Fuzzy Small Submodule and Jacobson 𝐿-Radical

Abstract

Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and sufficient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions. Finally, a relation between small 𝐿-submodules and Jacobson 𝐿-radical is established.

1. Introduction

After the introduction of fuzzy sets by Zadeh [1], a number of applications of this fundamental concept have come up. Rosenfeld [2] was the first one to define the concept of fuzzy subgroups of a group. Since then many generalizations of this fundamental concept have been done in the last three decades. Naegoita and Ralescu [3] applied this concept to modules and defined fuzzy submodules of a module. Consequently, fuzzy finitely generated submodules, fuzzy quotient modules [4], radical of fuzzy submodules, and primary fuzzy submodules [5, 6] were investigated. Saikia and Kalita [7] defined fuzzy essential submodules and investigated various characteristics of such submodules. These modules play a prominent role in fuzzy Goldie dimension of modules.

In this paper we fuzzify various properties of small (or superfluous) submodules of a module. We define fuzzy small ephimorphism and fuzzy coessential extension of a fuzzy submodule. We investigate various characteristics of fuzzy small submodules. Necessary and sufficient conditions for fuzzy small submodules are established. We also investigate the nature of fuzzy small submodule under fuzzy direct sum. A relation regarding fuzzy small submodule of a module and fuzzy quotient module is also established. We attempt to fuzzify the well-known relation between the Jacobson radical and the small submodules of a module. In [8] Basnet et. al. have shown that the relation between the Jacobson radical and the small submodules of a module does not hold in fuzzy setting whereas we have tried to achieve the relation. It is established that the Jacobson 𝐿-radical is the sum of all the small 𝐿-submodules of a module. In case of a finitely generated module, the Jacobson 𝐿-radical is also a small 𝐿-submodule of the module under the condition that 𝐿-{1} possesses a maximal element.

2. Basic Definitions and Notations

By 𝑅 we mean a commutative ring with unity 1 and 𝑀 denotes a 𝑅-module. The zero elements of 𝑅 and 𝑀 are 0 and 𝜃, respectively. A complete Heyting algebra 𝐿 is a complete lattice such that for all 𝐴⊆𝐿 and for all 𝑏∈𝐿, ∨{𝑎∧𝑏∣𝑎∈𝐴}=(∨{𝑎∣𝑎∈𝐴})∧𝑏 and ∧{𝑎∨𝑏∣𝑎∈𝐴}=(∧{𝑎∣𝑎∈𝐴})∨𝑏.

Definition 2.1. A submodule 𝑆 of a module 𝑀 over a ring 𝑅 is said to be a small submodule of 𝑀 if for every submodule 𝑁 of 𝑀 with 𝑁≠𝑀 implies 𝑆+𝑁≠𝑀.
The notation 𝑆≪𝑀 indicates that 𝑆 is a small submodule of 𝑀.

Fuzzy set on a nonempty set was introduced by Zadeh [1] in 1965. It is defined as follows.

Definition 2.2. By a fuzzy set of a module 𝑀, we mean any mapping 𝜇 from 𝑀 to [0,1]. By [0,1]^𝑀 we will denote the set of all fuzzy subsets of 𝑀. If 𝜇 is a mapping from 𝑀 to 𝐿, where 𝐿 is a complete Heyting algebra then 𝜇 is called an 𝐿-subset of 𝑀. By 𝐿^𝑀 we will denote the set of all 𝐿-subsets of 𝑀.
For each fuzzy set 𝜇 in 𝑀 and any 𝛼∈[0,1], we define two sets 𝑈(𝜇,𝛼)={𝑥∈𝑀∣𝜇(𝑥)≥𝛼}, 𝐿(𝜇,𝛼)={𝑥∈𝑀∣𝜇(𝑥)≤𝛼}, which are called an upper level cut and a lower level cut of 𝜇, respectively. The complement of 𝜇, denoted by 𝜇^𝑐, is the fuzzy set on 𝑀 defined by 𝜇^𝑐(𝑥)=1−𝜇(𝑥). The support of a fuzzy set 𝜇, denoted by 𝜇^∗, is a subset of 𝑀 defined by 𝜇^∗={𝑥∈𝑀∣𝜇(𝑥)>0}. The subset 𝜇_∗ of 𝑀 is defined as 𝜇_∗={𝑥∈𝑀∣𝜇(𝑥)=𝜇(𝜃)}.

Definition 2.3 (see [9]). If 𝑁⊆𝑀 and 𝛼∈[0,1]^𝑀 then 𝛼_𝑁 is defined as, 𝛼_𝑁(𝑥)=𝛼if0𝑥∈𝑁,otherwise. (2.1)

If 𝑁= {𝑥} then 𝛼_{𝑥} is often called a fuzzy point and is denoted by 𝑥_𝛼. When 𝛼=1 then 1_𝑁 is known as the characteristic function of 𝑁. From now onwards, we will denote the characteristic function of 𝑁 as 𝜒_𝑁.

If 𝜇,𝜎∈[0,1]^𝑀 then(1)𝜇⊆𝜎, if and only if 𝜇(𝑥)≤𝜎(𝑥);(2)(𝜇∪𝜎)(𝑥)=max{𝜇(𝑥),𝜎(𝑥)}=𝜇(𝑥)∨𝜎(𝑥);(3)(𝜇∩𝜎)(𝑥)=min{𝜇(𝑥),𝜎(𝑥)}=𝜇(𝑥)∧𝜎(𝑥).

For any family {𝜇_𝑖∣𝑖∈𝐼} of fuzzy subsets of 𝑀, where 𝐼 is any nonempty index set,(4)(⋃_𝑖∈𝐼𝜇_𝑖)(𝑥)=sup_𝑖∈𝐼𝜇_𝑖(𝑥)=⋁_𝑖∈𝐼𝜇_𝑖(𝑥);(5)(⋂_𝑖∈𝐼𝜇_𝑖)(𝑥)=inf_𝑖∈𝐼𝜇_𝑖(𝑥)=⋀_𝑖∈𝐼𝜇_𝑖(𝑥) for all 𝑥∈𝑀;(6)(𝜇+𝜎)(𝑥)=∨{𝜇(𝑦)∧𝜎(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥}.

Definition 2.4 (see [9]). Let 𝑋 and 𝑌 be any two nonempty sets, and 𝑓∶𝑋→𝑌 be a mapping. Let 𝜇∈[0,1]^𝑋 and 𝜎∈[0,1]^𝑌 then the image 𝑓(𝜇)∈[0,1]^𝑌 and the inverse image 𝑓⁻¹(𝜎)∈[0,1]^𝑋 are defined as follows: for all 𝑦∈𝑌𝑓𝜇𝑦=∨𝜇(𝑥)∣𝑥∈𝑋,𝑓(𝑥)=𝑦,if𝑓⁻¹𝑦≠𝜙,0,otherwise. (2.2) and 𝑓⁻¹(𝜎)(𝑥)=𝜎(𝑓(𝑥)) for all 𝑥∈𝑋.

Definition 2.5 (see [9]). Let 𝜁∈[0,1]^𝑅 and 𝜇∈[0,1]^𝑀. Then 𝜁⊙𝜇 is a fuzzy subset of 𝑀 and it is defined by 𝜁⊙𝜇(𝑥)=∨^𝑛_𝑖=1𝜁𝑟_𝑖∧𝜇𝑥_𝑖∣𝑟_𝑖∈𝑅,𝑥_𝑖∈𝑀,1≤𝑖≤𝑛,𝑛∈𝑁,^𝑛_𝑖=1𝑟_𝑖𝑥_𝑖=𝑥 (2.3) for all 𝑥∈𝑀.

Definition 2.6 (see [9]). A fuzzy set 𝜇 of 𝑅 is called a fuzzy ideal, if it satisfies the following properties:(1)𝜇(𝑥−𝑦)≥𝜇(𝑥)∧𝜇(𝑦),(2)𝜇(𝑥𝑦)≥𝜇(𝑥)∨𝜇(𝑦), for all 𝑥,𝑦∈𝑅.

The following definition is given by Naegoita and Ralescu [3].

Definition 2.7 (see [3]). Let 𝑀 be a module over a ring 𝑅 and 𝐿 be a Complete Heyting algebra. An 𝐿 subset 𝜇 in 𝑀 is called an 𝐿-submodule of 𝑀, if for every 𝑥,𝑦∈𝑀 and 𝑟∈𝑅 the following conditions are satisfied:(1)𝜇(𝜃)=1,(2)𝜇(𝑥−𝑦)≥𝜇(𝑥)∧𝜇(𝑦),(3)𝜇(𝑟𝑥)≥𝜇(𝑥).We denote the set of all 𝐿-submodules of 𝑀 by 𝐿(𝑀). If 𝐿=[0,1], then 𝜇 is called a fuzzy submodule of 𝑀. The set of all fuzzy submodules of 𝑀 are denoted by 𝐹(𝑀).

Definition 2.8 (see [9]). Let 𝜇,𝜈∈𝐹(𝑀) be such that 𝜇⊆𝜈. Then the quotient of 𝜈 with respect to 𝜇, is a fuzzy submodule of 𝑀/𝜇^∗, denoted by 𝜈/𝜇, and is defined as follows: 𝜈𝜇([𝑥])=∨{𝜈(𝑧)∣𝑧∈[𝑥]},∀𝑥∈𝜈^∗, (2.4) where [𝑥] denotes the coset 𝑥+𝜇^∗.

Definition 2.9 (see [9]). Let 𝜇∈[0,1]^𝑀. Then ∩{𝜈∣𝜇⊆𝜈,𝜈∈𝐹(𝑀)} is a fuzzy submodule of 𝑀, and it is called the fuzzy submodule generated by the fuzzy subset 𝜇. We denote this by ⟨𝜇⟩, that is, 𝜇=∩𝜈∣𝜇⊆𝜈,𝜈∈𝐹(𝑀). (2.5) Let 𝜉∈𝐹(𝑀). If 𝜉=⟨𝜇⟩ for some 𝜇∈[0,1]^𝑀, then 𝜇 is called a generating fuzzy subset of 𝜉.

Remark 2.10. (a) If 𝐴 is a nonempty subset of 𝑀, then ⟨𝜒_𝐴⟩=𝜒_⟨𝐴⟩, where ⟨𝐴⟩ is the submodule of 𝑀 generated by 𝐴.
(b) If 𝑥∈𝑀, then 𝜒_𝑅⊙𝜒_{𝑥} is a fuzzy submodule of 𝑀 generated by 𝜒_{𝑥}, and in this case, 𝜒_𝑅⊙𝜒_{𝑥}=𝜒_{𝑥}=𝜒_⟨{𝑥}⟩=𝜒_𝑅𝑥. (2.6)

3. Preliminaries

This section contains some preliminary results that are needed in the sequel.

Lemma 3.1 (see [10]). Let 𝑀 be a module and suppose that 𝐾≤𝑁≤𝑀 and 𝐻≤𝑀. Then(a)𝐻+𝐾≪𝑀 if and only if 𝐻≪𝑀 and 𝐾≪𝑀;(b)if 𝐾≪𝑁, then 𝐾≪𝑀;(c)if 𝑁 is a direct summand of 𝑀, then 𝐾≪𝑀 if and only if 𝐾≪𝑁;(d)if 𝑀=𝑀₁⊕𝑀₂ and 𝐾_𝑖≤𝑀_𝑖 for 𝑖=1,2, then 𝐾₁⊕𝐾₂≪𝑀₁⊕𝑀₂ if and only if 𝐾₁≪𝑀₁ and 𝐾₁≪𝑀₁.

Lemma 3.2 (see [9]). Let 𝜇,𝜈∈𝐹(𝑀). Then 𝜇+𝜈∈𝐹(𝑀).

Lemma 3.3 (see [9]). Let 𝜇_𝑖∈𝐹(𝑀), for each 𝑖∈𝐼, where |𝐼|>1. Then ∑_𝑖∈𝐼𝜇_𝑖∈𝐹(𝑀) and ⟨⋃_𝑖∈𝐼𝜇_𝑖⟩=∑_𝑖∈𝐼𝜇_𝑖.

Lemma 3.4 (see [5]). Let 𝜇∈[0,1]^𝑀. Then the level subset 𝜇_𝑡={𝑥∈𝑀∶𝜇(𝑥)≥𝑡}, 𝑡∈Im(𝜇) is a submodule of 𝑀 if and only if 𝜇 is a fuzzy submodule of 𝑀.

Corollary 3.5. 𝜇_∗ is a submodule of 𝑀 if and only if 𝜇 is a fuzzy submodule of 𝑀.

In the next two sections we present our main results.

4. Fuzzy Small Submodule

Definition 4.1. Let 𝑀 be a module over a ring 𝑅 and let 𝜇∈𝐿(𝑀). Then 𝜇 is said to be a Small 𝐿-Submodule of 𝑀, if for any 𝜈∈𝐿(𝑀) satisfying 𝜈≠𝜒_𝑀 implies 𝜇+𝜈≠𝜒_𝑀. The notation 𝜇≪_𝐿𝑀 indicates that 𝜇 is a small 𝐿-submodule of 𝑀.
If 𝐿=[0,1], then 𝜇 is called a fuzzy small submodule of 𝑀 and it is indicated by the notation 𝜇≪_𝑓𝑀. It is obvious that 𝜒_{𝜃} is always a fuzzy small submodule of 𝑀.

Definition 4.2. Let 𝜇∈𝐹(𝑀). If 𝜇≪_𝑓𝑀, then we say 𝜒_𝑀 (or 𝑀) is a fuzzy small cover of 𝜒_𝑀/𝜇∗ or 𝑀/𝜇.

Definition 4.3. Let 𝑀 and 𝐿 be any two modules over a ring 𝑅. Then an ephimorphism 𝑓∶𝑀→𝐿 is called a fuzzy small ephimorphism, if 𝑓⁻¹(𝜒_{𝜃})≪_𝑓𝑀.

It is obvious that 𝑀 is a fuzzy small cover of 𝑀/𝜇 if and only if the canonical projection 𝑀→𝑀/𝜇_∗ is a fuzzy small ephimorphism.

Definition 4.4. A fuzzy ideal 𝜇 of 𝑅 with 𝜇(0)=1 is called a fuzzy small ideal of 𝑅 if it is a fuzzy small submodule of _𝑅𝑅.

Let 𝜇 and 𝜎 be any two fuzzy submodules of 𝑀 such that 𝜇⊆𝜎, then 𝜇 is called a fuzzy submodule of 𝜎. And 𝜇 is called a fuzzy small submodule in 𝜎, denoted by 𝜇≪_𝑓𝜎, if 𝜇≪_𝑓𝜎^∗ in the sense that for every submodule 𝛾 in 𝑀 satisfying 𝛾_|𝜎∗≠𝜒_𝜎∗ implies 𝜇_|𝜎∗+𝛾_|𝜎∗≠𝜒_𝜎∗ (by 𝜇_|𝜎∗,𝛾_|𝜎∗ we mean the restriction mapping of 𝜇, 𝛾 on 𝜎^∗ resp.).

Definition 4.5. Let 𝑀 be a module over a ring 𝑅 and suppose that 𝜇,𝜈∈𝐹(𝑀) with 𝜇⊆𝜈. Then we say 𝜈 lies above 𝜇 in 𝑀 or 𝜇 is a coessential extension of 𝜈 if 𝑀/𝜇 is a fuzzy small cover of 𝑀/𝜈, that is, 𝜈/𝜇≪_𝑓𝑀/𝜇(=𝜒_𝑀/𝜇∗).

Example 4.6. Consider 𝑀=𝑍₈={0,1,2,3,4,5,6,7} under addition modulo 8. Then 𝑀 is a module over the ring 𝑍. Let 𝑆={0,2,4,6}. Define 𝜇∈[0,1]^𝑀 as follows: 𝜇(𝑥)=1if𝛼𝑥∈𝑆,otherwise. (4.1) where 0≤𝛼<1. Then 𝜇 is a fuzzy small submodule of 𝑀.

Remark 4.7. Let 𝐾={0,4}. Clearly 𝑆,𝐾 are the only proper submodules of 𝑀 and 𝑆+𝐾={0,2,4,6}+{0,4}≠𝑀. Therefore 𝑆≪𝑀. Also 𝜇_∗={0,2,4,6}=𝑆. Moreover, if we take 𝛼=0 then 𝜇 becomes the characteristic function of 𝑆.

The above remark inspires us to state the following two theorems.

Theorem 4.8. Let 𝑀 be a module and 𝑁≤𝑀. Then 𝑁≪𝑀 if and only if 𝜒_𝑁≪_𝑓𝑀.

Proof. Let 𝑁≪𝑀. We assume 𝜒_𝑁 is not a fuzzy small submodule of 𝑀. Thus there exists, 𝜈∈𝐹(𝑀), 𝜈≠𝜒_𝑀 such that 𝜒_𝑁+𝜈=𝜒_𝑀. Let 𝑥∈𝑀. Then 1=𝜒_𝑁+𝜈(𝑥)=∨𝜒_𝑁𝑦∧𝜈(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥. (4.2) So, there exist 𝑦₀,𝑧₀∈𝑀 with 𝑦₀+𝑧₀=𝑥 such that 𝜒_𝑁(𝑦₀)∧𝜈(𝑧₀)=1. Thus we have 𝜒_𝑁(𝑦₀)=1 and 𝜈(𝑧₀)=1, and so 𝑦₀∈𝑁, 𝑧₀∈𝜈_∗. This implies that 𝑥=𝑦₀+𝑧₀∈𝑁+𝜈_∗. Since 𝑥∈𝑀 is arbitrary, so this implies 𝑀=𝑁+𝜈_∗. But 𝑁≪𝑀. So, we must have 𝑀=𝜈_∗ and this implies 𝜈=𝜒_𝑀, a contradiction. Therefore, 𝜒_𝑁≪_𝑓𝑀.
Conversely, we assume 𝜒_𝑁≪_𝑓𝑀. If possible let 𝑁 be not a small submodule of 𝑀. Thus there exists 𝑇≤𝑀, 𝑇≠𝑀, but 𝑁+𝑇=𝑀. Thus 𝜒_𝑁,𝜒_𝑇∈𝐹(𝑀) and 𝜒_𝑁≠𝜒_𝑀, 𝜒_𝑇≠𝜒_𝑀. Let 𝑥∈𝑀. Since 𝑁+𝑇=𝑀, so there exist, 𝑛∈𝑁, 𝑙∈𝑇 such that 𝑥=𝑛+𝑙. Now, 𝜒_𝑁+𝜒_𝑇(𝑥)=∨𝜒_𝑁𝑦∧𝜒_𝑇(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥≥𝜒_𝑁(𝑛)∧𝜒_𝑇(𝑙)=1. (4.3) This implies 𝜒_𝑁+𝜒_𝑇=𝜒_𝑀 and it contradicts the fact that 𝜒_𝑁≪_𝑓𝑀. Hence 𝑁≪𝑀.

Here we present an alternative proof of the following theorem.

Theorem 4.9 (see [8]). Let 𝜇∈𝐹(𝑀). Then 𝜇≪_𝑓𝑀 if and only if 𝜇_∗≪𝑀.

Proof. Suppose, 𝜇≪_𝑓𝑀. Let 𝑁≤𝑀 and 𝑁≠𝑀. We claim 𝜇_∗+𝑁≠𝑀. Now, 𝑁≠𝑀 implies 𝜒_𝑁≠𝜒_𝑀. Since 𝜇≪_𝑓𝑀, so we must have 𝜇+𝜒_𝑁≠𝜒_𝑀.⇒ there exists 𝑥₀∈𝑀 such that (𝜇+𝜒_𝑁)(𝑥₀)<1,⇒∨{𝜇(𝑦)∧𝜒_𝑁(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥₀}<1,⇒ either 𝜇(𝑦)<1 or 𝑧∉𝑁, for all 𝑦,𝑧∈𝑀, 𝑦+𝑧=𝑥₀,⇒ either 𝑦∉𝜇_∗ or 𝑧∉𝑁, for all 𝑦,𝑧∈𝑀, 𝑦+𝑧=𝑥₀,⇒𝑥₀=𝑦+𝑧∉𝜇_∗+𝑁,⇒𝜇_∗+𝑁≠𝑀,⇒𝜇_∗≪𝑀.Conversely, we assume 𝜇_∗≪𝑀. Let 𝜈∈𝐹(𝑀) be such that 𝜈≠𝜒_𝑀. This implies that 𝜈_∗≠𝑀. Thus 𝜇_∗+𝜈_∗≠𝑀 (since 𝜇_∗≪𝑀). This implies that there exists 𝑥₀∈𝑀 such that 𝑥∉𝜇_∗+𝜈_∗. Thus for every 𝑦,𝑧∈𝑀 with 𝑦+𝑧=𝑥₀ implies either 𝑦∉𝜇_∗ or 𝑧∉𝜈_∗ otherwise 𝑥₀∈𝜇_∗+𝜈_∗. Therefore,
𝜇(𝑦)<1 or 𝜈(𝑧)<1 for every 𝑦,𝑧∈𝑀 and 𝑦+𝑧=𝑥₀,⇒𝜇(𝑦)∧𝜈(𝑧)<1, for every 𝑦,𝑧∈𝑀 and 𝑦+𝑧=𝑥₀,⇒∨{𝜇(𝑦)∧𝜈(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥₀}<1,⇒(𝜇+𝜈)(𝑥₀)<1,⇒𝜇+𝜈≠𝜒_𝑀,⇒𝜇≪_𝑓𝑀.

Corollary 4.10. Let 𝜇,𝜎∈𝐹(𝑀). Then 𝜇≪_𝑓𝜎 if and only if 𝜇_∗≪𝜎^∗.

Proof. Let 𝜇≪_𝑓𝜎⇒𝜇≪_𝑓𝜎^∗. So, by above theorem we get 𝜇_∗≪𝜎^∗. Conversely if 𝜇_∗≪𝜎^∗. So, by above theorem 𝜇≪_𝑓𝜎^∗. Hence 𝜇≪_𝑓𝜎.

Theorem 4.11 (see [8]). Let 𝜇,𝜈∈𝐹(𝑀). Then 𝜇≪_𝑓𝑀, 𝜈≪_𝑓𝑀 if and only if 𝜇+𝜈≪_𝑓𝑀.

As a consequence, we obtain the following.

Theorem 4.12. Any finite sum of fuzzy small submodules of 𝑀 is also a fuzzy small submodule in 𝑀.

Theorem 4.13. Let 𝑁≤𝑀 and 𝜇∈𝐹(𝑀) be such that 𝜇⊆𝜒_𝑁. If 𝜇_|𝑁≪_𝑓𝑁, then 𝜇≪_𝑓𝑀.

Proof. Let 𝜈∈𝐹(𝑀) be such that 𝜇+𝜈=𝜒_𝑀. Let 𝑥∈𝑁. Then
(𝜇_|𝑁+(𝜈_|𝑁∩𝜒_𝑁))(𝑥),=∨{𝜇(𝑦)∧(𝜈_|𝑁∩𝜒_𝑁)(𝑧)∣𝑦,𝑧∈𝑁,𝑦+𝑧=𝑥},=∨{𝜇(𝑦)∧𝜈(𝑧)∣𝑦,𝑧∈𝑁,𝑦+𝑧=𝑥},=(𝜇+𝜈)(𝑥)=1 (since 𝜇+𝜈=𝜒_𝑀). Therefore, (𝜇_|𝑁+(𝜈_|𝑁∩𝜒_𝑁))=𝜒_𝑁. Since 𝜇_|𝑁≪_𝑓𝑁, so 𝜈_|𝑁∩𝜒_𝑁=𝜒_𝑁. This implies that 𝜒_𝑁⊆𝜈 and 𝜇⊆𝜒_𝑁⊆𝜈. Thus 𝜇+𝜈⊆𝜈⇒𝜒_𝑀⊆𝜈⇒𝜒_𝑀=𝜈. Hence 𝜇≪_𝑓𝑀.

Corollary 4.14. Let 𝜇,𝜈∈𝐹(𝑀) and 𝜇⊆𝜈. If 𝜇≪_𝑓𝜈, then 𝜇≪_𝑓𝑀.

Proof. By definition, 𝜇≪_𝑓𝜈 means 𝜇≪_𝑓𝜈^∗. Therefore, from above theorem we get 𝜇≪_𝑓𝑀.

Theorem 4.15. Let 𝜇,𝜈∈𝐹(𝑀). Then 𝜇≪_𝑓𝜈 if and only if 𝜇_∗≪𝜈_∗.

Proof. Suppose, 𝜇≪_𝑓𝜈. Let 𝑁≤𝜈_∗ and 𝑁≠𝜈_∗. This implies 𝜒_𝑁≠𝜒_𝜈∗. Since 𝜇≪_𝑓𝜈, so 𝜇+𝜒_𝑁≠𝜒_𝜈∗. This ensures that there exists 𝑥₀ in 𝜈_∗ such that 𝑥₀∉𝜇_∗+𝑁. Thus 𝜇_∗+𝑁≠𝜈_∗ and hence 𝜇_∗≪𝜈_∗.
Conversely, we assume 𝜇_∗≪𝜈_∗. This implies 𝜇_∗≪𝜈_∗≤𝜈^∗. Therefore, 𝜇_∗≪𝜈^∗ (Lemma 3.1(b)). So, by Corollary 4.10 we have 𝜇≪_𝑓𝜈.

Definition 4.16. A fuzzy submodule 𝜎 in 𝑀 is called a fuzzy direct sum of two fuzzy submodules 𝜇 and 𝜈 if 𝜎=𝜇+𝜈 and 𝜇∩𝜈=𝜒_𝜃.

Definition 4.17. Any 𝜇∈𝐹(𝑀) is called a fuzzy direct summand of 𝑀 if there exists 𝜈∈𝐹(𝑀) such that 𝜒_𝑀 is a fuzzy direct sum of 𝜇,𝜈.

Theorem 4.18. Let 𝜇,𝜈 be fuzzy submodules of M with 𝜇⊆𝜈 and 𝜈 be a direct summand of 𝑀. Then 𝜇≪_𝑓𝑀 if and only if 𝜇≪_𝑓𝜈.

Proof. Suppose, 𝜇≪_𝑓𝑀. Since 𝜈 is a direct summand of 𝑀, so there exists 𝛾∈𝐹(𝑀) such that 𝜒_𝑀=𝜈+𝛾,𝜈∩𝛾=𝜒_𝜃. (4.4) First we prove 𝑀=𝜈^∗⊕𝛾^∗. Now, 𝜈∩𝛾=𝜒_𝜃 implies 𝜈^∗∩𝛾^∗=𝜃. Also, from 𝜒_𝑀=𝜈+𝛾 we have 𝑀=(𝜈+𝛾)^∗. We claim (𝜈+𝛾)^∗=𝜈^∗+𝛾^∗. For this let 𝑥∈(𝜈+𝛾)^∗. Then
(𝜈+𝛾)(𝑥)=∨{𝜈(𝑦)∧𝛾(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥}>0, ⇒𝜈(𝑦)>0 and 𝛾(𝑧)>0 for some 𝑦,𝑧∈𝑀, 𝑦+𝑧=𝑥,⇒𝑥=𝑦+𝑧, for some 𝑦∈𝜈^∗, 𝑧∈𝛾^∗,⇒𝑥∈𝜈^∗+𝛾^∗,⇒(𝜈+𝛾)^∗⊆𝜈^∗+𝛾^∗, On the other hand if 𝑥∈𝜈^∗+𝛾^∗, then 𝑥=𝑦+𝑧 for some 𝑦,𝑧∈𝑀 with 𝜈(𝑦)>0, 𝛾(𝑧)>0. This implies 0<∨{𝜈(𝑦)∧𝛾(𝑧)∣𝑦,𝑧∈𝑀,𝑦+𝑧=𝑥}=(𝜈+𝛾)(𝑥). Thus 𝑥∈(𝜈+𝛾)^∗ and so, we have (𝜈+𝛾)^∗=𝜈^∗+𝛾^∗ and hence 𝑀=(𝜈+𝛾)^∗=𝜈^∗+𝛾^∗. Therefore, 𝜈^∗ is a direct summand of 𝑀. But, we know 𝜇≪_𝑓𝑀 if and only if 𝜇_∗≪𝑀. Thus 𝜇_∗≪𝜈^∗ and 𝜈^∗ is a direct summand of 𝑀 and so, by Lemma 3.1(c) we get 𝜇_∗≪𝜈^∗. Hence, by Corollary 4.10 we have 𝜇≪_𝑓𝜈.
The proof of the converse part follows from Corollary 4.14.

Theorem 4.19. Let 𝜎₁,𝜎₂∈𝐹(𝑀) and 𝜒_𝑀=𝜎₁⊕𝜎₂. Also, let 𝜇₁,𝜇₂∈𝐹(𝑀) be such that 𝜇₁⊆𝜎₁ and 𝜇₂⊆𝜎₂. Then 𝜇₁≪_𝑓𝜎₁ and 𝜇₂≪_𝑓𝜎₂ if and only if 𝜇₁⊕𝜇₂≪_𝑓𝜎₁⊕𝜎₂, that is, if and only if 𝜇₁⊕𝜇₂≪_𝑓𝑀.

Proof. Suppose, 𝜇₁≪_𝑓𝜎₁ and 𝜇₂≪_𝑓𝜎₂. Then 𝜇_1∗≪_𝑓𝜎^∗₁ and 𝜇_2∗≪_𝑓𝜎^∗₂ (Corollary 4.10). Therefore, we get 𝜇_1∗⊕𝜇_2∗≪𝜎^∗₁⊕𝜎^∗₂ (Lemma 3.1(d)). Now, 𝜒_𝑀=𝜎₁⊕𝜎₂ implies 𝑀=(𝜎₁⊕𝜎₂)^∗=𝜎^∗₁⊕𝜎^∗₂ and since 𝜇_1∗⊕𝜇_2∗=(𝜇₁⊕𝜇₂)_∗, therefore we get (𝜇₁⊕𝜇₂)_∗≪(𝜎₁⊕𝜎₂)^∗. Hence 𝜇₁⊕𝜇₂≪_𝑓𝜎₁⊕𝜎₂ (Corollary 4.10).
Conversely, we assume 𝜇₁⊕𝜇₂≪𝜎₁⊕𝜎₂ if and only if 𝜇₁⊕𝜇₂≪_𝑓𝑀. Now, 𝜇₁≤𝜇₁⊕𝜇₂ and 𝜇₁⊕𝜇₂≪_𝑓𝑀 imply 𝜇₁≪_𝑓𝑀. Again, since 𝜎₁ is a fuzzy direct summand of 𝑀 and 𝜇₁⊆𝜎₁, so by Theorem 4.18 we get 𝜇₁≪_𝑓𝜎₁. Similarly, it can be proved that 𝜇₂≪_𝑓𝜎₂.

Corollary 4.20. Let 𝑀₁≤𝑀, 𝑀₂≤𝑀 and 𝑀=𝑀₁⊕𝑀₂. Let 𝜇₁∈𝐹(𝑀₁), 𝜇₂∈𝐹(𝑀₂). Define 𝜇_𝑖(𝑥)=𝜇_𝑖(𝑥)if𝑥∈𝑀_𝑖,0otherwise. (4.5) for all 𝑥∈𝑀, 𝑖=1,2. Then 𝜇₁⊕𝜇₂≪_𝑓𝑀 if and only if 𝜇₁≪_𝑓𝑀₁ and 𝜇₂≪_𝑓𝑀₂.

Theorem 4.21 (see [8]). Let 𝑀, 𝑀 be any two modules over the same ring R and let 𝑓∶𝑀→𝑀 be a module homomorphism. If 𝜇≪_𝑓𝑀, then 𝑓(𝜇)≪_𝑓𝑀.

Theorem 4.22. Let 𝜇,𝜈∈𝐹(𝑀) be such that 𝜇⊆𝜈. Then 𝜈≪_𝑓𝑀 if and only if 𝜇≪_𝑓𝑀 and 𝜈/𝜇≪_𝑓𝜒_𝑀/𝜇, that is, if and only if 𝜇≪_𝑓𝑀 and 𝜈/𝜇≪_𝑓𝑀/𝜇^∗.

Proof. It is obvious that, 𝜒_𝑀/𝜇=𝜒_𝑀/𝜇∗. So, it is sufficient to show 𝜈≪_𝑓𝑀 if and only if 𝜇≪_𝑓𝑀 and 𝜈/𝜇≪_𝑓𝑀/𝜇^∗.
Suppose, 𝜈≪_𝑓𝑀. Then since 𝜇⊆𝜈, so 𝜇≪_𝑓𝑀. Next, we will prove 𝜈/𝜇≪_𝑓𝑀/𝜇^∗. Consider, the natural homomorphism, 𝑓∶𝑀→𝑀/𝜇^∗, defined by 𝑓(𝑥)=[𝑥], where [𝑥] denotes the coset 𝑥+𝜇^∗. Since 𝜈≪_𝑓𝑀, so by Theorem 4.21 we get, 𝑓(𝜈)≪_𝑓𝑀/𝜇^∗. Now, 𝑓(𝜈)([𝑥])=∨{𝜈(𝑦)∣𝑦∈𝑀,𝑓(𝑦)=[𝑥]},=∨{𝜈(𝑦)∣𝑦∈𝑀,[𝑦]=[𝑥]},=∨{𝜈(𝑦)∣𝑦∈𝑀,𝑦−𝑥∈𝜇^∗},=∨{𝜈(𝑦)∣𝑦∈𝑀,𝑦−𝑥=𝑚,𝑚∈𝜇^∗},=∨{𝜈(𝑥+𝑚)∣𝑚∈𝜇^∗},=∨{𝜈(𝑢)∣𝑢=𝑥+𝑚,𝑚∈𝜇^∗},=∨{𝜈(𝑦)∣𝑢∈[𝑥]},=𝜈/𝜇([𝑥]) for all 𝑥∈𝜈^∗.Thus 𝜈/𝜇=𝑓(𝜈) on 𝜈^∗ and 𝑓(𝜈)≪_𝑓𝑀/𝜇^∗. Therefore, 𝜈/𝜇≪_𝑓𝑀/𝜇^∗.
Conversely, we assume 𝜇≪_𝑓𝑀 and 𝜈/𝜇≪_𝑓𝑀/𝜇^∗. To show 𝜈≪_𝑓𝑀, let 𝜎∈𝐹(𝑀) be such that 𝜎≠𝜒_𝑀. Since 𝜇≪_𝑓𝑀 and 𝜎≠𝜒_𝑀, so we have 𝜇+𝜎≠𝜒_𝑀. This implies (𝜇+𝜎)/𝜇≠𝜒_𝑀/𝜇(=𝜒_𝑀/𝜇∗). Again, since 𝜈/𝜇≪_𝑓𝑀/𝜇^∗ and (𝜇+𝜎)/𝜇≠𝜒_𝑀/𝜇∗, so we must have(𝜈/𝜇)+(𝜇+𝜎)/𝜇≠𝜒_𝑀/𝜇∗,⇒(𝜈+𝜇+𝜎)/𝜇≠𝜒_𝑀/𝜇∗,⇒(𝜇+𝜈+𝜎)/𝜇≠𝜒_𝑀/𝜇∗,⇒(𝜈+𝜎)/(𝜇∩(𝜈+𝜎))≠𝜒_𝑀/𝜇∗, (see [9, Theorem 4.2.5])⇒(𝜈+𝜎)/𝜇≠𝜒_𝑀/𝜇∗, (since 𝜇⊆𝜈)⇒𝜈+𝜎≠𝜒_𝑀, (since 𝜒_𝑀/𝜇=𝜒_𝑀/𝜇∗). Therefore, 𝜈≪_𝑓𝑀.

Corollary 4.23. Let 𝜇,𝜈∈𝐹(𝑀) be such that 𝜇⊆𝜈. If 𝜈≪_𝑓𝑀, then 𝜇 is a coessential extension of 𝜈 in 𝑀.

Theorem 4.24. Let 𝜇,𝜈∈𝐹(𝑀) be such that 𝜇⊆𝜈. Then 𝜇 is a coessential extension of 𝜈 in 𝑀 if and only if 𝜇+𝜎=𝜒_𝑀 holds for all 𝜎∈𝐹(𝑀) with 𝜈+𝜎=𝜒_𝑀.

Proof. Suppose, 𝜇 is a coessential extension of 𝜈, that is, 𝜈/𝜇≪_𝑓𝜒_𝑀/𝜇. If for all 𝜎∈𝐹(𝑀) with 𝜈+𝜎=𝜒_𝑀, then 𝜒_𝑀𝜇=(𝜈+𝜎)𝜇=𝜈+𝜎+𝜇𝜇=𝜈𝜇+𝜎+𝜇𝜇. (4.6) So, 𝜒_𝑀/𝜇=(𝜎+𝜇)/𝜇 (since 𝜈/𝜇≪_𝑓𝜒_𝑀/𝜇). Thus 𝜇+𝜎=𝜒_𝑀.
Conversely, we assume 𝜇+𝜎=𝜒_𝑀 holds for all 𝜎∈𝐹(𝑀) with 𝜈+𝜎=𝜒_𝑀. If there exists 𝜎∈𝐹(𝑀) containing 𝜇 such that 𝜈/𝜇+𝜎/𝜇=𝜒_𝑀/𝜇, then 𝜒_𝑀=𝜈+𝜎. This yields 𝜒_𝑀=𝜇+𝜎=𝜎, (since 𝜇⊆𝜎) and so 𝜈/𝜇≪_𝑓𝜒_𝑀/𝜇. Hence 𝜇 is a coessential extension of 𝜈.

As a consequence, we obtain the following.

Theorem 4.25. Let 𝛾,𝜇,𝜈∈𝐹(𝑀) be such that 𝛾⊆𝜇⊆𝜈. Then 𝛾 is a coessential extension of 𝜈 in 𝑀 if and only if 𝜇 is a coessential extension of 𝜈 in 𝑀 and 𝛾 is a coessential extension of 𝜇 in 𝑀.

5. Jacobson 𝐿-Radical

Definition 5.1. Let 𝜇∈𝐿(𝑀). Then 𝜇 is called a maximal 𝐿-submodule of 𝑀 if 𝜇≠𝜒_𝑀 (i.e., 𝜇 is proper 𝐿-submodule of 𝑀) and if 𝜎 any other proper 𝐿-submodules of 𝑀 containing 𝜇, then 𝜇=𝜎. Equivalently 𝜇 is a maximal element in the set of all nonconstant 𝐿-submodules of 𝑀 under point wise partial ordering.

The intersection of all maximal 𝐿-submodules of 𝑀 is known as Jacobson 𝐿-radical of 𝑀 and is denoted by JLR(𝑀).

Theorem 5.2. Let 𝜇∈𝐿^𝑀. Then 𝜇 is a maximal 𝐿-submodule of 𝑀 if and only if 𝜇 can be expressed as 𝜇=𝜒_𝜇∗∪𝛼_𝑀, where 𝜇_∗ is a maximal submodule of 𝑀 and 𝛼 is a maximal element of 𝐿-{1}.

Proof. Proof is similar to the proof of Theorem 3.4.3 of [9] and so it is omitted.

Lemma 5.3. Let M be a module over R and let 𝑥∈ M. Then 𝜒_𝑅⊙𝜒_{𝑥}≪_𝐿𝑀 if and only if 𝜒_{𝑥} is in the sum of all small 𝐿-submodules of M.

Proof. Suppose, 𝜒_𝑅⊙𝜒_{𝑥}≪_𝐿𝑀. Then by Theorem 4.9 we have, (𝜒_𝑅⊙𝜒_{𝑥})_∗≪𝑀. But, from Remark 2.10(b) we have 𝜒_𝑅⊙𝜒_{𝑥}=𝜒_{𝑥}=𝜒_⟨{𝑥}⟩=𝜒_𝑅𝑥. (5.1) Therefore, 𝜒_{𝑥}⊆𝜒_𝑅⊙𝜒_{𝑥} as 𝜒_{𝑥}⊆𝜒_𝑅𝑥. This implies 𝜒_{𝑥} is in the sum of all small 𝐿-submodules of 𝑀.
Conversely, we assume 𝜒_{𝑥} is in the sum of all small 𝐿-submodules of 𝑀. Then 𝜒_{𝑥}⊆∑𝜇_𝑖 (finite), where 𝜇_𝑖≪_𝐿𝑀 and so, 𝑥∈∑(𝜇_𝑖)_∗(finite),(𝜇_𝑖)_∗≪𝑀⇒𝑥=∑𝑥_𝑖(finite), where 𝑥_𝑖∈(𝜇_𝑖)_∗. Now, 𝜒_𝑅⊙𝜒_{{𝑥𝑖}∗}=𝜒_{𝑅𝑥𝑖∗}=𝑅𝑥_𝑖≤𝜇_𝑖∗ (5.2) for all 𝑖. Therefore, (𝜒_𝑅⊙𝜒_{𝑥𝑖})_∗≪𝑀 (since (𝜇_𝑖)_∗≪𝑀). Therefore, by Theorem 4.9 we have 𝜒_𝑅⊙𝜒_{𝑥𝑖}≪_𝐿𝑀. This implies ∑𝜒_𝑅⊙𝜒_{𝑥𝑖}(finite)≪_𝐿𝑀. But, 𝜒_{𝑥}⊆∑𝜒_𝑅⊙𝜒_{𝑥𝑖}(finite), and so, we must have, 𝜒_𝑅⊙𝜒_{𝑥}⊆∑𝜒_𝑅⊙𝜒_{𝑥𝑖}(finite). Since ∑𝜒_𝑅⊙𝜒_{𝑥𝑖}(finite)≪_𝐿𝑀. Therefore, we have 𝜒_𝑅⊙𝜒_{𝑥}≪_𝐿𝑀.

Definition 5.4 (see [9]). Let 𝐿 be a complete Heyting algebra. Then 𝑎∈𝐿-{1} is called a maximal element, if there does not exist 𝑐∈𝐿-{1} such that 𝑎<𝑐<1.

Theorem 5.5. For any module 𝑀, JLR(𝑀) (the Jacobson 𝐿-radical of 𝑀), is the sum of all small 𝐿-submodules of 𝑀.

Proof. In view of Lemma 5.3, it is sufficient to show that 𝜒_𝑅⊙𝜒_{𝑥}≪_𝐿𝑀 if and only if 𝜒_{𝑥}⊆JLR(𝑀); or equivalently: 𝜒_{𝑥} is not a subset of JLR(𝑀) if and only if 𝜒_𝑅⊙𝜒_{𝑥} is not a small 𝐿-submodule of 𝑀. We will proof the later one.
Suppose, 𝜒_{𝑥} is not a subset of JLR(𝑀). Then there exists a maximal 𝐿-submodule 𝜈 of 𝑀 such that 𝜒_{𝑥} is not a subset of 𝜈. This implies 𝜒_𝑅⊙𝜒_{𝑥}=⟨𝜒_{𝑥}⟩ which is not a submodule of 𝜈. Since 𝜈 is maximal, so 𝜈≠𝜒_𝑀. Therefore, 𝜈 is a maximal 𝐿-submodule of 𝑀 which is properly contained in 𝜒_𝑅⊙𝜒_{𝑥}+𝜈. This implies 𝜒_𝑅⊙𝜒_{𝑥}+𝜈=𝜒_𝑀. Thus 𝜈≠𝜒_𝑀 and 𝜒_𝑅⊙𝜒_{𝑥}+𝜈=𝜒_𝑀. So, by definition we have, 𝜒_𝑅⊙𝜒_{𝑥} is not a small 𝐿-submodule of 𝑀.
Conversely, we assume 𝜒_𝑅⊙𝜒_{𝑥} is not a small 𝐿-submodule of 𝑀. Then there exists a 𝜈∈𝐿(𝑀) with 𝜈≠𝜒_𝑀 such that 𝜒_𝑅⊙𝜒_{𝑥}+𝜈=𝜒_𝑀. Let 𝑆 be the collection of all such 𝜈. Then 𝑆≠Φ because, 𝜈∈𝑆. Now, for each 𝜎∈𝑆, 𝜎≠𝜒_𝑀 and 𝜒_{𝑥} is not a subset of 𝜎. Moreover, any proper fuzzy submodule containing 𝜈 is also in 𝑆. Also, (𝑆,⊆) forms a poset and the union of members of a chain in 𝑆 is again a member of 𝑆. Therefore, by Zorn's lemma, 𝑆 has a maximal element, (say) 𝜇. Since 𝜇∈𝑆 so, 𝜒_{𝑥} is not a subset of 𝜇. Now, let 𝛾∈𝐿(𝑀) be such that 𝜇⊆𝛾. If 𝛾≠𝜒_𝑀, then 𝛾 is also in 𝑆. So, by maximality of 𝜇, we have 𝜇=𝛾. This shows that 𝜇 is a maximal 𝐿-submodule of 𝑀. Since 𝜒_{𝑥} is not a subset of 𝜇, so we must have 𝜒_{𝑥} is not a subset of JLR(𝑀).

Corollary 5.6. If JLR(𝑀) is a small 𝐿-submodule of 𝑀, then it is the largest small 𝐿-submodule of 𝑀.

Corollary 5.7. Let 𝑀 be finitely generated module, then JLR(𝑀) exists and is a small 𝐿-submodule of 𝑀 provided 𝐿-{1} has a maximal element.

Remark 5.8. However, if we take 𝐿=[0,1], then 𝐿-{1} does not possess any maximal element, and so by Theorem 5.2, maximal 𝐿-submodule of 𝑀 does not exist. Since the existence of the JLR(𝑀) depends on the existence of maximal 𝐿-submodule of 𝑀, therefore, the assumption of the existence of a maximal element of 𝐿-{1} in corollary 5.5 is necessary.

Example 5.9. Let 𝐿={0,0.25,0.5,0.75,1}. Then 𝐿 is a Complete Heyting algebra together with the operations minimum(meet), maximum(join) and ≤(partial ordering), then 0.75 is a maximal element of 𝐿-{1}. Consider 𝑀=𝑍₈={0,1,2,3,4,5,6,7} under addition modulo 8. Then 𝑀 is a module over the ring 𝑍. Let 𝑆={0,2,4,6}. Define 𝜇∈[0,1]^𝑀 as follows: 𝜇(𝑥)=1if𝑥∈𝑆,0.75,otherwise. (5.3) Then 𝜇_∗={0,2,4,6}=𝑆, which is a maximal submodule of 𝑍₈. Also, 𝜇=𝜒_𝜇∗∪0.75_𝑀, where 0.75 is a maximal element of 𝐿-{1}. So, by Theorem 5.2 we have 𝜇 as a maximal 𝐿-submodule of 𝑍₈. In fact, 𝜇 is the only maximal 𝐿-submodule of 𝑍₈ and so JLR(𝑀)=𝜇. Since 𝜇_∗≪𝑍₈ and hence by Theorem 4.9, we get 𝜇≪_𝐿𝑍₈. Thus JLR(𝑀)≪_𝐿𝑍₈. However, if we consider 𝐿=[0,1], then [0,1) does not have a maximal element and so by Theorem 5.2 there does not exist any maximal 𝐿-submodule(maximal fuzzy submodule). So, JLR(𝑀) does not exist when 𝐿=[0,1].

6. Conclusion

In this paper some aspects and properties of fuzzy small submodules have been introduced which dualize the notion of fuzzy essential submodules. This concept has opened a new avenue toward the study of fuzzy Goldie dimension, for example using the notion of fuzzy small submodules one can define hollow submodules and discrete submodules. In our future study we may investigate various aspects of (i) spanning dimension of fuzzy submodules, (ii) corank of fuzzy submodules, (iii) fuzzy lifting modules with chain condition on fuzzy small submodules, and (iv) Noetherian and Artinian conditions on fuzzy Jacobson radical of a module.