Singular fuzzy ideals of commutative rings

Abstract

Notion of singular fuzzy ideals of commutative rings is introduced in this paper. These singular fuzzy ideals are defined via essential fuzzy ideals. Various characteristic features of such ideals to establish relationship between fuzzy non singularity and fuzzy semiprime character of commutative rings are presented.

Keywords

Fuzzy ideal essential fuzzy ideal singular ideal semiprime

1 Introduction

Algebraic structures are used in various disciplines such as computer science, control engineering, information technology, theoretical physics, coding theory, topological spaces. Since the introduction of the notion of fuzzy sets by Zadeh [17] in 1965, this concept of fuzzy sets has been applied to many algebraic structures like groups, rings, modules and topologies. Concept of fuzzy subgroups of a group was first defined by Rosenfeld [13]. Based on such foundation, Liu [6] initiated study on fuzzy subrings and fuzzy ideals of rings around year 1982. Subsequently, Liu himself and various research groups such as Mukherjee et al. [9], Swami et al. [16] contributed to some new concepts of fuzzy sub-rings and ideals. These studies were further carried forward by many researchers Dixit [2], Kumar [3], Zahedi [18], Malik and Mordeson [8], Kumbhokar et al. [4, 5] etc. Recently, Navarro et al. [10] introduced prime fuzzy ideals over a non-commutative ring. They have shown that notion of primeness is equivalent to level cuts being crisp prime ideals. This study also generalized results put forward by Kumbhojkar and Bapat [4] in year 1993.

Notion of singularity plays a very important role in study of algebraic structures. It was remarked in [7] that from study on properties of rings one can say with affirmation that the considered rings are either singular or nonsingular. It is natural to ask whether one can reduce studies of arbitrary rings to these two particular cases. Thus, concept of singularity is considered to be remarkable in studies of rings. Singular ideals can be defined via essential ideals of rings. Investigation of this concept in fuzzy setting has not been explored till now. So, we have investigated here fuzzy aspects of singular ideals. This study is also to bring about correspondence between certain ideals and ideals arising from annihilation as well as essential ideals which play key roles in various structure theorems of rings. Earlier we introduced concept of fuzzy annihilators of fuzzy subsets of modules [14]. We also defined and characterized essential submodules [15]. Recently, Rahman et al defined and characterized fuzzy small submodules, dual notion of essential submodules [12]. In this paper using concept of essentiality we define singular fuzzy ideals and investigate different characteristics of such ideals. Motivation of this work is thus to study finiteness conditions on fuzzy substructures of rings.

Let R be a commutative ring with unity, and let Z denote ∪ {γ|γ ∈ [0, 1] ^R} such that γσ ⊆ χ₀, for some essential fuzzy ideal σ of R}, where χ₀ denotes the characteristic function on {0}, the zero ideal of R. It is established that Z is a fuzzy ideal of R, termed as singular fuzzy ideal of R. If γ is a fuzzy ideal of R then γ ⊆ Z if and only if annihilator of γ is an essential fuzzy ideal of R. For a fuzzy point r_α on R, r_α ∈ Z, such a relation is valid if and only if annihilator of r_α is an essential fuzzy ideal of R. It is also proved that χ₀ is a fuzzy semiprime ideal of R if and only if Z = χ₀.

2 Basic definitions and notations

Throughout our discussions in the manuscript R denotes a commutative ring with unity. Unless and otherwise stated M denotes an R-module. Class of all fuzzy subsets of X is denoted by [0, 1] ^x.

Let μ ∈ [0, 1] ^X. Then μ^* = {x ∈ X|μ (x) >0} is called the support of μ and for t ∈ [0, 1], μ_t = {x ∈ X|μ (x) ≥ t} is called the cut set of μ.

Let μ ∈ [0, 1] ^X. Then a fuzzy point x_t, x ∈ X, t ∈ (0, 1] is defined as the fuzzy subset x_t of X such that x_t (x) = t and x_t (y) =0 for all y ∈ X - {x}. We write x_t ∈ μ if and only if x ∈ μ_t.

If Y ⊆ X, then χ_Y is the characteristic function on Y.

Let μ ∈ [0, 1] ^R. Then μ is called a fuzzy ideal of R if it satisfies

μ (x - y) ≥ μ (x) ∧ μ (y) ∀ x, y ∈ R

μ (xy) ≥ χ_Y

μ (x) \lor μ (y) \forall x, y \in R .

The class of all fuzzy ideals of R is denoted by FI(R).

Let μ, σ ∈ [0, 1] ^R. Then sum of μ and σ is defined as (μ + σ) (x) = ∨ {μ (y) ∧ σ (z) | y, z ∈ R, x = y + z}.

And the product of μ and σ is defined as (σμ) (x) = ∨ {μ (y) ∧ σ (z) | y, z ∈ R, x = yz}.

Let γ₁, γ₂, σ ∈ F1 (R) be such that γ₁ ⊆ σ and γ₂ ⊆ σ. Then γ₁ + γ₂ ⊆ σ .

Let γ ∈ FI (R). Then γ is called a semiprime fuzzy ideal if γ is non-constant and if μ² ⊆ γ, where μ ∈ FI (R), implies μ ⊆ γ. From the definition, it is obvious that γ is a semiprime fuzzy ideal, if for all x ∈ R, for all α ∈ [0, 1], (x_α) ² ⊆ γ implies x_α ⊆ γ.

3 Essential fuzzy ideals and annihilators of fuzzy subsets of R

In this section we study two important fuzzy substructures namely essential fuzzy ideals and annihilators of fuzzy subsets of R. Both concepts play fundamental roles in the sequel. Essential fuzzy submodules of a module were earlier defined in literature [15]:

Definition 3.1. [15] A fuzzy sub-module μ of an R-module M is called an essential fuzzy sub-module of M, denoted by μ ⊆ _eM, if for every non zero fuzzy sub-module σ of M, μ ∩ σ ≠ χ_θ, where θ denotes the zero element of M.

Definition 3.2. A fuzzy ideal μ of R with μ (0) =1 is called an essential fuzzy ideal of R if it is essential as a fuzzy submodule of R as a module over itself. In other words μ is an essential fuzzy ideal of R if and only if μ ∩ σ ≠ χ_θ, for every non-zero fuzzy ideal σ of R.

Definition 3.3. Let μ and σ be two non-zero fuzzy ideals of R such that μ ⊆ σ. Then μ is called essential in σ, denoted by μ ⊆ _e σ, if for every non zero fuzzy ideal γ of R with γ ⊆ σ, μ ∩ γ ≠ χ_o.

Following results in [15] justify our introduction of the concept of essential fuzzy ideals:

Theorem 3.1.Letμbe an essential fuzzy ideal of R. Thenμ_tis an essential ideal of R for some t ∈ (0, 1].

Theorem 3.2.Letμbe a fuzzy ideal of R withμ (0) =1. Ifμ_t ⊆ _eR, for all t ∈ (0, 1], thenμ ⊆ _eR.

Theorem 3.3.Letμ ∈ [0, 1] ^R. Thenμ ⊆ _e Rif and only ifμ^* ⊆ _eR

Theorem 3.4.Letμ and σbe two non-zero fuzzy ideals of R such thatμ ⊆ σ. Thenμ ⊆ _eσif and only ifμ^* ⊆ _eσ^*.

We construct a fuzzy subset σ of R in the following way:

Definition 3.4. Let a (≠0) ∈ R, and γ be an essential fuzzy ideal of R. We define a fuzzy set σ ∈ [0, 1] ^R as σ = ∪ {δ|δ ∈ [0, 1] ^R, δa_p ⊆ γ}, where p ∈ (0, 1].

Lemma 3.5.The fuzzy setσ ∈ [0, 1] ^Rdefined as in definition 3.4 can equivalently be constructed as follows: $σ = \cup {r_{α} | r \in R, α \in [0, 1], r_{α} a_{p} \subseteq γ} .$ Proof: It is obvious that {r_α|r ∈ R, α ∈ [0, 1]} ⊆ [0, 1] ^R.

Therefore, {r_α|r ∈ R, α ∈ [0, 1] , r_αa_p ⊆ γ } ⊆ {δ|δ ∈ [0, 1] ^R, δa_p ⊆ γ}.

This implies ∪ {r_α|r ∈ R, α ∈ [0, 1] , r_αa_p ⊆ γ} ⊆ ∪ {δ|δ ∈ [0, 1] ^R, δa_p ⊆ γ} = σ.

Let δ ∈ [0, 1] ^R such that δa_P ⊆ γ .

We consider r ∈ R such that δ (r) = α. $\begin{matrix} Now, (r_{α} a_{p}) (x) \\ = \lor {r_{α} (s) \land a_{p} (y) | s, y \in R, s y = x} \\ = \lor {δ (r) \land a_{p} (y) | y \in R, r y = x} \\ \leq \lor {δ (s) \land a_{p} (y) | s \in R, s y = x} \\ = (δ a_{p}) (x) \leq γ (x), for all x \in R \end{matrix}$

Thus, r_αa_p ⊆ γ.

So σ ⊆ ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_αa_p ⊆ γ } .

Therefore, σ = ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_αa_p ⊆ γ } .

Following result gives another equivalent construction of the fuzzy set σ ∈ [0, 1] ^R as defined in definition 3.4:

Lemma 3.6.σ = ∪ { δ|δ ∈ FI (R) , δa_P ⊆ γ } .

Proof: Clearly ∪ { δ|δ ∈ FI (R) , δa_P ⊆ γ } ⊆ ∪ { δ|δ ∈ [0, 1] ^R, δa_P ⊆ γ } = σ .

Let r ∈ R, α ∈ [0, 1] , and r_α a_p ⊆ γ .

Let δ =< r_α >

Now,

$\begin{matrix} < r_{α} > a_{P} = (χ_{0} \cup α_{< r >}) a_{P} \\ = χ_{0} a_{P} \cup α_{< r >} a_{P} \\ \subseteq χ_{o} \cup α_{< r >} a_{P} \end{matrix}$

Again, (α_<r> a_p) (x) = ∨ {(α_<r> (s) ∧ a_p (y) |s, y ∈ R, sy = x} = ∨ {(α ∧ a_p (y) |s ∈ < r > , y ∈ R, sy = x} $\begin{matrix} \leq \lor {(r_{a} a_{p}) (r y) | t, y \in R, t (ry) = x} \\ \leq \lor {γ (ry) | t, y \in R, t (ry) = x} \\ \leq \lor {γ (t (ry)) | t, y \in R, t (ry) = x} \\ \leq γ (x), for all x \in R \end{matrix}$

Therefore, α_<r> a_P ⊆ γ .

So <r_α > a_p ⊆ χ₀ ∪ γ = γ

Hence, ∪ { δ|δ ∈ FI (R) , δ a_P ⊆ γ } ⊇ ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_α a_p ⊆ γ } = σ .

Therefore, σ = ∪ { δ|δ ∈ FI (R) , δa_P ⊆ γ } .

In the following result we establish the ideal character of σ ∈ [0, 1] ^R which is defined in definition 4.1

Lemma 3.7. If σ is defined as in 3.4., then σ ∈ FI (R) and σ (0) = 1 .

Proof: Clearly χ₀ a_P = χ₀ ⊆ γ .

So, χ₀ ⊆ σ . $\begin{matrix} Now, σ (r_{1}) \land σ (r_{2}) \\ = (\lor {δ_{1} (r_{1}) | δ_{1} \in FI (R), \\ δ_{1} a_{P} \subseteq γ}) \land (\lor {δ_{2} (r_{r}) | δ_{2} \in FI (R), \\ δ_{2} a_{P} \subseteq γ}) \\ = (\lor {δ_{1} (r_{1}) \land δ_{2} (r_{2}) | δ_{1}, δ_{2} \in FI (R), δ_{1} a_{P} \subseteq γ, \\ δ_{2} a_{P} \subseteq γ}) \\ \leq \lor {(δ_{1} + δ_{2}) (r_{1}) \land (δ_{1} + δ_{2}) (r_{2}) | δ_{1}, \\ δ_{2} \in FI (R), δ_{1} a_{P} \subseteq γ, δ_{2} a_{P} \subseteq γ} \\ \leq \lor {(δ_{1} + δ_{2}) (r_{1} - r_{2}) | δ_{1}, δ_{2} \in FI (R), \\ (δ_{1} + δ_{2}) a_{P} \subseteq δ_{1} a_{P} + δ_{2} a_{P} \subseteq γ + γ \subseteq γ} \\ \leq \lor {δ (r_{1} - r_{2}) | δ \in FI (R), δ a_{p} \subseteq γ} \\ = σ (r_{1} - r_{2}) \forall r_{1}, r_{2} \in R \end{matrix}$ $\begin{matrix} Now, σ (sr) = \lor {δ (sr) | δ \in FI (R), δ a_{p} \subseteq γ} \\ \geq \lor {δ (r) | δ \in FI (R), δ a_{p} \subseteq γ} \\ = δ (r) \forall s, r \in R \end{matrix}$

Therefore, σ ∈ FI (R)

Existence of essential fuzzy ideals for every non-zero element in R, is exhibited in the following result.

Theorem 3.8.Ifγis an essential fuzzy ideal ofRwithγ (0) =1 and a ≠ 0, then there exists an essential fuzzy idealσsuch that a_pσ ≠ χ₀and a_Pσ ⊆ γwherep ∈ (0, 1] .

Proof. We set σ = ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_αa_p ⊆ γ } .

By Lemma 3.7: σ ∈ FI (R) .

Now γ being an essential fuzzy ideal, γ^* is an essential ideal of R. So for a ≠ 0 and γ^*, by Lemma 1.1 of [1], there exists an essential ideal L ={ r ∈ R|ra ∈ γ^* } such that aL ≠ {0} and aL ⊆ γ^* .

Now, (r_αa_p) (z) = ∨ {r_α (x) ∧ a_p (y) |x, y ∈ R, xy= z} $= {\begin{matrix} 0 & if z \neq ra \\ α \land p & if z = ra \end{matrix}$

Therefore, r_αa_P = (ra) _α∧p .

So σ = ∪ { r_α|r ∈ R, α ∈ [0, 1] , (ra) _α ∈ γ } .

Also x ∈ σ^* implies that there exists α ∈ (0, 1] such that (xa) _α∧p ∈ γ .

This gives γ (xa) ≥ α ∧ p > 0 ⇒ xa ∈ γ^* ⇒ x ∈ L

Therefore, σ^* ⊆ L .

Conversely, x ∈ L implies xa ∈ γ^*

Therefore, γ (xa) = α, say, where α ≠ 0 ⇒ γ (xa) ≥ α ∧ p ⇒ (xa) _α∧p ∈ γ

This implies x_α ∈ { x_α|x ∈ R, α ∈ [0, 1] , (xa) α_α∧p∈ γ }

Thus, x_α ∈ σ. This implies σ (x) ≥ α > 0. Hence x ∈ σ^*

Therefore, L ⊆ σ^* . So σ^* ⊆ L .

Therefore from above aσ^* ≠ {0}, aσ^* ⊆ γ^* and σ^* is an essential ideal of R. Hence σ is an essential fuzzy ideal of R. From the definition of σ, σ a_p ⊆ γ

Since a σ^* ≠ {0}, there exists r ∈ σ^*, r a ≠ {0}.

Now, (σa_p) (ra) ≥ σ (r) ∧ a_p (a) > 0 . So σ a_p≠χ₀.

Hence the result follows.

Notion of annihilators of fuzzy subsets of modules and various characteristics were introduced earlier in reference [14]. Analogously, we define annihilators of fuzzy subsets of rings and obtain similar results as proved earlier [14]:

Definition 3.5. Let μ ∈ [0, 1] ^R. Then annihilator ofμ, denoted by ann (μ), is defined as $ann (μ) = \cup {η | η \in [0, 1]^{R}, η μ \subseteq χ_{0}} .$

It is seen that ann (μ) ∈ FI (R) and ann (χ₀) = χ_R

The following results can be established in the same line as in [8]:

Theorem 3.9.Ifμ ∈ [0, 1] ^Rthen

χ₀ ⊆ ann (μ).

ann (μ) = ∪ {r_α|r ∈ R, α ∈ [0, 1] , r_αμ ⊆ χ₀}.

ann (μ) μ ⊆ χ₀. If μ (0) =1, then ann (μ) μ = χ₀.

Theorem 3.10.Letσ, μ ∈ [0, 1] ^R. Thenσμ ⊆ χ₀, if and only ifσ ⊆ ann (μ).

If σ (0) =1 and μ (0) =1, then σμ = χ₀ if and only if if σ ⊆ ann (μ).

Theorem 3.11.Letμ ∈ [0, 1] ^R. Then ann (μ) = ∪ {σ|σ ∈ FI (R) , σμ ⊆ χ₀}.

Theorem 3.12. Let μ, σ ∈ [0, 1] ^R. If μ ⊆ σ, then ann (σ) ⊆ ann (μ).

4 Singular fuzzy ideals

In this section we study the notion of singularity in fuzzy ideals and obtain various characteristics of these ideals.

Definition 4.1. We define a fuzzy set Z as

Z = ∪ {γ|γ ∈ [0, 1] ^R such that γσ ⊆ χ₀, for some essential fuzzy ideal σ of R}.

The fuzzy set Z defined in definition 4.1 can be characterized in the following ways:

Theorem 4.1. Z = ∪ {r_α|r ∈ R, α ∈ [0, 1] , r_ασ ⊆ χ₀ for some essential fuzzy ideal σ of R}.

Proof. Clearly {r_α|r ∈ R, α ∈ [0, 1] , r_ασ ⊆ χ₀, for some essential fuzzy ideal σ} $\begin{matrix} \subseteq {γ | γ \in [0, 1]^{R}, γ σ \subseteq χ_{0}, \\ for some essential fuzzy ideal σ of R} \end{matrix}$

Therefore, ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_ασ ⊆ χ₀, for some essential fuzzy ideal σ of R} $\begin{matrix} \subseteq \cup {γ | γ \in [0, 1]^{R}, γ σ \subseteq χ_{0}, \\ for some essential fuzzy ideal σ of R} \end{matrix}$ = Z

Let γ ∈ [0, 1] ^R be such that γσ ⊆ χ₀ for some essential fuzzy ideal σ.

Let r ∈ R such that γ (r) = α

$\begin{matrix} Now, (r_{α} σ) (x) \\ = \lor {r_{α} (s) \land σ (y) | x = sy; s, y \in R} \\ = \lor {γ (r) \land σ (z) | x = r z} \\ \leq \lor {γ (s) \land σ (y) | x = sy; s, y \in R} \\ = (γ σ) (x) \subseteq χ_{0} (x) \forall x \in R . \end{matrix}$

Thus, r_ασ ⊆ χ₀ .

So Z ⊆ ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_ασ ⊆ χ₀, for some essential fuzzy ideal σ of R}

Hence we get the result.

Fuzzy subset Z defined in 4.1 can also equivalently be constructed as follows:

Theorem 4.2. Z = ∪ {γ|γ ∈ FI (R) , γσ ⊆ χ₀ for some essential fuzzy ideal σ of R}.

Proof. Clearly {γ|γ ∈ FI (R) , γ σ ⊆ χ₀ for some essential fuzzy ideal σ of R}

⊆ {γ|γ ∈ [0, 1] ^R, γ σ ⊆ χ₀ for some essential fuzzy ideal σ of R}.

Therefore, ∪ {γ|γ ∈ FI (R) , γ σ ⊆ χ₀ for some essential fuzzy ideal σ of R} ⊆ Z

Let r ∈ R, α ∈ [0, 1] r_ασ ⊆ χ₀ for some essential fuzzy ideal σ of R.

Let γ = 〈 r_α 〉 .

Now

$\begin{array}{l} 〈 r_{α} 〉 σ = (χ_{0} \cup α_{〈 r 〉}) σ \\ = χ_{0} σ \cup α_{〈 r 〉} σ \\ \subseteq χ_{0} \cup α_{〈 r 〉} σ \end{array}$

$\begin{array}{l} Again, (α_{〈 r 〉} σ) (x) \\ = \lor {α_{〈 r 〉} (s) \land σ (y) | s, y \in R, s y = x} \\ = \lor {α \land σ (y) | s \in < r >, y \in R, s y = x} \\ \leq \lor {r_{α} σ (ry) | t, r, y \in R, x = t (ry)} \\ \leq \lor {χ_{0} (ry) | t, r, y \in R, x = t (ry)} \\ \leq \lor {χ_{0} (t (ry)) | t, r, y \in R, x = t (ry)} \\ = χ_{0} (x) \forall x \in R \end{array}$

So $α_{〈 r 〉} σ \subseteq χ_{0} \cup χ_{0} = χ_{0} .$

Hence, ∪ {γ|γ ∈ FI (R) , γ σ ⊆ χ₀ for some essential fuzzy ideal σ of R}

⊇ ∪ { r_α|r ∈ R, α ∈ [0, 1] , r_ασ ⊆ χ₀, for some essential fuzzy ideal σ of R}

= Z

Therefore Z = ∪ {γ|γ ∈ FI (R) , γσ ⊆ χ₀}, for some essential fuzzy ideal σ of R.

Following result exhibits the ideal structure of Z.

Theorem 4.3.Z ∈ FI (R) .

Proof. Clearly χ_R is an essential fuzzy ideal of R.

Since, χ₀χ_R = χ₀, so χ₀ ⊆ Z .

Now, Z (r₁) ∧ Z (r₂) = (∨ {γ₁ (r₁) |γ₁ ∈ FI (R) , γ₁σ₁ ⊆ χ₀, for some essential fuzzy ideal σ₁}) ∧ (∨ {γ₂ (r₂) |γ₂ ∈ FI (R) , γ₂σ₂ ⊆ χ₀ for some essential fuzzy ideal σ₂ } .

= (∨ {γ₁ (r₁) ∧ γ₂ (r₂) |γ₁, γ₂ ∈ FI (R) , γ₁σ₁ ⊆ χ₀, γ₂σ₂ ⊆ χ₀, for some essential fuzzy ideals σ₁ and σ₂} $\begin{matrix} \leq \lor {(γ_{1} + γ_{2}) (r_{1}) \land (γ_{1} + γ_{2}) (r_{2}) | γ_{1}, γ_{2} \in FI (R), \\ γ_{1} σ_{1} \subseteq χ_{0}, γ_{2} σ_{2} \subseteq χ_{0}} \\ \leq \lor {(γ_{1} + γ_{2}) (r_{1} - r_{2}) | (γ_{1} + γ_{2}) (σ_{1} \cap σ_{1}) \subseteq \\ γ_{1} (σ_{2} \cap σ_{2}) + γ_{2} (σ_{1} \cap σ_{2}) \\ \subseteq γ_{1} σ_{1} + γ_{2} σ_{2} \subseteq χ_{0} + χ_{0} \subseteq χ_{0}} \end{matrix}$ = ∨ {(γ₁ + γ₂) (r₁ - r₂) | (γ₁ + γ₂) σ ⊆ χ₀, for some essential fuzzy ideal σ} $= Z (r_{1} - r_{2}) \forall r_{1}, r_{2} \in R .$ Now, Z (sr) = ∨ {γ (sr) |γ ∈ FI (R) , γσ ⊆ χ₀ for some essential fuzzy ideal σ}

≥ ∨ {γ (r) |γ ∈ FI (R) , γσ ⊆ χ₀, for some essential fuzzy ideal σ} $= Z (r) \forall s, r \in R .$

Therefore, Z ∈ FI (R).

Definition 4.2. Fuzzy set Z defined in Definition 4.1, which is also equivalently defined in Theorems 4.1 and 4.2, is called the singular fuzzy ideal of R.

Theorem 4.4.Forγ ∈ FI (R) , γ ⊆ Zif and only if ann (γ) is an essential fuzzy ideal of R.

Proof. Let γ ⊆ Z. Then γσ ⊆ χ₀, for some essential fuzzy ideal σ of R.

So σ⊆ ann (γ).

Now σ being an essential fuzzy ideal of R, ann (γ) is also an essential fuzzy ideal of R.

Conversely, let ann (γ) be an essential fuzzy ideal of R.

Then γ ann (γ) ⊆ χ₀ .

By Definition 4.1 of Z: γ ⊆ Z .

Theorem 4.5.Forr_α ∈ [0, 1] ^R, r_α ∈ Zif and only if ann (r_α) is an essential fuzzy ideal of R.

Proof. Let r_α ∈ Z (χ_R). Then r_ασ ⊆ χ₀, for some essential fuzzy ideal σ of R.

So σ ⊆ ann (r_α).

Now σ being an essential fuzzy ideal of R, ann (r_α) is also an essential fuzzy ideal of R.

Conversely, let ann (r_α) be essential.

Now, r_αann (r_α) ⊆ χ₀

By Definition 4.1 of Z: r_α ∈ Z

Note: Let δ be a fuzzy ideal of R. We define a fuzzy subset Z (δ) by:

Z (δ) = ∪ {μ|μ ∈ [0, 1] ^R, μ ⊆ δ, μσ ⊆ χ₀}, for some essential fuzzy ideal σ of R.

It is seen that Z (δ) is a fuzzy ideal of δ and can be equivalently defined as follows:

Z (δ) = ∪ {r_α|r_α ∈ δ, r_ασ ⊆ χ_θ, for some essential fuzzy ideal σ of R},

and Z (δ) = ∪ {γ|γ ∈ FI (R) , γ ⊆ δ, γσ ⊆ χ₀ for some essential fuzzy ideal σ of R}.

Lemma 4.6. For r ∈ R, α ∈ [0, 1] , r_αχ_R is a fuzzy ideal of R.

Proof. (r_αχ_R) (x) = ∨ { r_α (y) ∧ χ_R) (z) |y, z ∈ R, x = yz} $\begin{matrix} = \lor {r_{α} (y) | y \in R, x = yz} \\ = {\begin{matrix} 0 & if x \notin < r > \\ α & if x \in < r > \end{matrix} \end{matrix}$

Therefore, (r_αχ_R) = < r > _α .

Let x, y ∈ R.

If <r > _α (x) =0, then <r > _α (x) ∧ < r > _α (y)≤ < r > _α (x - y) .

If <r > _α (x) ≠0 (i . e . = α) and <r> _α (y) ≠0 (i . e . = α),

Then, x = rz₁ and y = rz₂ for some z₁, z₂ ∈ R .

So, x - y = r (z₁ - z₂)

Hence, <r > _α (x - y) = α = < r > _α (x) ∧ < r>_α (y)

Similarly, <r > _α (xy) ≥ < r > _α (x) ∀ x, y ∈ R .

Therefore, <r > _αi . e . r_αχ_R is a fuzzy ideal of R.

Lemma 4.7.χ₀ ∪ r_αχ_Ris a fuzzy ideal, wherer ∈ R, α ∈ [0, 1] .

Proof. Let σ = r_αχ_R and γ = χ₀ ∪ σ.

Let t ∈ [0, 1] and t ≤ α. Then γ_t = σ_t which is an ideal of R.

Let t > α. Then γ_t = {0} which is an ideal of R.

Hence, γ is a fuzzy ideal of R.

Theorem 4.8.The fuzzy setχ₀is semiprime if and only ifZ = χ₀ .

Proof. We suppose that Z = χ₀ .

Let r_α ∈ χ_R be such that $r_{α}^{2} \subseteq χ_{0} .$

Let σ be a fuzzy ideal and x_{α
^′} ∈ σ where $x_{α^{'}} \neq χ_{0}$ .

Now, $(x_{α^{'}} χ_{R}) (y) = \lor {x_{α^{'}} (a) \land χ_{R} (b) | y = a b}$ $\begin{matrix} = {_{0 if y \neq xb, for any b}^{α if y = xb, for some b} \\ = < x >_{α} (y) \end{matrix}$

Therefore, x_α′χ_R = < x > _α′ .

If y∉ < x > then <x > _α′ (y) ⊆ σ (y) .

If y∈ < x > then y = xz for some z ∈ R .

Now, σ (y) = σ (xz) ≥ σ (x) ≥ α′ = < x > _α′ (y) .

Therefore, <x > _α′ ⊆ σ .

That is, x_α′χ_R ⊆ σ .

Now either, r_αx^_α′ ⊆ χ₀ or $r_{α} x_{α'} \underline{⊄} χ_{0} .$

r_αx^_α′ ⊆ χ₀ implies x^_α′ ∈ ann (r_α) .

$r_{α} x_{α'} \underline{⊄} χ_{0}$ implies r_α (r_αx^_α′ ) ⊆ χ₀, that is r_αx^_α′ ∈ ann (r_α) .

In both the cases x_α′χ_R ∩ ann (r_α) ≠ χ₀ where x_α′ ∈ σ.

Therefore, σ ∩ ann (r_α) ≠ χ₀ .

So ann (r_α) is an essential fuzzy ideal of R.

This implies, r_α ∈ Z = χ₀ .

Hence, χ₀ is semiprime.

Conversely, we suppose that χ₀ is semiprime.

Clearly, χ₀ ⊆ Z .

Let r_α ∈ Z where r ≠ 0 .

We set, μ = r_αχ_R ∩ ann (r_α) with μ (0) =1 .

Now, (r_αχ_R) ann(r_α) ⊆ χ₀ .

Therefore, μ² = μμ ⊆ (r_αχ_R)ann(r_α) ⊆ χ₀ .

This implies, μ ⊆ χ₀, as χ₀ is semiprime.

Again, r_α ∈ Z implies ann (r_α) is essential.

Therefore, (r_αχ_R) (x) =0 for x ≠ 0 .

That is, (< r > _α) (x) =0 for x ≠ 0 .

In particular, (< r > _α) (r) =0 .

This implies, α = 0 .

Hence, Z = χ₀.

5 Conclusions

Since, rings with chain conditions on fuzzy substructures, other than ideals, have not yet been explored in details, this study provides a foundation for further study on such problems. In our present investigation on rings with finiteness conditions, we build the foundation by introducing essential fuzzy ideals and annihilators of fuzzy subsets. In this paper these two notions help us to develop the concept of singularity which in turn will lead us to many aspects of rings with chain conditions on various fuzzy substructures.

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