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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.

The fundamental concept of fuzzy set was introduced by Zadeh [

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 [

Let

We assume throughout that all rings are commutative with

Also, for

A fuzzy subset

Let

If

Let

A fuzzy ideal

Let

A fuzzy ideal

Let

Let

If

Let

A nonzero proper ideal

A proper ideal

If

An element

Let

Every prime fuzzy ideal of

The proof is straightforward.

Let

Let

The following example shows that the converse of the lemma need not be true.

Let

Then

Let

Every primary fuzzy ideal of

It is clear from the definition of primary fuzzy ideal.

Let

Assume that

Every 2-absorbing fuzzy ideal of

The proof is straightforward by the definition of the 2-absorbing fuzzy ideal.

The following example shows that the converse of Theorem

Let

Let

If

Note that if

If

Let

Let

Let

Suppose that

In the following example, we show that if

Let

Let

Assume that

If

Let

Let

Suppose

Let

Assume that

If

Let

Suppose that

Thus,

By a similar way, it is easy to see that

(i) A nonconstant fuzzy ideal

(ii) A nonconstant fuzzy ideal

Every weakly completely 2-absorbing fuzzy ideal of

The proof is straightforward.

The following example shows that the converse of Theorem

Let

Assume that

Every primary fuzzy ideal of

Let

Let

Assume that

Conversely, assume that

If

If

Let

Every weakly completely 2-absorbing primary fuzzy ideal is a

Assume that

Note that the following example shows that a

Let

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

Since every weakly completely prime fuzzy ideal is primary fuzzy ideal, by Proposition

Every

The proof is straightforward.

The following example shows that the converse of Theorem

Define the fuzzy ideal

Let

Assume that

Let

Assume that

We state the following corollary without proof. Its proof is a result of Theorems

Let

Note that the following diagram shows the transition between definitions of fuzzy ideals:

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.

The authors declare that they have no conflicts of interest.