MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2017/5485839 5485839 Research Article On 2-Absorbing Primary Fuzzy Ideals of Commutative Rings http://orcid.org/0000-0002-7574-4245 Sönmez Deniz 1 http://orcid.org/0000-0002-7279-9275 Yeşilot Gürsel 1 http://orcid.org/0000-0003-3084-7694 Onar Serkan 1 http://orcid.org/0000-0002-8307-9644 Ersoy Bayram Ali 1 Davvaz Bijan 2 Shaikhet Leonid 1 Department of Mathematics Yildiz Technical University Davutpaşa Istanbul Turkey yildiz.edu.tr 2 Department of Mathematics Yazd University Yazd Iran yazd.ac.ir 2017 2382017 2017 06 04 2017 20 07 2017 2382017 2017 Copyright © 2017 Deniz Sönmez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The fundamental concept of fuzzy set was introduced by Zadeh  in 1965. In 1982, Liu introduced the notion of fuzzy ideal of a ring . Mukherjee and Sen have continued the study of fuzzy ideals by introducing the notion of prime fuzzy ideals . 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 , and the concept of 2-absorbing primary ideals, which is a generalization of primary ideals , 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.

Let R be a commutative ring with identity. Recall that a proper ideal I of R is called a 2-absorbing ideal if whenever a,b,cR and abcI, then either abI or acI or bcI  and a proper ideal I of R is called a 2-absorbing primary ideal if whenever a,b,cR and abcI, then either abI or acI or bcI . Based on these definitions, a nonconstant fuzzy ideal μ of R is called a 2-absorbing fuzzy ideal of R if for any fuzzy points xr,ys,zt of R, xrysztμ implies that either xrysμ or xrztμ or ysztμ and a nonconstant fuzzy ideal μ of R is said to be a 2-absorbing primary fuzzy ideal of R if for any fuzzy points xr,ys,zt of R, xrysztμ implies that either xrysμ or xrztμ or ysztμ. 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 of weakly completely 2-absorbing fuzzy ideal and weakly 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:RS, 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 with 10. 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,ξL(R), we say μξ if and only if μ(x)ξ(x) for all xR. When rL, x,yR we define xrL(R) as follows: (1)xry=rx=y,0otherweise,and xr is referred to as fuzzy point of R.

Also, for μL(R) and tL, define μt as follows: (2)μt=xR:μxt.

Definition 1 (see [<xref ref-type="bibr" rid="B5">2</xref>]).

A fuzzy subset μ of a ring R is called a fuzzy ideal of R if for all x,yR the following conditions are satisfied:

μ(x-y)μ(x)μ(y),x,yR.

μ(xy)μ(x)μ(y),x,yR.

Let μ be any fuzzy ideal of R; x,yR, and let 0 be the additive identity of R. Then it is easy to verify the following:

μ(0)μ(x), μ(x)=μ(-x) and μtμs, where s,tIm(μ) and t>s.

If μ(0)=μ(x-y), then μ(x)=μ(y), μ(x)=s iff xμs, and xμt,t>s.

Definition 2 (see [<xref ref-type="bibr" rid="B4">7</xref>]).

Let μ be any fuzzy ideal of R. The ideals μt, (μ(0)t) are called level ideals of μ.

Definition 3 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

A fuzzy ideal μ of R is called prime fuzzy ideal if for any two fuzzy points xr,ys of R, xrysμ implies either xrμ or ysμ.

Definition 4 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

Let μ be a fuzzy ideal of R. Then μ, called the radical of μ, is defined by μ(x)=n1μ(xn).

Definition 5 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

A fuzzy ideal μ of R is called primary fuzzy ideal if for x,yR, μ(xy)>μ(x) implies μ(xy)μ(yn) for some positive integer n.

Theorem 6 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

Let μ be fuzzy ideal of a ring R. Then μ is a fuzzy ideal of R.

Definition 7 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Let R be a ring. Then a nonconstant fuzzy ideal μ is said to be weakly completely prime fuzzy ideal iff for x,yR, μ(xy)=max{μ(x),μ(y)}.

Theorem 8 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

If μ and ξ are two fuzzy ideals of R, then μξ=μξ.

Theorem 9 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let f:RS 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,

f(μ)=f(μ),

f-1(ξ)=f-1(ξ).

Definition 10 (see [<xref ref-type="bibr" rid="B1">4</xref>]).

A nonzero proper ideal I of a commutative ring R with 10 is called a 2-absorbing ideal if whenever a,b,cR with abcI, then either abI or acI or bcI.

Definition 11 (see [<xref ref-type="bibr" rid="B2">5</xref>]).

A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a,b,cR with abcI, then either abI or acI or bcI.

Theorem 12 (see [<xref ref-type="bibr" rid="B2">5</xref>]).

If I is a 2-absorbing primary ideal of R, then I is a 2-absorbing ideal of R.

Definition 13 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

An element 1>αL is called a 2-absorbing element if for any x,y,zL, xyz<α implies either xy<α or xz<α or yz<α.

3. 2-Absorbing Primary Fuzzy Ideals Definition 14.

Let μ 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, xrysztμ implies that either xrysμ or xrztμ or ysztμ.

Theorem 15.

Every prime fuzzy ideal of R is a 2-absorbing fuzzy ideal.

Proof.

The proof is straightforward.

Lemma 16.

Let μ 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.

Let μ be a 2-absorbing fuzzy ideal and t[0,μ(0)]. If xyzμt for any x,y,zR, then μ(xyz)t. Thus, (xyz)t(xyz)=tμ(xyz) and (xyz)t=xtytztμ. Since μ is a 2-absorbing fuzzy ideal, we have xtyt=(xy)tμ or xtzt=(xz)tμ or ytzt=(yz)tμ. Hence, xyμt or xzμt or yzμt. 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.

Let R=Z, the ring of integers. Define the fuzzy ideal μ of Z by (3)μx=1,x=0,14,x4Z-0,0,xZ-4Z.

Then μt is (0), 4Z, Z in case t1, t1/4, t0, respectively. Thus, it is seen that μt is 2-absorbing ideal for all tImμ. Since 21/221/211/4(4)=1/4μ(4)=1/4 so 21/221/211/4μ 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.

Let μ be a nonconstant fuzzy ideal of R. Then μ is said to be a 2-absorbing primary fuzzy ideal of R if xrysztμ implies that either xrysμ or xrztμ or ysztμ for any fuzzy points xr,ys,zt.

Theorem 19.

Every primary fuzzy ideal of R is a 2-absorbing primary fuzzy ideal of R.

Proof.

It is clear from the definition of primary fuzzy ideal.

Proposition 20.

Let I be a 2-absorbing primary ideal of R and 1αL a 2-absorbing element. If μ is the fuzzy ideal of R defined by(4)μx=1,xI,α,xI,for all xR, then μ is a 2-absorbing primary fuzzy ideal of R.

Proof.

Assume that xrysztμ but xrysμ and xrztμ and ysztμ for any x,y,zR. Then μ(xy)<rs and μ((yz)n)μ(yz)<st and μ((xz)n)μ(xz)<rt for all n1. In this case, μ(xy)=α and xyI, μ((yz)n)=α and (yz)nI so yzI, μ((xz)n)=α, and (xz)nI so xzI. Since I is a 2-absorbing primary ideal of R, then we get xyzI and so μ(xyz)=α. By our assumption we get (xyz)rst=xrysztμ and rstμ(xyz)=α. Thus, rsα or stα or rtα, 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 of R 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 Theorem 21 is not true.

Example 22.

Let μLI(R) be defined as(5)μx=1,x8Z,0,x8Z.By Theorem 19 and Proposition 20  μ is a 2-absorbing primary fuzzy ideal. For 2Z, 212121μ but 2121(4)=1>μ(4)=0 so 2121μ. Thus, μ is not a 2-absorbing fuzzy ideal.

Lemma 23.

Let μ 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.

If xyzμt for any x,y,zR, then μ(xyz)t. Thus, (xyz)t(xyz)=tμ(xyz) and (xyz)t=xtytztμ. Since μ is a 2-absorbing primary fuzzy ideal, we have xtyt=(xy)tμ or xtzt=(xz)tμ or ytzt=(yz)tμ. Hence, xyμt or xzμt=μt or yzμt=μt. Therefore, μt is a 2-absorbing primary ideal.

Note that if μt 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.

If μ is a 2-absorbing primary fuzzy ideal of R, then μ is a 2-absorbing fuzzy ideal of R.

Proof.

Let xr,ys,zt be any fuzzy points of R such that xrysztμ and xrysμ. Since xrysztμ, we get rst=xryszt(xyz)μ(xyz). By the definition of μ, μ(xyz)=n1μ(xnynzn)rst. Then there is a kZ+ such that rstμ(xkykzk)=μ((xyz)k). It implies that (xryszt)kμ. If xrysμ, then, for all kZ+, (xrys)k=xrkyskμ. Since μ is a 2-absorbing primary fuzzy ideal, we conclude xrztμ or ysztμ. Thus, μ is a 2-absorbing fuzzy ideal.

Definition 25.

Let μ 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.

Let μ1,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 that xrysztμ and xrysμ. Then xrysμj for some nj1 and xrysztμj for all nj1. Since μj is a γ-2-absorbing primary fuzzy ideal, we have ysztμj=γ=i=1nμi=i=1nμi=μ or xrztμj=γ=i=1nμi=i=1nμi=μ. Thus, μ is a γ-2-absorbing primary ideal of R.

In the following example, we show that if μ1,μ2 are 2-absorbing primary fuzzy ideals of a ring R, then μ1μ2 need not to be a 2-absorbing primary fuzzy ideal of R.

Example 27.

Let R=Z, the ring of integers. Define the fuzzy ideals μ1 and μ2 of Z by(6)μ1x=1,x50Z,0,x50Z,and by(7)μ2x=1,x75Z,0,x75Z.Here μ1 and μ2 are 2-absorbing primary fuzzy ideals of Z by Proposition 20. But it is not difficult to show that μ1μ2 is not a 2-absorbing primary fuzzy ideal of Z. Since (8)μ1μ2x=1,x150Z,0,x150Z,then 251.31.21μ1μ2 but 25131μ1μ2, 25121μ1μ2, and 2131μ1μ2. Moreover, by the definition of (9)μ1μ2=1,x30Z,0,x30Z,we conclude that 251.31.21μ1μ2 but 25131μ1μ2, 25121μ1μ2, and 2131μ1μ2. Hence, μ1μ2 is not a 2-absorbing primary fuzzy ideal of Z.

Theorem 28.

Let μ 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 that xrysztμ and xrysμ for any x,y,zR and r,s,t[0,1]. Since xrysztμ and R is commutative ring, we have xrysztzt=(xrzt)(yszt)μμ. Thus, xrztμ or ysztμ 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.

If μ is a prime fuzzy ideal of R, then μn is 2-absorbing primary fuzzy ideal of R for any nZ+.

Proof.

Let μ be a prime fuzzy ideal and xrysztμn but xrysμn for any nZ+. Since xrysztμn and R is commutative ring, we conclude that xrysztzt=(xrzt)(yszt)μnμ. Hence, xrztμ=μn or ysztμ=μn since μ is prime fuzzy ideal of R.

Theorem 30.

Let {μi:iI} be a directed collection of 2-absorbing primary fuzzy ideals of R. Then the fuzzy ideal μ=iIμi is a 2-absorbing primary fuzzy ideal of R.

Proof.

Suppose xrysztμ and xrysμ for some xr,ys,zt fuzzy points of R. Then there are some jI such that xrysztμj and xrysμj for all jI. Since μj is 2-absorbing primary fuzzy ideal, we have ysztμj or xrztμj. Thus, ysztμjiIμi=iIμi=μ or xrztμjiIμi=iIμi=μ.

Theorem 31.

Let f:RS 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 that xrysztf-1(ξ), where xr,ys,zt are any fuzzy points of R. Then rstf-1(ξ)(xyz)=ξ(f(xyz))=ξ(f(x)f(y)f(z)). Let f(x)=a, f(y)=b, and f(z)=cS. Thus, we get that rstξ(abc) and arbsctξ. Since ξ is a 2-absorbing primary fuzzy ideal, we conclude that arbsξ or arctξ or bsctξ. If arbsξ, then rsξ(ab)=ξ(f(x)f(y))=ξ(f(xy))=f-1(ξ)(xy); hence, we conclude that xrysf-1(ξ).

If arctξ, then, for some nZ+, rtξ(ancn)=ξ(f(x)nf(z)n)n1ξ(f(x)nf(z)n)=ξ(f(x)f(z))=ξ(f(xz))=f-1(ξ)(xz)=f-1(ξ). Thus, we get that xrztf-1(ξ). By a similar way, it can be see that ysztf-1(ξ).

Theorem 32.

Let f:RS 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 that arbsctf(μ), where ar,bs,ct are any fuzzy points of S. Since f is a surjective ring homomorphism, there exist x,y,zR such that f(x)=a, f(y)=b,f(z)=c. Thus, arbsct(abc)=rstf(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 xrysztμ. Since μ is a 2-absorbing primary fuzzy ideal, we conclude xrysμ or xrztμ or ysztμ.

Thus, rsμ(xy)=f(μ)(f(xy))=f(μ)(f(x)f(y))=f(μ)(ab) so arbsf(μ) or rtμ(xz)=n1μ(xnzn)=n1f(μ)(f(xn)f(zn))=n1f(μ)(ancn)=f(μ)(ac) so arctf(μ).

By a similar way, it is easy to see that bsctf(μ) if ysztμ.

4. Weakly Completely 2-Absorbing Primary Fuzzy Ideals Definition 33.

(i) A nonconstant fuzzy ideal μ of R is called a weakly completely 2-absorbing fuzzy ideal of R if for all a,b,cR, μ(abc)μ(ab) or μ(abc)μ(ac) or μ(abc)μ(bc).

(ii) A nonconstant fuzzy ideal μ of R is called a weakly completely 2-absorbing primary fuzzy ideal of R if for all a,b,cR, μ(abc)μ(ab) or μ(abc)μ(ac) or μ(abc)μ(bc).

Theorem 34.

Every weakly completely 2-absorbing fuzzy ideal of R is a weakly completely 2-absorbing primary fuzzy ideal.

Proof.

The proof is straightforward.

The following example shows that the converse of Theorem 34 is not necessarily true.

Example 35.

Let R=Z, the ring of integers. Define the fuzzy ideal μ of Z by(10)μx=1,x8Z,0,x8Z.

Assume that μ(xyz)>μ(xy) for any x,y,zR. Thus, μ(xyz)=1 and 0=μ(xy) so we get xyz8Z and xy8Z. Since 8Z is primary ideal of Z, we get z2Z. By the definition of radical μ(11)μx=1,x2Z,0,x2Z,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 of R is a weakly completely 2-absorbing primary fuzzy ideal.

Proof.

Let μ be a primary fuzzy ideal of R. Assume that μ(xyz)>μ(xy) for any x,y,zR. 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.

Let μ 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 that xyzμt and xyμt for any x,y,zR. We show that yzμt or xzμt. 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, xzμt=μt or yzμt=μt. So we conclude that μt is a 2-absorbing primary ideal of R.

Conversely, assume that μt is a 2-absorbing primary ideal of R for all t[0,μ(0)]. If μ(xyz)>μ(xy) for any x,y,zR, then there is a k[0,μ(0)] such that μ(xyz)=k and k=μ(xyz)>μ(xy). So xyzμk and xyμk. Since μk is a 2-absorbing primary ideal of R, we get that xzμk=μk or yzμk=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.

If μ is a weakly completely 2-absorbing primary fuzzy ideal of R, then μ is a weakly completely 2-absorbing fuzzy ideal of R.

Proof.

If μ 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.

Let μ be fuzzy ideal of R. Then μ is called a K-2-absorbing primary fuzzy ideal of R if for all x,y,zR, 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 a K-2-absorbing primary fuzzy ideal.

Proof.

Assume that μ is a weakly completely 2-absorbing primary fuzzy ideal. If μ(xyz)=μ(0) for any x,y,zR, 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 a K-2-absorbing primary fuzzy ideal need not be a weakly completely 2-absorbing primary fuzzy ideal.

Example 41.

Let R=Z, the ring of integers. Define the fuzzy ideal μ of Z by(12)μx=1,x=0,12,x30Z-0,13,xZ-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 Proposition 36 every weakly completely prime fuzzy ideal is a weakly completely 2-absorbing primary fuzzy ideal.

Theorem 43.

Every K-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 Theorem 43 is not true.

Example 44.

Define the fuzzy ideal μ of Z by (15)μx=1,x8Z,0,x8Z.Then μ is a K-2-absorbing primary fuzzy ideal but since μ(2.2.2)=μ(2.2)=n1μ(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.

Let f:RS 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 that f-1(ξ)(xyz)>f-1(ξ)(xy) for any x,y,zR. 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.

Let f:RS 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 that f(μ)(abc)>f(μ)(ab) for any a,b,cS. Since f is surjective ring homomorphism, f(x)=a, f(y)=b, and f(z)=c for some x,y,zR. 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 Theorems 45 and 46.

Corollary 47.

Let f 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-absorbingK-2-absorbing  primaryweakly  completely  primew.  c.  2-absorbingw.  c.  2-absorbing  primaryprime2-absorbing2-absorbing  primaryprimary

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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Zadeh L. A. Fuzzy sets Information and Control 1965 8 3 338 353 2-s2.0-34248666540 10.1016/S0019-9958(65)90241-X Liu W. J. Fuzzy invariant subgroups and fuzzy ideals Fuzzy Sets and Systems 1982 8 2 133 139 10.1016/0165-0114(82)90003-3 MR666626 Zbl0488.20005 2-s2.0-0020172677 Mukherjee T. K. Sen M. K. Prime fuzzy ideals in rings Fuzzy Sets and Systems 1989 32 3 337 341 10.1016/0165-0114(89)90266-2 MR1018724 Badawi A. On 2-absorbing ideals of commutative rings Bulletin of the Australian Mathematical Society 2007 75 3 417 429 10.1017/S0004972700039344 MR2331019 2-s2.0-34547275279 Badawi A. Tekir U. Yetkin E. On 2-absorbing primary ideals in commutative rings Bulletin of the Korean Mathematical Society 2014 51 4 1163 1173 10.4134/BKMS.2014.51.4.1163 MR3248714 Zbl1308.13001 2-s2.0-84905643723 Mukherjee T. K. Sen M. K. Primary fuzzy ideals and radical of fuzzy ideals Fuzzy Sets and Systems 1993 56 1 97 101 10.1016/0165-0114(93)90189-O MR1223198 Dixit V. N. Kumar R. Ajmal N. Fuzzy ideals and fuzzy prime ideals of a ring Fuzzy Sets and Systems 1991 44 1 127 138 10.1016/0165-0114(91)90038-R MR1133988 Sidky F. I. Khatab S. A. Nil radical of fuzzy ideal Fuzzy Sets and Systems 1992 47 1 117 120 10.1016/0165-0114(92)90068-F MR1170009 Çallialp F. Yetkin E. Tekir U. On 2-absorbing primary and weakly 2-absorbing primary elements in multiplicative lattices Italian Journal of Pure and Applied Mathematics 2015 34 263 276 MR3393088