∉I and so μ(xyz)=α. By our assumption we get (xyz)r∧s∧t=xryszt∈μ and r∧s∧t≤μ(xyz)=α. Thus, r∧s≤α or s∧t≤α or r∧t≤α, since α is 2-absorbing element, which is a contradiction. Hence, μ is a 2-absorbing primary fuzzy ideal of R.Theorem 21.
Every 2-absorbing fuzzy ideal ofR is a 2-absorbing primary fuzzy ideal.Proof.
The proof is straightforward by the definition of the 2-absorbing fuzzy ideal.The following example shows that the converse of Theorem21 is not true.Example 22.
Letμ∈LI(R) be defined as(5)μx=1,x∈8Z,0,x∉8Z.By Theorem 19 and Proposition 20  μ is a 2-absorbing primary fuzzy ideal. For 2∈Z, 212121∈μ but 2121(4)=1>μ(4)=0 so 2121∉μ. Thus, μ is not a 2-absorbing fuzzy ideal.Lemma 23.
Letμ be a fuzzy ideal and t∈[0,μ(0)]. If μ is a 2-absorbing primary fuzzy ideal, then μt is a 2-absorbing primary ideal of R.Proof.
Ifxyz∈μt for any x,y,z∈R, then μ(xyz)≥t. Thus, (xyz)t(xyz)=t≤μ(xyz) and (xyz)t=xtytzt∈μ. Since μ is a 2-absorbing primary fuzzy ideal, we have xtyt=(xy)t∈μ or xtzt=(xz)t∈μ or ytzt=(yz)t∈μ. Hence, xy∈μt or xz∈μt=μt or yz∈μt=μt. Therefore, μt is a 2-absorbing primary ideal.Note that ifμt is a 2-absorbing primary ideal of R, then μ need not be 2-absorbing primary fuzzy ideal of R. In Example 17, μ is not 2-absorbing fuzzy ideal and also it is not 2-absorbing primary fuzzy ideal by Theorem 21, although μt is a 2-absorbing primary ideal of R.Proposition 24.
Ifμ is a 2-absorbing primary fuzzy ideal of R, then μ is a 2-absorbing fuzzy ideal of R.Proof.
Letxr,ys,zt be any fuzzy points of R such that xryszt∈μ and xrys∉μ. Since xryszt∈μ, we get r∧s∧t=xryszt(xyz)≤μ(xyz). By the definition of μ, μ(xyz)=⋁n≥1μ(xnynzn)≥r∧s∧t. Then there is a k∈Z+ such that r∧s∧t≤μ(xkykzk)=μ((xyz)k). It implies that (xryszt)k∈μ. If xrys∉μ, then, for all k∈Z+, (xrys)k=xrkysk∉μ. Since μ is a 2-absorbing primary fuzzy ideal, we conclude xrzt∈μ or yszt∈μ. Thus, μ is a 2-absorbing fuzzy ideal.Definition 25.
Letμ be a 2-absorbing primary fuzzy ideal of R. Then γ=μ is a 2-absorbing fuzzy ideal by Proposition 24. We say that μ is a γ-2-absorbing primary fuzzy ideal of R.Theorem 26.
Letμ1,μ2,…,μn be γ-2-absorbing primary fuzzy ideals of R for some 2-absorbing fuzzy ideal γ of R. Then μ=⋂i=1nμi is a γ-2-absorbing primary fuzzy ideal of R.Proof.
Suppose thatxryszt∈μ and xrys∉μ. Then xrys∉μj for some n≥j≥1 and xryszt∈μj for all n≥j≥1. Since μj is a γ-2-absorbing primary fuzzy ideal, we have yszt∈μj=γ=⋂i=1nμi=⋂i=1nμi=μ or xrzt∈μj=γ=⋂i=1nμi=⋂i=1nμi=μ. Thus, μ is a γ-2-absorbing primary ideal of R.In the following example, we show that ifμ1,μ2 are 2-absorbing primary fuzzy ideals of a ring R, then μ1∩μ2 need not to be a 2-absorbing primary fuzzy ideal of R.Example 27.
LetR=Z, the ring of integers. Define the fuzzy ideals μ1 and μ2 of Z by(6)μ1x=1,x∈50Z,0,x∉50Z,and by(7)μ2x=1,x∈75Z,0,x∉75Z.Here μ1 and μ2 are 2-absorbing primary fuzzy ideals of Z by Proposition 20. But it is not difficult to show that μ1∩μ2 is not a 2-absorbing primary fuzzy ideal of Z. Since (8)μ1∩μ2x=1,x∈150Z,0,x∉150Z,then 251.31.21∈μ1∩μ2 but 25131∉μ1∩μ2, 25121∉μ1∩μ2, and 2131∉μ1∩μ2. Moreover, by the definition of (9)μ1∩μ2=1,x∈30Z,0,x∉30Z,we conclude that 251.31.21∈μ1∩μ2 but 25131∉μ1∩μ2, 25121∉μ1∩μ2, and 2131∉μ1∩μ2. Hence, μ1∩μ2 is not a 2-absorbing primary fuzzy ideal of Z.Theorem 28.
Letμ be a fuzzy ideal of R. If μ is a prime fuzzy ideal of R, then μ is a 2-absorbing primary fuzzy ideal of R.Proof.
Assume thatxryszt∈μ and xrys∉μ for any x,y,z∈R and r,s,t∈[0,1]. Since xryszt∈μ and R is commutative ring, we have xrysztzt=(xrzt)(yszt)∈μ⊆μ. Thus, xrzt∈μ or yszt∈μ since μ is a prime fuzzy ideal of R. Hence, we conclude that μ is a 2-absorbing primary fuzzy ideal of R.Corollary 29.
Ifμ is a prime fuzzy ideal of R, then μn is 2-absorbing primary fuzzy ideal of R for any n∈Z+.Proof.
Letμ be a prime fuzzy ideal and xryszt∈μn but xrys∉μn for any n∈Z+. Since xryszt∈μn and R is commutative ring, we conclude that xrysztzt=(xrzt)(yszt)∈μn⊆μ. Hence, xrzt∈μ=μn or yszt∈μ=μn since μ is prime fuzzy ideal of R.Theorem 30.
Let{μi:i∈I} be a directed collection of 2-absorbing primary fuzzy ideals of R. Then the fuzzy ideal μ=⋃i∈Iμi is a 2-absorbing primary fuzzy ideal of R.Proof.
Supposexryszt∈μ and xrys∉μ for some xr,ys,zt fuzzy points of R. Then there are some j∈I such that xryszt∈μj and xrys∉μj for all j∈I. Since μj is 2-absorbing primary fuzzy ideal, we have yszt∈μj or xrzt∈μj. Thus, yszt∈μj⊆⋃i∈Iμi=⋃i∈Iμi=μ or xrzt∈μj⊆⋃i∈Iμi=⋃i∈Iμi=μ.Theorem 31.
Letf:R→S be a ring homomorphism. If ξ is a 2-absorbing primary fuzzy ideal of S, then f-1(ξ) is a 2-absorbing primary fuzzy ideal of R.Proof.
Assume thatxryszt∈f-1(ξ), where xr,ys,zt are any fuzzy points of R. Then r∧s∧t≤f-1(ξ)(xyz)=ξ(f(xyz))=ξ(f(x)f(y)f(z)). Let f(x)=a, f(y)=b, and f(z)=c∈S. Thus, we get that r∧s∧t≤ξ(abc) and arbsct∈ξ. Since ξ is a 2-absorbing primary fuzzy ideal, we conclude that arbs∈ξ or arct∈ξ or bsct∈ξ. If arbs∈ξ, then r∧s≤ξ(ab)=ξ(f(x)f(y))=ξ(f(xy))=f-1(ξ)(xy); hence, we conclude that xrys∈f-1(ξ).
Ifarct∈ξ, then, for some n∈Z+, r∧t≤ξ(ancn)=ξ(f(x)nf(z)n)≤⋁n≥1ξ(f(x)nf(z)n)=ξ(f(x)f(z))=ξ(f(xz))=f-1(ξ)(xz)=f-1(ξ). Thus, we get that xrzt∈f-1(ξ). By a similar way, it can be see that yszt∈f-1(ξ).Theorem 32.
Letf:R→S be a surjective ring homomorphism. If μ is a 2-absorbing primary fuzzy ideal of R which is constant on Kerf, then f(μ) is a 2-absorbing primary fuzzy ideal of S.Proof.
Suppose thatarbsct∈f(μ), where ar,bs,ct are any fuzzy points of S. Since f is a surjective ring homomorphism, there exist x,y,z∈R such that f(x)=a, f(y)=b,f(z)=c. Thus, arbsct(abc)=r∧s∧t≤f(μ)(abc)=f(μ)(f(x)f(y)f(z))=f(μ)(f(xyz))=μ(xyz) because μ is constant on Kerf. Then we get xryszt∈μ. Since μ is a 2-absorbing primary fuzzy ideal, we conclude xrys∈μ or xrzt∈μ or yszt∈μ.
Thus,r∧s≤μ(xy)=f(μ)(f(xy))=f(μ)(f(x)f(y))=f(μ)(ab) so arbs∈f(μ) or r∧t≤μ(xz)=⋁n≥1μ(xnzn)=⋁n≥1f(μ)(f(xn)f(zn))=⋁n≥1f(μ)(ancn)=f(μ)(ac) so arct∈f(μ).
By a similar way, it is easy to see thatbsct∈f(μ) if yszt∈μ.


## 4. Weakly Completely 2-Absorbing Primary Fuzzy Ideals

Definition 33.
(i) A nonconstant fuzzy idealμ of R is called a weakly completely 2-absorbing fuzzy ideal of R if for all a,b,c∈R, μ(abc)≤μ(ab) or μ(abc)≤μ(ac) or μ(abc)≤μ(bc).
(ii) A nonconstant fuzzy idealμ of R is called a weakly completely 2-absorbing primary fuzzy ideal of R if for all a,b,c∈R, μ(abc)≤μ(ab) or μ(abc)≤μ(ac) or μ(abc)≤μ(bc).Theorem 34.
Every weakly completely 2-absorbing fuzzy ideal ofR is a weakly completely 2-absorbing primary fuzzy ideal.Proof.
The proof is straightforward.The following example shows that the converse of Theorem34 is not necessarily true.Example 35.
LetR=Z, the ring of integers. Define the fuzzy ideal μ of Z by(10)μx=1,x∈8Z,0,x∉8Z.
Assume thatμ(xyz)>μ(xy) for any x,y,z∈R. Thus, μ(xyz)=1 and 0=μ(xy) so we get xyz∈8Z and xy∉8Z. Since 8Z is primary ideal of Z, we get z∈2Z. By the definition of radical μ(11)μx=1,x∈2Z,0,x∉2Z,μxz=1,μyz=1,hence, μ(xz)≥μ(xyz) or μ(yz)≥μ(xyz). Therefore, μ is a weakly completely 2-absorbing primary fuzzy ideal. But since μ(2.2.2)=1>0=μ(2.2), we conclude that μ is not a weakly completely 2-absorbing fuzzy ideal.Proposition 36.
Every primary fuzzy ideal ofR is a weakly completely 2-absorbing primary fuzzy ideal.Proof.
Letμ be a primary fuzzy ideal of R. Assume that μ(xyz)>μ(xy) for any x,y,z∈R. Since μ is primary fuzzy ideal, we conclude μ(z)≥μ(xyz). Since μ is a fuzzy ideal, we have μ(xz)≥μ(z)≥μ(xyz) or μ(yz)≥μ(z)≥μ(xyz). So μ is a weakly completely 2-absorbing primary fuzzy ideal.Lemma 37.
Letμ be a fuzzy ideal of R. Then μ is a weakly completely 2-absorbing primary fuzzy ideal of R if and only if μt is a 2-absorbing primary ideal of R for all t∈[0,μ(0)].Proof.
Assume thatxyz∈μt and xy∉μt for any x,y,z∈R. We show that yz∈μt or xz∈μt. Note that μ(xyz)≥t>μ(xy). Since μ is a weakly completely 2-absorbing primary fuzzy ideal of R, we have μ(xz)≥μ(xyz)≥t or μ(yz)≥μ(xyz)≥t. Thus, xz∈μt=μt or yz∈μt=μt. So we conclude that μt is a 2-absorbing primary ideal of R.
Conversely, assume thatμt is a 2-absorbing primary ideal of R for all t∈[0,μ(0)]. If μ(xyz)>μ(xy) for any x,y,z∈R, then there is a k∈[0,μ(0)] such that μ(xyz)=k and k=μ(xyz)>μ(xy). So xyz∈μk and xy∉μk. Since μk is a 2-absorbing primary ideal of R, we get that xz∈μk=μk or yz∈μk=μk. Hence, μ(xz)≥k=μ(xyz) or μ(yz)≥k=μ(xyz). Therefore, μ a is weakly completely 2-absorbing primary fuzzy ideal of R.Theorem 38.
Ifμ is a weakly completely 2-absorbing primary fuzzy ideal of R, then μ is a weakly completely 2-absorbing fuzzy ideal of R.Proof.
Ifμ is a weakly completely 2-absorbing primary fuzzy ideal, then by Lemma 37  μt is a 2-absorbing primary ideal of R for any t∈[0,μ(0)]. By [5, Theorem 2.2], μt=μt is 2-absorbing ideal of R. Then it is easy to see that from Definition 33  μt is a 2-absorbing ideal of R if and only if μ is a weakly completely 2-absorbing fuzzy ideal.Definition 39.
Letμ be fuzzy ideal of R. Then μ is called a K-2-absorbing primary fuzzy ideal of R if for all x,y,z∈R, μ(xyz)=μ(0) implies that μ(xy)=μ(0) or μ(yz)=μ(0) or μ(xz)=μ(0).Proposition 40.
Every weakly completely 2-absorbing primary fuzzy ideal is aK-2-absorbing primary fuzzy ideal.Proof.
Assume thatμ is a weakly completely 2-absorbing primary fuzzy ideal. If μ(xyz)=μ(0) for any x,y,z∈R, then μ(0)≥μ(xy)≥μ(xyz)=μ(0) or μ(0)=μ(0)≥μ(xz)≥μ(xyz)=μ(0) or μ(0)=μ(0)≥μ(yz)≥μ(xyz)=μ(0) since μ is a weakly completely 2-absorbing primary fuzzy ideal. Hence, μ(xy)=μ(0) or μ(xz)=μ(0) or μ(yz)=μ(0). We conclude that μ is a K-2-absorbing primary fuzzy ideal.
Note that the following example shows that aK-2-absorbing primary fuzzy ideal need not be a weakly completely 2-absorbing primary fuzzy ideal.Example 41.
LetR=Z, the ring of integers. Define the fuzzy ideal μ of Z by(12)μx=1,x=0,12,x∈30Z-0,13,x∈Z-30Z.Then μ is a K-2-absorbing primary fuzzy ideal. But since(13)μ2.3.5=12>⋁μ2.3,μ2.5,μ3.5=13or(14)μ2.3.5=12>⋁μ2.3,μ2.5,μ3.5=13,then μ is not a weakly completely 2-absorbing primary fuzzy ideal.Corollary 42.
Every weakly completely prime fuzzy ideal is a weakly completely 2-absorbing primary fuzzy ideal.Proof.
Since every weakly completely prime fuzzy ideal is primary fuzzy ideal, by Proposition36 every weakly completely prime fuzzy ideal is a weakly completely 2-absorbing primary fuzzy ideal.Theorem 43.
EveryK-2-absorbing fuzzy ideal is a K-2-absorbing primary fuzzy ideal.Proof.
The proof is straightforward.The following example shows that the converse of Theorem43 is not true.Example 44.
Define the fuzzy idealμ of Z by (15)μx=1,x∈8Z,0,x∉8Z.Then μ is a K-2-absorbing primary fuzzy ideal but since μ(2.2.2)=μ(2.2)=⋁n≥1μ(4n)=1=μ(0) and μ(2.2.2)=1=μ(0)≠μ(4)=0, we have that μ is not a K-2-absorbing fuzzy ideal.Theorem 45.
Letf:R→S be a ring homomorphism. If ξ is a weakly completely 2-absorbing primary fuzzy ideal of S, then f-1(ξ) is a weakly completely 2-absorbing primary fuzzy ideal of R.Proof.
Assume thatf-1(ξ)(xyz)>f-1(ξ)(xy) for any x,y,z∈R. Then f-1(ξ)(xyz)=ξ(f(xyz))=ξ(f(x)f(y)f(z))>f-1(ξ)(xy)=ξ(f(xy))=ξ(f(x)f(y)). Since ξ is a weakly completely 2-absorbing primary fuzzy ideal of S, we conclude that ξ(f(x)f(y)f(z))=f-1(ξ)(xyz)≤ξ(f(x)f(z))=ξ(f(xz))=f-1(ξ(xz))=f-1(ξ)(xz) or ξ(f(x)f(y)f(z))=f-1(ξ)(xyz)≤ξ(f(y)f(z))=ξ(f(yz))=f-1(ξ(yz))=f-1(ξ)(yz). Thus, f-1(ξ) is a weakly completely 2-absorbing primary fuzzy ideal of R.Theorem 46.
Letf:R→S be a surjective ring homomorphism. If μ is a weakly completely 2-absorbing primary fuzzy ideal of R which is constant on Kerf, then f(μ) is a weakly completely 2-absorbing primary fuzzy ideal of S.Proof.
Assume thatf(μ)(abc)>f(μ)(ab) for any a,b,c∈S. Since f is surjective ring homomorphism, f(x)=a, f(y)=b, and f(z)=c for some x,y,z∈R. Thus, f(μ)(abc)=f(μ)(f(x)f(y)f(z))=f(μ)(f(xyz))>f(μ)(ab)=f(μ)(f(x)f(y))=f(μ)(f(xy)). So, as μ is constant on Kerf, f(μ)(f(xyz))=μ(xyz) and f(μ)(f(xy))=μ(xy). This means that f(μ)(abc)=μ(xyz)>μ(xy)=f(μ)(ab). Since μ is a weakly completely 2-absorbing primary fuzzy ideal of R, we have μ(xyz)=f(μ)(f(x)f(y)f(z))=f(μ)(abc)≤μ(xz)=f(μ)(f(xz))=f(μ)(ac)=f(μ)(ac) or μ(xyz)=f(μ)(f(x)f(y)f(z))=f(μ)(abc)≤μ(yz)=f(μ)(f(yz))=f(μ)(bc)=f(μ)(bc). Hence, f(μ) is a weakly completely 2-absorbing primary fuzzy ideal of R.We state the following corollary without proof. Its proof is a result of Theorems45 and 46.Corollary 47.
Letf be a homomorphism of a ring R onto a ring S. Then f induces a one-one inclusion preserving correspondence between the weakly completely 2-absorbing primary fuzzy ideals of R which is constant on Kerf and the weakly completely 2-absorbing primary fuzzy ideals of S in such a way that if μ is a weakly completely 2-absorbing primary fuzzy ideal of R constant on Kerf, then f(μ) is the corresponding weakly completely 2-absorbing primary fuzzy ideal of S, and if ξ is a weakly completely 2-absorbing primary fuzzy ideal of S, then f-1(ξ) is the corresponding weakly completely 2-absorbing primary fuzzy ideal of R.Remark 48.
Note that the following diagram shows the transition between definitions of fuzzy ideals:(16)K-2-absorbing⟶K-2-absorbing  primary↑↑weakly  completely  prime⟶w.  c.  2-absorbing⟶w.  c.  2-absorbing  primary↑↑↑prime⟶2-absorbing⟶2-absorbing  primary↘primary↗


## 5. Conclusion

This article investigates the weakly completely 2-absorbing primary fuzzy ideal and 2-absorbing primary fuzzy ideal as a generalization of primary fuzzy ideal in commutative rings. Also some characterizations of 2-absorbing primary fuzzy ideal are obtained. Moreover, we see that a 2-absorbing primary fuzzy ideal by a 2-absorbing primary ideal of a commutative ring is established, so the transition between the two structures can be analyzed.



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