ON WEAKLY PRIME FUZZY IDEALS OF COMMUTATIVE RINGS

Citation: Sönmez, D., Yeşilot, G., "On Weakly Pri me Fuzzy Ideals of Commutative


Introduction
The concept of weakly prime ideals of a commutative ring was introduced and studied by D.D. Anderson and E. Smith in [1].Subsequently, S. E. Atani and F. Farzalipour introduced the notion of weakly primary ideals [2].These studies brought a new concept to the literature of ring theory.Here we aim to define the concept of weakly prime fuzzy ideals in fuzzy ring theory.
We assume throughout this paper that all rings are commutative with non-zero identity.Unless stated otherwise [0,1] L = will be a complete lattice.
( ) LI R denotes the set of fuzzy ideals of R. I λ denotes the characteristic function of I .
Recall from [1] that a proper ideal I of R is called a weakly prime ideal if whenever 0 ab I ≠ ∈ , then either a I ∈ or b I ∈ .Note that every prime ideal is a weakly prime ideal.However, the converse is not true.Recall from [7] that a fuzzy ideal µ of R is called fuzzy prime ideal if for any two fuzzy points , r s x y of R , r s x y µ ∈ implies either r x µ ∈ or s y µ ∈ .From [7] we know that a nonconstant fuzzy ideal µ is said to be weakly completely prime fuzzy ideal if and only if for any , , x y R ∈ ( ) { ( ), ( )} xy max x y µ µ µ = .
Motivated from this concept, in Section 2, we define weakly fuzzy ideals and we investigate many properties of weakly prime fuzzy ideals.A nonconstant fuzzy ideal µ of R is called weakly prime fuzzy ideal if 0 t r s x y µ . Among many results in this study, it is proved (Example 2.3) that every prime fuzzy ideal is a weakly prime fuzzy ideal but converse is not true.It is shown that how to construct a weakly prime fuzzy ideal by a weakly prime ideal (Theorem 2.6).Also, it is shown (in Theorem 2.7) if (0) 1 µ = then µ is a weakly prime fuzzy ideal if and only if * µ is a weakly prime ideal and In theorem 2.9, we show that a nonconstant fuzzy ideal µ is a weakly .Morever, we investigate to properties of cartesian product of weakly prime fuzzy ideals.Finally, it is defined partial weakly prime fuzzy ideal and associated with partial weakly prime fuzzy ideals and weakly prime ideals.

Weakly prime fuzzy ideals
Since µ is a weakly prime fuzzy ideal and 0 ( ) Note that the converse of the theorem is not true.
is a weakly prime ideal.Now we show µ is not a weakly prime fuzzy ideal.For fuzzy points 3 1 Hence µ is not weakly prime fuzzy ideal.

Theorem 2.6
Let I be a weakly prime ideal of R and 1 L α ≠ ∈ .If µ is the fuzzy ideal defined by 1 ( ) then µ is a weakly prime fuzzy ideal of R .
Proof. because I is a weakly prime ideal.Hence µ is a weakly prime fuzzy ideal.
Theorem 2.7 Let µ be a nonconstant fuzzy ideal of R and (0) 1 µ = .Then µ is a weakly prime fuzzy ideal of R if and only if * µ is a weakly prime ideal of R and 2 Imµ = .
Proof.Let * µ be a weakly prime ideal and { ,1} We show that µ is a weakly prime fuzzy ideal.By the previous theorem, if we get I µ * = then µ is a weakly prime fuzzy ideal.
For fuzzy points    x y Note that if µ and α are fuzzy ideals of R then µ α × is a fuzzy ideal of R R × .
Theorem 2.11 Let 1 R and 2 R be rings and 1 µ and 2 µ be weakly prime fuzzy ideals, respectively.Then Proof.It is easy to see that weakly prime fuzzy ideal and Similary, it can be easily shown that Definition 2.12 Let µ be a nonconstant fuzzy ideal of R .If ( ) xy ≠ then µ is called partial weakly prime fuzzy ideal.
Theorem 2.13 If µ is a weakly prime fuzzy ideal then µ is a partial weakly prime fuzzy ideal.
Proof.Assume that µ is a weakly prime fuzzy ideal and µ µ = .
The following example shows the converse of the theorem is not true.

Example 2.14 Let
It is easy to prove that µ is a partial weakly prime fuzzy ideal .However µ is not weakly prime fuzzy ideal.Assume that Theorem 2.15 Let µ be a nonconstant fuzzy ideal of R .Then µ is a partial weakly prime fuzzy ideal if and only if t µ is a weakly prime ideal of R for all (0, (0)] t µ ∈ .
Proof.Assume that µ is a partial weakly prime fuzzy ideal.
Hence we conclude that 1 ( ) is a partial weakly prime fuzzy ideal.
Theorem 2.17 Let : f R S → be a surjective ring homorphism.If µ is a partial weakly prime fuzzy ideal of R which is constant on Kerf then ( ) f µ is a partial weakly prime fuzzy ideal of S .
Proof.Assume that 0 xy ≠ where , x y S ∈ .Since f is an epimorphism then there exist , r u R ∈ such that ( ) , ( ) f r x f u s = = . Because f is constant on Kerf so ( ) ( ) Hence we get that ( ) f µ is a partial weakly prime fuzzy ideal.

Conclusions
In this paper, we have characterized weakly prime fuzzy ideals.Also the notions of partial weakly prime fuzzy ideals and their properties are proposed.Furthermore, we have given generalization of definitions of weakly prime fuzzy ideals.
To extend this study, one could study other algebraic structures and do some further studies of their properties.

1 )( 2 )
Let µ and α be two fuzzy ideals of R .The cartesian product of µ and α is defined by µ α × such that ( )( Let µ and α be two fuzzy ideals of R .If ( , )

×
are weakly prime fuzzy ideals of 1 2 R R × .
µ is not weakly prime fuzzy ideal.

Theorem 2.2 Every prime fuzzy ideal is a weakly prime fuzzy ideal. Proof. The result follows by the definition of prime fuzzy ideal.
Definition 2.1 A nonconstant fuzzy ideal µ of R is said to be weakly prime if 0 t