On 2-Absorbing Primary Fuzzy Ideals of Commutative Rings

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.

2-Absorbing Semiprimary Fuzzy Ideal of Commutative Rings

“…Let I be a 2-absorbing primary ideal of R. Assume that x r y s z t ∈ µ but x r y s / ∈ √ µ and x r z t / ∈ √ µ and y s z t / ∈ √ µ for any x, y, z ∈ R. Proof. Let ξ be a 2-absorbing semiprimary fuzzy ideal of S. We show that f −1 (ξ ) is a 2-absorbing fuzzy ideal of R. Since f −1 (ξ ) = f −1 ( ξ ) and ξ is a 2-absorbing fuzzy ideal then the inverse image of ξ is also 2-absorbing fuzzy ideal by [7,Theorem 31]. Hence f −1 (ξ ) is a 2-absorbing semiprimary fuzzy ideal of R. Theorem 3.22.…”

“…Proof. If µ is a 2-absorbing semiprimary then √ µ is a 2-absorbing fuzzy ideal of R. By [7,Lemma 3.3], √ µ t = √ µ t is also 2-absorbing ideal. Hence µ t is 2-absorbing semiprimary ideal of R.…”

“…[9] If µ and ξ are two fuzzy ideals of R, then µ ∩ ξ = √ µ ∩ ξ = µξ Theorem 2.17. [7] f : R → S be a ring homomorphism . If ξ is a 2-absorbing fuzzy ideal of S then f −1 (ξ ) is a 2-absorbing fuzzy ideal of R.…”

“…Theorem 2.18. [7] Let f : R → S be a surjective ring homomorphism. If µ is a 2-absorbing fuzzy ideal of R which is constant on Ker f then f (µ) is a 2-absorbing fuzzy ideal of S.…”

“…Recall from [4,5] 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 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. Recall also from [7] that a nonconstant fuzzy ideal µ of R is called a 2-absorbing fuzzy ideal of R if for any fuzzy points x r , y s , z t of R, x r y s z t ∈ µ implies that either x r y s ∈ µ or x r z t ∈ µ or y s z t ∈ µ and a nonconstant fuzzy ideal µ of R is called a 2-absorbing primary fuzzy ideal of R if for any fuzzy points x r , y s , z t of R, x r y s z t ∈ µ implies that either x r y s ∈ µ or x r z t ∈ √ µ or y s z t ∈ √ µ. Based on these definitions, a nonconstant fuzzy ideal µ is called a 2-absorbing semiprimary fuzzy ideal if √ µ is a 2-absorbing fuzzy ideal of R.…”