Fuzzy prime ideals redefined

R E S E A R C H

Open Access

Fuzzy prime ideals redefined

Bingxue Yao

*_{Correspondence:} [email protected] School of Mathematics Science, Liaocheng University, Liaocheng, Shandong 252059, China

Abstract

In order to generalize the notions of a (∈,∈ ∨q)-fuzzy subring and various

(∈,∈ ∨q)-fuzzy ideals of a ring, a (λ,

μ)-fuzzy subring and a (λ,

μ)-fuzzy ideal of a ring

μ)-fuzzy semiprime, prime, semiprimary and primary

MSC: 16L99; 03E72

Keywords: (λ,

μ)-fuzzy subring; (λ,

μ)-fuzzy ideal; (λ,

μ)-fuzzy prime ideal;

μ)-fuzzy primary ideal

1 Introduction

The concept of a fuzzy set introduced by Zadeh [] was applied to the group theory by Rosenfeld [] and the ring theory by Liu []. Since then, many scholars have studied the theories of fuzzy subrings and various fuzzy prime ideals [–]. It is worth pointing out that Bhakat and Das introduced the concept of an (α,β)-fuzzy subgroup by using the ‘be-longs to’ relation and ‘quasi-coincident with’ relation between a fuzzy point and a fuzzy subset, and gave the concepts of an (∈,∈ ∨q)-fuzzy subgroup and an (∈,∈ ∨q)-fuzzy sub-ring [, ]. It is well known that a fuzzy subsetAof a groupGis a Rosenfeld’s fuzzy sub-group if and only ifAt={x∈G|A(x)≥t}is a subgroup ofGfor allt∈(, ] (for our convenience, here∅is regarded as a subgroup ofG). Similarly,Ais an (∈,∈ ∨q)-fuzzy subgroup if and only ifAt is a subgroup ofGfor allt∈(, .]. A corresponding result should be considered whenAt is a subgroup ofGfor allt∈(a,b], where (a,b] is an ar-bitrary subinterval of [,]. Motivated by the above problem, Yuanet al.[] introduced a fuzzy subgroup with the thresholds of a group. In order to generalize the concepts of an (∈,∈ ∨q)-fuzzy subring and an (∈,∈ ∨q)-fuzzy ideal of a ring, Yao [] introduced the notions of a (λ,μ)-fuzzy subring and a (λ,μ)-fuzzy ideal and discussed their fundamen-tal properties. In this paper, we will introduce the concepts of (λ,μ)-fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals of a ring.

2 (

λ

μ

LetXbe a nonempty set. By a fuzzy subsetAofX, we mean a map fromXto the interval [, ],A:X→[, ]. IfAis a fuzzy subset ofXandt∈[, ], then the cut setAt and the open cut setAtofAare defined as follows:

At=

x∈X|A(x)≥t, At=

x∈X|A(x) >t.

First, we recall some definitions and results for the sake of completeness.

Definition  Letx∈X,t∈(, ]. A fuzzy subsetAofXof the form

A(y) =

⎧ ⎨ ⎩

t, ify=x,

, ify =x

is said to be a fuzzy point with supportxand valuetand is denoted byxt.

Definition [] A fuzzy pointxt is said to belong to (resp. be quasi-coincident with) a fuzzy subsetA, written asxt∈A(resp.xtqA), if

A(x)≥t resp.A(x) +t> ,

xt∈AorxtqAwill be denoted byxt∈ ∨qA.

In the following discussions,Ralways stands for an associate ring,λandμare constant numbers such that ≤λ<μ≤, andNdenotes the set of all positive integers.

Definition [] A fuzzy subsetAofRis said to be an (∈,∈ ∨q)-fuzzy subring ofRif for allx,y∈Randt,r∈(, ],

() xt,yr∈A=⇒(x+y)t∧r∈ ∨qA,

() xt∈A=⇒(–x)t∈ ∨qA,

() xt,yr∈A=⇒(xy)t∧r∈ ∨qA.

Definition [] A fuzzy subsetAofRis said to be an (∈,∈ ∨q)-fuzzy ideal ofRif () Ais an(∈,∈ ∨q)-fuzzy subring ofR,

() xt∈A,y∈R=⇒(xy)t, (yx)t∈ ∨qA,∀t∈(, ].

According to Definition  and Definition , we have that a fuzzy subsetAofRis a (λ,μ )-fuzzy subring (ideal) ofRif and only if for allx,y∈R,

() A(x–y)≥A(x)∧A(y)∧.,

() A(xy)≥A(x)∧A(y)∧.(()A(xy)≥(A(x)∨A(y))∧.).

In order to give more general concepts of a fuzzy subring and a fuzzy ideal of R, we introduce the following definitions.

Definition  A fuzzy subsetAofRis said to be a (λ,μ)-fuzzy addition subgroup ofRif for allx,y∈R,

A(x+y)∨λ≥A(x)∧A(y)∧μ, A(–x)∨λ≥A(x)∧μ.

Clearly, a fuzzy subsetAofRis a (λ,μ)-fuzzy addition subgroup ofRif and only if for allx,y∈R,A(x–y)∨λ≥A(x)∧A(y)∧μ.

Definition [] A fuzzy subsetAofRis said to be a (λ,μ)-fuzzy subring ofRif for all

x,y∈R,

Definition [] A fuzzy subsetAofRis said to be a (λ,μ)-fuzzy left ideal (resp. fuzzy right ideal) ofRif for allx,y∈R,

() A(x–y)∨λ≥A(x)∧A(y)∧μ,

() A(xy)∨λ≥A(y)∧μ(resp.A(xy)∨λ≥A(x)∧μ).

Ais said to be a (λ,μ)-fuzzy ideal ofRif it is both a (λ,μ)-fuzzy left ideal and a (λ,μ)-fuzzy right ideal ofR.

According to the above definitions, a (λ,μ)-fuzzy left ideal or a (λ,μ)-fuzzy right ideal ofRmust be a (λ,μ)-fuzzy subring. A fuzzy subsetAofRis a (λ,μ)-fuzzy ideal ofRif and only if for allx,y∈R,

() A(x–y)∨λ≥A(x)∧A(y)∧μ, () A(xy)∨λ≥(A(x)∨A(y))∧μ.

Obviously, an (∈,∈ ∨q)-fuzzy subring (fuzzy ideal) ofRis a (λ,μ)-fuzzy subring (fuzzy ideal) ofRwithλ=  andμ= ..

The following theorem is obvious.

Theorem  Let A,B be(λ,μ)-fuzzy left ideals(fuzzy right ideals,fuzzy ideals,fuzzy

sub-rings)of R.Then A∩B is also a fuzzy left ideal(fuzzy right ideal,fuzzy ideal,fuzzy subring)

of R.

Theorem  Let A,B be(λ,μ)-fuzzy left ideals(fuzzy right ideals,fuzzy ideals)of R.Then A+B is also a(λ,μ)-fuzzy left ideal(fuzzy right ideal,fuzzy ideal)of R,where

(A+B)(x) =supA(x)∧B(x)|x=x+x

, ∀x∈R.

Proof We only prove the case of a (λ,μ)-fuzzy left ideal. For allx,y∈R, we have

(A+B)(x–y)∨λ

≥supA(x–x)∧B(y–y)|x=x+y,y=x+y

∨λ

≥supA(x)∧A(x)∧B(y)∧B(y)∧μ|x=x+y,y=x+y

=supA(x)∧B(y)|x=x+y

∧supA(x)∧B(y)|y=x+y

∧μ

= (A+B)(x)∧(A+B)(y)∧μ,

(A+B)(xy)∨λ

≥supA(xy)∧B(xy)|y=y+y

∨λ

=supA(xy)∨λ

∧B(xy)∨λ

|y=y+y

≥supA(y)∧μ

∧B(y)∧μ

|y=y+y

=supA(y)∧B(y)|y=y+y

∧μ

= (A+B)(y)∧μ.

LetA,Bbe fuzzy subsets ofR. Then the fuzzy subsetABis defined as follows:∀x∈R,

(AB)(x) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sup{inf≤i≤n(A(xi)∧B(yi))|x=ni=xiyi,xi,yi∈R,n∈N},

ifxcan be expressed asx=xiyi,xi,yi∈R,

, otherwise.

Theorem  Let A be a(λ,μ)-fuzzy left ideal,and let B be a fuzzy subset of R.Then AB

is a(λ,μ)-fuzzy left ideal of R.

Proof For allz,z∈R, we have

(AB)(z–z)∨λ

≥sup

inf

≤i≤n,≤j≤mA(xi)∧B(yi)∧A

–x_j∧By_j|,

z= n

i=

xiyi,z= m

j=

x_jy_j,m,n∈N

∨λ

≥sup

inf

≤i≤nA(xi)∧B(yi) z=

n

i=

xiyi,n∈N

∧sup

inf

≤j≤mA

x_j∧By_j z= m

j=

x_jy_j,m∈N

∧μ

= (AB)(z)∧(AB)(z)∧μ,

(AB)(zz)∨λ≥sup

inf

≤i≤nA(zxi)∧B(yi) z=

n

i=

xiyi,n∈N

∨λ

≥sup

inf

≤i≤nA(xi)∧B(yi) z=

n

i=

xiyi,n∈N

∧μ

= (AB)(z)∧μ.

So,ABis a (∈,∈ ∨q)-fuzzy left ideal ofR.

Similarly, we have the following theorem.

Theorem  Let A be a fuzzy subset,and let B be a(λ,μ)-fuzzy right ideal of R.Then AB is a(λ,μ)-fuzzy right ideal of R.

The following theorem is an immediate consequence of Theorem  and Theorem .

Theorem  Let A be a(λ,μ)-fuzzy left ideal,and let B be a(λ,μ)-fuzzy right ideal of R.

Then AB is a(λ,μ)-fuzzy ideal of R.

Theorem [] A fuzzy subset A of R is a(λ,μ)-fuzzy subring(fuzzy ideal)of R if and only if for all t∈(λ,μ],Atis a subring(ideal)of R or At=∅.

Theorem  A fuzzy subset A of R is a(λ,μ)-fuzzy subring(fuzzy ideal)of R if and only if for all t∈[λ,μ),Atis a subring(ideal)of R or At=∅.

Proof We only prove the case of a (λ,μ)-fuzzy subring.

LetAbe a (λ,μ)-fuzzy subring ofR, and lett∈[λ,μ). Ifx,y∈At, thenA(x)∧A(y) >t,

A(x–y)∨λ≥A(x)∧A(y)∧μ>t. Consideringλ≤t, we haveA(x–y) >tandx–y∈At. Similarly,xy∈At. It follows thatAtis a subring ofR.

Conversely, assume that Atis a subring of Rfor allt∈[λ,μ). If possible, letA(x– y)∨λ<A(x)∧A(y)∧μfor somex,y∈R. Putt=A(x–y)∨λ, thent∈[λ,μ), and A(x–y)≤t,A(x)∧A(y) >t. So,x,y∈At, andx–y∈/At. This is a contradiction to the fact thatAtis a subring ofR. This shows thatA(x–y)∨λ≥A(x)∧A(y)∧μholds for allx,y∈R.

Similarly,A(xy)∨λ≥A(x)∧A(y)∧μ,∀x,y∈R. That is,Atis a subring ofR.

In the following theorems, it is shown that the homomorphism image (preimage) of a (λ,μ)-fuzzy subring is also a (λ,μ)-fuzzy subring. Similar result can be obtained for a (λ,μ)-fuzzy ideal under some conditions.

Theorem [] Let f:R→Rbe a homomorphism of rings.If A is a(λ,μ)-fuzzy subring of R,then f(A)is a(λ,μ)-fuzzy subring of R.Particularly,if A is a(λ,μ)-fuzzy ideal of R and f is onto,then f(A)is a(λ,μ)-fuzzy ideal of R,where

f(A)(y) =

⎧ ⎨ ⎩

sup{A(x)|f(x) =y}, if f–(y) =∅,

, otherwise ∀y∈R

_.

Theorem [] Let f :R→Rbe a homomorphism of rings.If B is a(λ,μ)-fuzzy subring

(fuzzy ideal)of R,then f–_{(B)is a(λ,μ)-fuzzy subring(fuzzy ideal)of R,where}

f–(B)(x) =Bf(x), ∀x∈R.

3 (

λ

μ

Based on the notion of an (∈,∈ ∨q)-level subset defined in [], we introduce the concept of a (λ,μ)-cut set of a fuzzy subset. LetAbe a fuzzy subset of a setXandt∈[, ]. Then the subsetA(tλ,μ)ofXdefined by

A(tλ,μ)=

x∈X|A(x)∨λ≥t∧μorA(x) > (μ–t)∨λ

is said to be a (λ,μ)-cut set ofA. We denotext∈(λ,μ)Aifx∈A(

λ,μ)

t .

Obviously, ifAis a fuzzy subset ofXandt∈[, ], then

A(tλ,μ)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

X, t≤λ,

At, λ<t≤μ,

Moreover, xt ∈ ∨qA coincides with xt ∈(,.) A, and xt ∈ ∨qkA [] coincides with xt∈_(,k

)A.

Lemma  Let A,B be fuzzy subsets of X.Then for all t∈[, ],

A⊆B =⇒ A(tλ,μ)⊆B(

λ,μ)

t .

Proof The proof is straightforward.

Theorem  Let A,B be fuzzy subsets of X and t∈[, ].Then

() (A∩B)(tλ,μ)=A(

λ,μ)

t ∩B(

λ,μ)

t ,

() (A∪B)(tλ,μ)=A(

λ,μ)

t ∪B(

λ,μ)

t .

Proof

() Obviously, we have (A∩B)(_tλ,μ)⊆A(_tλ,μ)∩B_t(λ,μ)from Lemma . Ifx∈A_t(λ,μ)∩B(_tλ,μ), thenx∈A(_tλ,μ)andx∈B(_tλ,μ). So, we have the following four cases.

Case , ifA(x)∨λ≥t∧μandB(x)∨λ≥t∧μ, then (A∩B)(x)∨λ≥t∧μ. So,x∈

(A∩B)(_tλ,μ).

Case , ifA(x) > (μ–t)∨λandB(x) > (μ–t)∨λ, then (A∩B)(x) > (μ–t)∨λ. So,

x∈(A∩B)(_tλ,μ).

Case , ifA(x)∨λ≥t∧μandB(x) > (μ–t)∨λ, then whent≤μ, we haveB(x)∨

λ> (μ–t)∨λ≥μ≥t∧μand hence (A∩B)(x)∨λ≥t∧μ. When t>μ, we have

A(x)≥μ> (μ–t)∨λand hence (A∩B)(x) > (μ–t)∨λ. It means thatx∈(A∩B)(_tλ,μ). Case , ifA(x) > (μ–t)∨λandB(x)∨λ≥t∧μ, then we can also obtain that x∈

(A∩B)(_tλ,μ)just as in Case .

Therefore, (A∩B)(_tλ,μ)=A_t(λ,μ)∩B(_tλ,μ).

() Obviously, we have (A∪B)(_tλ,μ)⊇A_t(λ,μ)∪B(_tλ,μ)from Lemma .

Letx∈(A∪B)_t(λ,μ), then eitherA(x)∨B(x)∨λ≥t∧μorA(x)∨B(x) > (μ–t)∨λ. It means thatA(x)∨λ≥t∧μ, orB(x)∨λ≥t∧μ, orA(x) > (μ–t)∨λ, orB(x) > (μ–t)∨λ. Sox∈A(_tλ,μ) orx∈B(_tλ,μ). That is,x∈A(_tλ,μ)∪B_t(λ,μ). Hence, (A∪B)(_tλ,μ)=A(_tλ,μ)∪B(_tλ,μ).

Theorem  Let A,B,C be fuzzy subsets of X and t∈[, ].Then

() [A∩(B∪C)]t(λ,μ)= (A∩B) (λ,μ)

t ∪(A∩C) (λ,μ)

t ,

() [A∪(B∩C)]t(λ,μ)= (A∪B) (λ,μ)

t ∩(A∪C) (λ,μ)

t .

Proof The proof can be obtained immediately from Theorem .

Theorem  Let A be a(λ,μ)-fuzzy subring(fuzzy ideal)of R.Then for all t∈[, ],A(_tλ,μ) is a subring(ideal)of R or A(_tλ,μ)=∅.

Proof We only prove the case of a (λ,μ)-fuzzy subring. If t≤λ, thenAt(λ,μ)=R. Ifλ<t≤μ, then A

(λ,μ)

t =At andAt is a subring of Rfrom

Theorem . Ift>μ, thenAt(λ,μ)=A(μ–t)∨λand (μ–t)∨λ∈[λ,μ). So,A(tλ,μ)is a subring

ofRfrom Theorem .

Proof The proof can be obtained from Theorem .

4 (

λ

μ

There are several deferent definitions of a fuzzy prime ideal and a fuzzy primary ideal ofR. In this section, by a prime idealSofR, we mean an ideal ofRsuch thatab∈S=⇒a∈Sor

b∈S.

Bhakat and Das [] defined fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals in a ring which must be (∈,∈ ∨q)-fuzzy ideals first.

Definition [] An (∈,∈ ∨q)-fuzzy idealAofRis said to be () fuzzy semiprime if for allx∈Randt∈(, ],(x₎

t∈A=⇒xt∈ ∨qA,

() fuzzy prime if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xt∈ ∨qAoryt∈ ∨qA,

() fuzzy semiprimary if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xmt ∈ ∨qAor yn_t ∈ ∨qAfor somem,n∈N,

() fuzzy primary if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xt∈ ∨qAorymt ∈ ∨qAfor

somem∈N.

In order to generalize these notions, we introduce (λ,μ)-fuzzy prime, (λ,μ)-fuzzy semiprime, (λ,μ)-fuzzy primary and (λ,μ)-fuzzy semiprimary ideals.

Definition  A (λ,μ)-fuzzy idealAofRis said to be () (λ,μ)-fuzzy semiprime if for allx∈Randt∈(, ],(x₎

t∈A=⇒xt∈(λ,μ)A,

() (λ,μ)-fuzzy prime if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xt∈(λ,μ)Aor yt∈(λ,μ)A,

() (λ,μ)-fuzzy semiprimary if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xtm∈(λ,μ)Aor yn

t ∈(λ,μ)Afor somem,n∈N,

() (λ,μ)-fuzzy primary if for allx,y∈Randt∈(, ],(xy)t∈A=⇒xt∈(λ,μ)Aor ym

t ∈(λ,μ)Afor somem∈N.

In the following four theorems, we give the equivalence condition of a (λ,μ)-fuzzy prime (semiprime, primary, semiprimary) ideal.

Theorem  A(λ,μ)-fuzzy ideal A of R is(λ,μ)-fuzzy prime if and only if for all x,y∈R,

λ∨A(x)∨A(y)≥A(xy)∧μ.

Proof LetAbe (λ,μ)-fuzzy prime. If possible, letλ∨A(x)∨A(y) <A(xy)∧μfor some x,y∈R. Putt=A(xy)∧μ, thenA(x)∨A(y) <tand (xy)t∈A. So,λ∨A(x) <t= t∧μandλ∨A(y) <t=t∧μ. Fromλ∨A(x) <t∧μ, we haveA(x)≤λ∨A(x) <t≤

μ≤μ–tand hencex∈/A(

λ,μ)

t . Similarly,y∈/A(

λ,μ)

t . This is a contradiction. Therefore,

for allx,y∈R, we haveλ∨A(x)∨A(y)≥A(xy)∧μ.

Conversely, if for allx,y∈R,λ∨A(x)∨A(y)≥A(xy)∧μand (xy)t∈A, thenλ∨A(x)∨ A(y)≥t∧μ. So,λ∨A(x)≥t∧μorλ∨A(y)≥t∧μ. That is,x∈(λ,μ)Aory∈(λ,μ)A. Hence,

A is (λ,μ)-fuzzy prime.

Theorem  A(λ,μ)-fuzzy ideal A of R is(λ,μ)-fuzzy semiprime if and only if for all x∈R,

Proof The proof is similar to that of Theorem .

Theorem  A(λ,μ)-fuzzy ideal A of R is(λ,μ)-fuzzy primary if and only if for all x,y∈R, ∃m∈N such that

λ∨A(x)∨Aym≥_A(xy)∧_μ.

Proof LetAbe (λ,μ)-fuzzy primary. If possible, there existx,y∈Rsuch thatλ∨A(x)∨ A(ym

) <A(xy)∧μfor allm∈N, thenλ∨A(x)∨A(ym) <t≤μand (xy)t∈A, where t=A(xy)∧μ. So,λ∨A(x) <t=t∧μandλ∨A(ym) <t=t∧μ. Fromλ∨A(x) <t, we

haveA(x)≤λ∨A(x) <t≤μ≤μ–tand hencex∈/At(λ,μ). Similarly,ym ∈/A (λ,μ) t . This is

a contradiction. Therefore, for allx,y∈R,∃m∈Nsuch thatλ∨A(x)∨A(ym)≥A(xy)∧μ.

Conversely, if for allx,y∈R,∃m∈Nsuch thatλ∨A(x)∨A(ym)≥A(xy)∧μ, then from

(xy)t∈A, we haveλ∨A(x)∨A(ym)≥t∧μ. So,λ∨A(x)≥t∧μorλ∨A(ym)≥t∧μ.

That is,xt∈(λ,μ)Aorytm∈(λ,μ)A. Hence, A is (λ,μ)-fuzzy primary.

Theorem  A(λ,μ)-fuzzy ideal A of R is(λ,μ)-fuzzy semiprimary if and only if for all x,y∈R,∃m,n∈N such that

λ∨Axm∨_Ayn≥_A(xy)∧_μ.

Proof The proof is similar to that of Theorem .

Now, we characterize the (λ,μ)-fuzzy prime (semiprimary) ideal by using its cut set.

Theorem  A fuzzy subset A of R is a(λ,μ)-fuzzy prime(fuzzy semiprime)ideal if and only if for all t∈(λ,μ],Atis a prime(semiprime)ideal of R or At=∅.

Proof We only prove the case of a (λ,μ)-fuzzy prime ideal.

LetAbe a (λ,μ)-fuzzy prime ideal ofR. ThenAis a (λ,μ)-fuzzy ideal ofR. So,At is an ideal ofRorAt=∅from Theorem . For allt∈(λ,μ], ifxy∈At, thenλ∨A(x)∨A(y)≥

A(xy)∧μ≥t∧μ=t. Consideringλ<t, we haveA(x)∨A(y)≥t. It follows thatx∈At or y∈At. Hence,Atis a prime ideal ofR.

Conversely, assumeAt is a prime ideal ofRfor allt∈(λ,μ] wheneverAt =∅, thenAt

is an ideal of R, and henceAis a (λ,μ)-fuzzy ideal from Theorem . Lett∈(, ] and (xy)t∈A. Ift≤λ, then it is clear thatxt∈(λ,μ)A. Ift∈(λ,μ], thenx∈Atory∈At since At is a prime ideal ofR. So,xt∈(λ,μ)Aoryt∈(λ,μ)A. Ift>μ, thenxy∈Aμ. It implies that

x∈Aμory∈Aμ, sinceAμis a prime ideal ofR. Furthermore, we have

x∈Aμ=⇒A(x)≥μ=t∧μ=⇒x∈A(tλ,μ)=⇒xt∈(λ,μ)A.

Similarly,

y∈Aμ=⇒yt∈(λ,μ)A.

Theorem  Let A be a(λ,μ)-fuzzy ideal of R such that Aμ =∅,and let B be a(λ,μ)-fuzzy

prime ideal of Aμ.Then A∩B is a(λ,μ)-fuzzy prime ideal of Aμ.

Proof From Theorem  and Theorem ,Aμis a subring ofRandA∩Bis a (λ,μ)-fuzzy ideal ofAμ. For allx,y∈Aμ,t∈(, ], we haveA(x)≥μandA(y)≥μ. Hence,x,y∈A(

λ,μ)

t .

If (xy)t∈A∩B, thenx∈B(tλ,μ)ory∈B (λ,μ)

t , sinceBis a (λ,μ)-fuzzy prime ideal ofAμ. So

x∈A(_tλ,μ)∩B_t(λ,μ)= (A∩B)(_tλ,μ), ory∈A(_tλ,μ)∩B(_tλ,μ)= (A∩B)(_tλ,μ). It follows thatA∩Bis a

(λ,μ)-fuzzy prime ideal ofAμ.

Similarly, we have the following theorem.

Theorem  Let A be a(λ,μ)-fuzzy ideal of R such that Aμ=∅,and let B be a(λ,μ)-fuzzy

semiprime(fuzzy primary,fuzzy semiprimary)ideal of Aμ.Then A∩B is a(λ,μ)-fuzzy

semiprime(fuzzy primary,fuzzy semiprimary)ideal of Aμ.

The following theorem gives the relation between a (λ,μ)-fuzzy prime ideal with its preimage under a ring homomorphism.

Lemma  Let f :R–→Rbe a homomorphism of rings,and let B be a(λ,μ)-fuzzy subring of R.Then for all t∈(, ],f–_(B)(λ,μ)

t =f–(B(

λ,μ)

t ).

Theorem  Let f:R–→Rbe a homomorphism of rings,and let B be a(λ,μ)-fuzzy prime ideal of R.Then f–_{(B)is a(λ,μ)-fuzzy prime ideal of R.}

Proof From Theorem ,f–_{(B) is a (λ,μ)-fuzzy ideal ofR. Letx,y∈Randt∈(, ]. If}

(xy)t∈f–(B), then (f(x)f(y))t ∈B. ConsideringBis a (λ,μ)-fuzzy prime ideal ofR, we

havef(x)∈Bt(λ,μ)orf(y)∈B(

λ,μ)

t . Hencex∈f–(B)(

λ,μ)

t ory∈f–(B)(

λ,μ)

t from Lemma . It

follows thatf–(B) is a (λ,μ)-fuzzy prime ideal ofR.

Similarly, we can obtain corresponding conclusions about a (λ,μ)-fuzzy semiprime ideal, a (λ,μ)-fuzzy primary ideal and a (λ,μ)-fuzzy semiprimary ideal. But in general, the homomorphism imagef(A) of a (λ,μ)-fuzzy prime idealAofRmay not be (λ,μ)-fuzzy prime even iff is a surjective homomorphism.

5 Conclusion

In this paper, we proposed the concept of a (λ,μ)-fuzzy ideal which can be regarded as the generalization of a common fuzzy ideal introduced by Liu []. In the meantime, we also proposed several concepts of various (λ,μ)-fuzzy ideals such as a (λ,μ)-fuzzy prime ideal and a (λ,μ)-fuzzy primary ideal, and then we characterized their properties and obtained several equivalence conditions of a (λ,μ)-fuzzy prime ideal and a (λ,μ)-fuzzy primary ideal.

Competing interests

The author declares that they have no competing interests.

Author details

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 71140008).

Received: 24 April 2012 Accepted: 3 September 2012 Published: 19 September 2012

References

1. Zadeh, LA: Fuzzy sets. Inf. Control8, 338-353 (1965)

2. Rosenfeld, A: Fuzzy groups. J. Math. Anal. Appl.35, 512-517 (1971)

3. Liu, W: Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst.8, 133-139 (1982)

4. Dixit, VN, Kumar, R, Ajmal, N: Fuzzy ideals and fuzzy prime ideals of a ring. Fuzzy Sets Syst.44, 127-138 (1990) 5. Kumar, R: Fuzzy semiprimary ideals of rings. Fuzzy Sets Syst.42, 263-272 (1991)

6. Malik, DS, Mordeson, N: Fuzzy prime ideal of a ring. Fuzzy Sets Syst.37, 93-98 (1990) 7. Bhakat, SK, Das, P: On the definitions of fuzzy subgroup. Fuzzy Sets Syst.51, 235-241 (1992) 8. Bhakat, SK, Das, P: Fuzzy subrings and ideals redefined. Fuzzy Sets Syst.81, 383-393 (1996)

9. Yuan, X, Zhang, C, Ren, Y: Generalized fuzzy groups and many-valued implications. Fuzzy Sets Syst.138, 205-211 (2003)

10. Yao, B: (λ,μ)-fuzzy subrings and (λ,μ)-fuzzy ideals. J. Fuzzy Math.15(4), 981-987 (2007)

11. Ming, PP, Ming, LY: Fuzzy topology 1: Neighbourhood structure of a fuzzy point and Moore-Smith cover. J. Math. Anal. Appl.76, 571-579 (1980)

12. Bhakat, SK, Das, P: (∈,∈ ∨q)-level subset. Fuzzy Sets Syst.103, 529-533 (1999)

doi:10.1186/1029-242X-2012-203