Microsoft Word - 89-100   89 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020       On the Space of Primary La-submodules Eman Yahea Habeeb Department of Mathematics, Faculty of Education for Girls, Kufa University Iman.habeeb@uokufa.edu.iq Abstract Suppose that F is a reciprocal ring which has a unity and suppose that H is an F-module. We topologize La-Prim(H), the set of all primary La-submodules of H , similar to that for FPrim(F), the spectrum of fuzzy primary ideals of F, and examine the characteristics of this topological space. Particularly, we will research the relation between La-Prim(H) and La- Prim(F/ Ann(H)) and get some results. Keywords primary La-submodules, Fuzzy primary spectrum, La-top modules. 1. Introduction Suppose that F is a reciprocal ring with a unity and H is a unitary F-module. The primary spectrum Prim (F) and the topological space acquired by inserting Zariski topology on the collection of primary ideals of a reciprocal ring with unity play an significant role in the fields of reciprocal algebra, algebraic geometry and lattice theory. As well, lately the concept of primary submodules and Zariski topology on Prim (H), the collection of all primary submodules of a module H on a reciprocal ring together identity F, were studied in a previous article [1]. As it is famous [2]. Inserted the concept of a fuzzy subset ϑ of a nonempty collection L as a mapping from L to [0,1]. Goguen JA [3]. Changed [0,1] by an entire lattice La in the definition of fuzzy collections while inserted the concept of La-fuzzy sets. Rosenfeld inserted the concept of fuzzy groups [4]. While fuzzy submodules of H over F were first inserted by [5]. Pan F-Z [6]. Elaborate fuzzy finitely created modules while fuzzy quotient modules (look at [7]). In previous years a large saucepan of labor has been completed on fuzzy ideals in common and primary fuzzy ideals in special, while several motivating topological features of the spectrum of fuzzy primary ideals of a ring were acquired (look at [8- 15]). Suppose that H is an F-module. By G H, we mean that G is a submodule of H. For any G H, we indicate the residual of G by H by [G:H], and define[G:H] ={ r ́∈ F \ r ́H ⊆G}. In special, [(0) :H] is called the annihilator of H and is indicated by Ann(H), that is Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.3.2476 Article history: Received 23 July 2019, Accepted 22 Septamber 2019, Published in July 2020.   90 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Ann(H)={r ́∈F \ r ́H=0}. A primary submodule (or a q-primary submodule) of H is a proper submodule Q with Q:H=q, such that r ́h ∈ Q for r ́∈ F and h ∈ H , either h ∈ Q or r ∈́ q. The collection of all primary submodules of H is called the primary spectrum of H or, artlessly the p spectra of H and is indicated by Prim(H). Note that the Prim(H) may be empty for some module H . Such a module is said to be primary less (cf. [1]). Clearly, zero module is primary less, but in [1]. Some nontrivial examples are shown. For example, the Prüfer group ℤ( p∞) as a ℤ-module has no primary submodule for any prime integer p. When Prim(H) ∅, the map φ:La-Prim(H)→ La-Prim(F/ Ann(H)) defined by φ(ϑ)= ϑ: 1H for ϑ ∈ La-Prim(H), φ will be called the standard map. In [1]. It is shown that for each multiplication module H,(An F-module H is called a multiplication module if every submodule B of H is of the form IH for some ideal I of F) the Prim(H) is non-empty. For any submodule G of H, V(G) indicates the collection of all primary submodules of H including G. Of course V(H) is just the empty set and V(0) is Prim(H). For any family of submodules Gj(j ∈ J ) of H , ⋂ V G V ∑ G∈∈ Thus if ω(H) indicates the set of all subsets V(G) of Prim(H), then ω(H) includes the empty set and Prim(H) and is closed beneath arbitrary intersection. If also ω(H) is closed beneath finite union, i.e. for any submodules G and K of H, there occurs a submodule J of H such that V(G)⋃ V(K)=V (J), for in this state ω(H) satisfies the axioms of closed subsets of a topological spaces, which is called Zariski topology. In [1]. A module with Zariski topology is called top module and it is shown that each multiplication module is a top module [1]. In [16,17]. Inserted the concept of primary La-submodules of a module H on a commutative ring together unity F, where La is a whole lattice. The collection of all primary La- submodules of H is called the primary La-spectrum of H or, artlessly the P-La-spectrum of H while is indicated via La-Prim(H). In this work, we follow [18]. And topologize La- Prim(H), which its surname is Zariski topology while examine the characteristics of this topological space. Thereafter, we discussed the relation between the topological spaces La- Prim(H) and La-Prim(F/Ann(H)). Finally, we located a basis for the Zariski topology on La-Prim(H). 2. Basic concepts During this article via F, we mean a reciprocal ring together unity, and H is a unital F- module and La indicates a whole lattice. Via an La-subset ϑ of Y ∅ , we mean a mapping ϑ from Y to La while if La=[0,1] , then ϑ is a surname of a fuzzy subset of Y. LaY indicates the collection of each La-subsets of Y . Suppose that C is a subset of Y and b ∈ La. Define bC ∈ LaY as follows: b y b if y ∈ C 0 otherwise In particular case if C={c} we indicate b{c} by bc , while its surname is an La-point of Y. For ϑ ∈ LaY while c ∈ La, locate ϑc as follows:   91 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 ϑc = {y ∈ Y | ϑ(y) ≥ c}, ϑc is called the c-level subset of ϑ. The image of ϑ is indicated via Ima(ϑ) or ϑ(Y). In [18]. It was proved that ϑ ⋃ c∈ . For ϑ, ϖ ∈ LaY we say that ϑ is included in ϖ while we write ϑ ⊆ ϖ if for every y ∈ Y , ϑ (y) ≤ ϖ (y). For ϑ, ϖ ∈ La , ϑ ∪ ϖ, ϑ ∩ ϖ ∈ LaY, are defined via (ϑ ∪ ϖ )(y) = ϑ(y) ∨ ϖ (y) and (ϑ ∩ ϖ)(y) = ϑ(y) ∧ ϖ (y), for each y ∈ Y . If g is a function from H into G, ϑ ∈ La and ϖ ∈ La , then the La-subsets g (ϑ) ∈ LaG and g (ϖ) ∈ LaH are defined as follows: ∀ d ∈ G , g (ϑ)(d)= ∨ ϑ y |y ∈ g d g d ∅; 0 otherwise and g (ϖ)(h)=ϖ ( g (h)) ∀ h ∈ H. Suppose that H,G are two F-modules while g: H→ G is an F-homomorphism. Then an La-subset ϑ of H is surname g-invariant if g (a)= g(b) then ϑ(a)=ϑ(b) for all a,b∈H . Definition 2.1 Suppose that ϑ ∈ La . Then ϑ is surname an La-ideal of F if for all a, b ∈ F the following situations are satisfied: (1) ϑ(a − b) ≥ ϑ(a) ∧ ϑ(b); (2) ϑ(ab) ≥ ϑ(a) ∨ ϑ(b). The collection of every La-ideals of F is indicated via LaI (F). For ϑ, ϖ ∈ LaI (F), ϑ ϖ (a) =∨{ϑ(b) ∧ ϖ(c)|b, c ∈ F, a = bc} ∀a ∈ F, and in [18]. It was confirmed that ϑ ϖ ∈ LaI (F). If La=[0, 1] , then an La-ideal is surname a fuzzy ideal while the collection of every fuzzy ideals of F is indicated via FI (F). Definition 2.2 [18]. Suppose that ϑ is a La-subset of F. The radical of ϑ is indicated by (√ϑ) and is defined by √ϑ (y)= ⋁ ϑ y∈ for all y ∈ F . Definition 2.3 η ∈ LaI (F) is surname a primary La-ideal of F if η is non-fixed and for all ϑ, ϖ ∈ LaI(F), if ϑ ϖ ⊆ η then ϑ ⊆ η or ϖ ⊆ η . Via La-Prim(F), we mean the collection of each primary La-ideals of F. Proposition 2.4 [18]. Suppose that F and S ́ are two rings while g: F ⟶ S ́ is an epimorphism. 1) Suppose that ϑ ∈ La-Prim(F) and g-invariant, then g(ϑ) ∈ La-Prim(S ́). 2) If ϖ ∈ La-Prim(S ́), then g (ϖ) ∈ La-Prim(F).   92 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 For γ ∈ LaI (F), V (γ) be the collection of all primary La-ideals of F such that includes γ, i.e V (γ) = { q ∈ La-Prim(F)| γ ⊆ q}. collection X(γ) = La-Prim(F)\V (γ), the La-Prim(F) together the collection 𝒯 ́= {X(γ)| γ ∈ LaI (F)} is a topological space while the collection ℬ ́= {X(yα)| y ∈ F, α ∈ (0, 1] formation a basis for 𝒯 ́ . As well, it can be shown that for two elements X(y ), X(x ́); X(y ) ∩ X(x ́) = X( yx ∧ ́ ,y,x ∈ F , α, α ́ ∈ La \{0}. Definition 2.5 An element z ∈ La\{1} is surname a prime element of La if for c, d ∈ La, c ∧ d ≤ z, then c ≤ z or d≤ z. Definition 2.6 Suppose that ε ∈ La and ϑ ∈ LaH . Define ε · ϑ ∈ LaH as follows: (ε. ϑ)(y)=∨{ ε(r ́)∧ ϑ(b)| r ́ ∈ F ,b ∈ H, r ́b = y} for all y ∈ H. Definition 2.7 An La-subset ϑ ∈LaH is a La-submodule of H if: 1) ϑ (0) = 1; 2) ϑ (r ́a) ≥ ϑ (a) for all r ́ ∈ F and a ∈ H ; 3) ϑ (a + b) ≥ ϑ (a) ∧ ϑ (b) for all a, b ∈ H. The collection of all La-submodules of H is indicated by La(H). Definition 2.8 [18]. Suppose that { ϑ j |j ∈ J }⊆La(H). Define the La-submodule ∑ ϑ ∈ of H by (∑ ϑ ∈ ) (y) = ∨{⋀ ϑ y∈ | y=∑ y ∈ , y ∈ H, ∀j ∈ J} ∀y ∈ H. It is easy to look that ∑ ϑ ∈ ∈ La(H). For ϑ , ϖ ∈ LaH and ε ∈ LaF, ϑ∶ ϖ ∈ La and ϑ∶ ε ∈ LaH are defined as follows: ϑ∶ ϖ = ⋃ γ|γ ∈ La , γ. ϖ ⊆ ϑ . ϑ∶ ε =⋃ ϖ|ϖ ∈ La , ε. ϖ ⊆ ϑ . In [18]. It was proved that if ϖ ∈ La , ϑ ∈ La(H), and ε ∈ LaI F ,then ϑ∶ ϖ = ⋃ γ|γ ∈ LaI F , γ. ϖ ⊆ ϑ and ϑ∶ ε =⋃ ϖ|ϖ ∈ La H , ε. ϖ ⊆ ϑ .Also it was shown that if ϑ ∈ La(H), ϖ ∈ La , ε ∈ LaI F , then ϑ∶ ϖ ∈ LaI F and ϑ ∶ ε ∈ La H . Theorem 2.9 [18]. If b∈ La and G are a submodule of H, then (1G⋃ b : 1H=1[G:H] ⋃ b . Definition 2.10[16]. A non- constant La-submodule ϑ of H is called primary if for ε ∈ LaI(F) and ϖ ∈ La(H) such that ε. ϖ ⊆ ϑ then either ϖ ⊆ ϑ or ε ⊆ ϑ: 1 . In the complement La-Prim(H) indicates the collection of all primary La-submodules of H. Theorem 2.11 [16]. ϑ ∈ La-Prim(H) if and only if ϑ=1ϑ∗ ∪ cH such that ϑ∗={ h∈H|ϑ(h)=1} be a primary submodule of H while z is a prime element of La.   93 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Theorem 2.12 [16]. If ϑ ∈ La-Prim(H), then ϑ : 1H is a primary La-ideal of F. 3. Topologies on La-Prim(H) In the complement via H we indicate a unitary module on a reciprocal ring together unity F. For ϑ ∈ LaH put V∗( ϑ) ={Q ∈ La-Prim(H)| ϑ ⊆ Q}. Proposition 3.1 For family { ϑ j}j∈J in La(H), the following situations are satisfied: 1- V∗(1{0})= La-Prim(H), V∗(1H) = ∅; 2- ⋂ ∈ V∗( ϑj)=V∗(∑ ϑj∈ ), for index collection J and ϑj ∈ La(H); 3- V∗(ϑ) ∪ V∗( ϖ) ⊆ V∗( ϑ ∩ ϖ), for ϑ , ϖ ∈ La(H). Proof (1) clearly. (2) Let Q ∈ ⋂ ∈ V∗( ϑj), then Q ∈ V∗( ϑj), ∀𝑗 ∈ 𝐽, and hence Q ⊆ ϑj , ∀𝑗 ∈ 𝐽. Moreover, we have (∑ ϑ ∈ ) (y) = ∨{⋀ ϑ y∈ | y=∑ y ∈ , y ∈ H, ∀j ∈ J} = ∨{⋀ Q y∈ | y=∑ y ∈ , y ∈ H, ∀j ∈ J} Q y . Then ∑ ϑ ∈ ⊆ Q implies that Q ∈ V∗(∑ ϑ ∈ ), and hence ⋂ ∈ V∗( ϑj) ⊆ V∗(∑ ϑ ∈ ) (i). For the converse, Q ∈ V∗(∑ ϑ ∈ ) then ∑ ϑ ∈ ⊆ Q, and so ϑj ⊆ ∑ ϑ ∈ , ∀j ∈ J. So ϑj ⊆Q, ∀j ∈ J. Therefore, Q ∈ V∗( ϑj) ∀𝑗 ∈ 𝐽, and hence Q ∈ ⋂ ∈ V∗( ϑj) then V∗(∑ ϑ ∈ ) ⊆ ⋂ ∈ V∗( ϑj) (ii). Now (2) instantly follows from (i) and (ii). For (3) let ϑ , ϖ ∈ La(H) and Q ∈ V∗( ϑ) ∪ V∗( ϖ). Then ϑ ⊆ Q, or ϖ ⊆ Q, and hence ϑ ∩ ϖ ⊆ Q. Thus Q ∈ V∗( ϑ ∩ ϖ), while so V∗( ϑ) ∪ V∗( ϖ) ⊆ V∗( ϑ ∩ ϖ). Suppose that ϑ ∈ LaH . The La-submodule generated by ϑ, indicated via ϑ , is the smallest La-submodule of H including ϑ. In fact, ϑ =∩{ϖ ∈ La(H) | ϑ ⊆ ϖ}. For ϑ ∈ La(H), put V( ϑ)={Q ∈ La-Prim(H) | ϑ:1H⊆Q: 1H}, while if ϖ ∈ LaH, by V(ϖ) we mean V( ϖ ). Then we have the next outcomes. Proposition 3.2. Suppose that ϑj ∈ , ϑj ∈ La(H). Then the following hold: )1( Prim(H); -)= La{0}(1 V ,∅)= H1( V )2( ; HLa∈ ϑ), for every ϑ)=V(ϑV( )3( ⋂ ∈ );∑ ϑ : 1 ∈ . 1V( =)jϑ V( )4( V(ϑ) ∪ V(ϖ) = V(ϑ ∩ ϖ), for ϑ, ϖ ∈ La(H). Proof (1) instant.   94 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 (2) It is an instant result of definition of 〈ϑ〉. For (3) let Q ∈ ⋂ ∈ V( ϑj), then ϑj : 1 ⊆Q: 1 , ∀𝑗 ∈ 𝐽. Thus for all 𝑗 ∈ 𝐽 we have (ϑj : 1 ) . 1 ⊆( Q : 1 ) . 1 ⊆Q ⟹ ∑ ϑj ∶ 1 ∈ . 1 ⊆ Q ⟹ ∑ ϑj ∶ 1 ∈ . 1 : 1 ⊆ Q : 1 So Q ∈ V ∑ ϑj ∶ 1 ∈ . 1 , and hence ⋂ ∈ V ( ϑj) ⊆ V ∑ ϑj ∶ 1 ∈ . 1 (a) Reciprocally, let Q ∈ V ∑ ϑj ∶ 1 ∈ . 1 , then ∑ ϑj ∶ 1 ∈ . 1 : 1 ⊆ Q : 1 . Clearly, we have ϑj ∶ 1 . 1 : 1 = ϑj : 1 , ∀𝑗 ∈ 𝐽. Also for each 𝑗 ∈ 𝐽, we get that ϑj ∶ 1 . 1 : 1 ⊆ ∑ ϑj ∶ 1 ∈ . 1 ∶ 1 ⊆ Q : 1 . Thus for any 𝑗 ∈ 𝐽 it deduces that ϑj ∶ 1 ⊆ Q : 1 ⟹ ∀𝑗 ∈ 𝐽, Q ∈ V( ϑj) ⟹ Q ∈ ⋂ ∈ V( ϑj). Thus V ∑ ϑj ∶ 1 ∈ . 1 ⊆ ⋂ ∈ V( ϑj) (2). Now (3) follows by (a) and (b). For(4) Let ϑ , ϖ ∈ La(H) and Q ∈ V( ϑ) ∪ V (ϖ). Then Q ∈ V( ϑ) or Q ∈ V(ϖ). Without loose of commonness, let Q ∈ V( ϑ) we have ϑ : 1 ⊆ Q : 1 ⟹ (ϑ ∩ ϖ) : 1 ⊆ ϑ: 1 ⊆ Q : 1 ⟹ Q ∈ V(ϑ ∩ ϖ). Thus V(ϑ) ∪ V(ϖ) ⊆ V(ϑ ∩ ϖ) (c) For the opposite, let Q ∈ V(ϑ ∩ ϖ) then (ϑ ∩ ϖ) : 1 ⊆ Q : 1 . But we have (ϑ ∩ ϖ) : 1 = (ϑ: 1 ) ∩ (ϖ: 1 ), and hence (ϑ: 1 ) (ϖ: 1 ) ⊆ (ϑ: 1 ) ∩ (ϖ: 1 ). Thus (ϑ: 1 ) (ϖ: 1 ) ⊆ Q : 1 . Since Q ∶ 1 is a primary La-ideal then ϑ : 1 ⊆ Q : 1 or ϖ : 1 ⊆ Q ∶ 1 ,since Q is La-primary submodule then Q ∶ 1 is prime thus Q ∶ 1 Q ∶ 1 . Thus Q ∈ V(ϑ or Q ∈ V(ϖ , so Q ∈ V(ϑ ∪ V(ϖ and hence V(ϑ ∩ ϖ) ⊆ V(ϑ ∪ V(ϖ . (d) Subsequently (2) follows via (c) and (d). Now, we set La- ε∗(H) = {V∗(ϑ | ϑ ∈ La(H)}; La- ε , (H) = {V∗(𝛾. 1 | 𝛾 ∈ LaI(F)}; La- ε (H) = {V∗(ϑ | ϑ ∈ La(H)}. We consider the topologies of La-Prim(H) produced, respectively, via these three collections. From Proposition 3.1, we can facilely look that there occurs a topology 𝜏 ∗ say, over La-Prim(H) having La- ε∗(H) as the set of every closed collections if and only if La- ε∗(H) is closed beneath finite union. In this state, we call the topology 𝜏 ∗ the near-Zariski topology on La-Prim(H). Following [17]. A module H is surname an La-P top module, if La- ε∗(H) result the topology 𝜏 ∗ over La- Prim(H). In contrast with La- ε∗(H), La- ε , (H), permanently occurs a topology 𝜏 , on La- Prim(H), since V∗(𝛾 . 1 ⋃ V∗(𝛾 . 1 = V∗( 𝛾 . 𝛾 .1 .   95 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Also, La- ε , (H) is closed beneath finite union. Obviously, 𝜏 , is coarser than the near-Zariski topology 𝜏 ∗, when H be an La-P top module. For each F-module H while ϑ ,ϑ ∈ La(H) we have the next outcome. Proposition 3.3 If ϑ1: 1H = ϑ2: 1H , then V(ϑ1) =V(ϑ2). The converse is true if both ϑ1 and ϑ2 are primary. Proof First let ϑ1: 1H = ϑ2 : 1H , and ϖ ∈ V(ϑ1). Then ϑ1 : 1H ⊆ ϖ : 1H and hence ϑ2 : 1H ⊆ ϖ : 1H , that is ϖ ∈ V(ϑ2). Therefore V(ϑ1) ⊆ V(ϑ2). Similarly we get that V(ϑ2) ⊆ V(ϑ1). Therefore V(ϑ1) = V(ϑ2). For the opposite, let ϑ1, ϑ2 ∈ La(H) are primary while V(ϑ1) = V(ϑ2). Then ϑ1 ⊆ V(ϑ1) = V(ϑ2) ⟹ ϑ2 : 1H ⊆ ϑ1 : 1H (a) and ϑ2 ⊆ V(ϑ2) = V(ϑ1) ⟹ ϑ1 : 1H ⊆ ϑ2 : 1H (b) Then by (a) and (b) we get that ϑ1: 1H = ϑ2: 1H. For q ∈ La-Prim(F), by La-Primq(H) we mean the collection of all ϑ ∈ La(H) such that ϑ : 1H = q . In other words La-Primq(H)= { ϑ ∈ La-Prim(H)| ϑ : 1H = q}. Proposition 3.4 (a) V(ϑ) = ⋃ q∈V ϑ: 1H) La-Primq(H) for ϑ ∈ La(H) (b) V(𝛾. 1 = V∗(𝛾. 1 for every La-ideal 𝛾 of F. Further, if ϑ ∈ La(H), then V(ϑ = V( ϑ: 1 . 1 = V∗( ϑ: 1 . 1 . Proof (1) : Suppose that ϖ ∈ V(ϑ . Then ϑ : 1 ⊆ ϖ : 1 = q, and hence q ∈ V(ϑ: 1 . Also, ϖ ∈ La-Primq(H) ⟹ ϖ ∈ ⋃ q∈V ϑ: 1H) La-Primq(H) ⟹ V(ϑ ⊆ ⋃ q∈V ϑ: 1H) La-Primq(H) (a) Now suppose that ϖ ∈ ⋃ q∈V ϑ: 1H) La-Primq(H). Then there occurs q ∈ La-Primq(H) such that ϑ: 1 ⊆ q and ϖ ∈ La-Primq(H). Thus ϖ : 1 = q ⟹ ϑ : 1 ⊆ ϖ : 1 ⟹ ϖ ∈ V(ϑ ⟹ ⋃ q∈V ϑ: 1H) La-Primq(H) ⊆V(ϑ (b) Now it follows from (a) and (b). (2) Suppose that Q ∈ V∗(𝛾. 1 . Then we have 𝛾. 1 ⊆ Q ⟹ 𝛾. 1 : 1 ⊆ Q : 1 ⟹ Q ∈ V(𝛾. 1 ⟹ V∗(𝛾. 1 ⊆ V(𝛾. 1 (c) Let Q ∈ V(𝛾. 1 , then 𝛾. 1 : 1 ⊆ Q : 1 . Clearly, 𝛾 ⊆ 𝛾. 1 . Thus 𝛾 ⊆ Q : 1 ⟹ 𝛾. 1 ⊆ Q ⟹ Q ∈ V∗(𝛾. 1 ⟹ V(𝛾. 1 ⊆ V∗(𝛾. 1 (d) Then from (c) , (d) the outcome satisfying. As well, via the preceding debate instantly we get that V( ϑ: 1 . 1 = V∗( ϑ: 1 . 1 ). Now for Q ∈ V(ϑ , it deduce that ϑ: 1 ⊆ Q : 1 . Then we get that ϑ: 1 . 1 ⊆ ϑ, and so ( ϑ: 1 . 1 : 1 ⊆ ϑ : 1 ⊆ Q : 1 ⟹ Q ∈ V( ϑ: 1 . 1 ⟹ V(ϑ ⊆ V( ϑ: 1 . 1 (e) Let Q ∈ V∗( ϑ: 1 . 1 ). Then ϑ: 1 . 1 ) ⊆ Q , so ϑ ∶ 1 ⊆ Q : 1 . Thus Q ∈ V(ϑ and   96 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 hence V∗( ϑ: 1 . 1 ) ⊆ V(ϑ (f) Consequently from (e) and (f), we get that V∗( ϑ: 1 . 1 ) ⊆ V(ϑ ⊆ V( ϑ: 1 . 1 . Thus V(ϑ = V∗( ϑ: 1 . 1 )= V( ϑ: 1 . 1 . Note that from Proposition 3.4 we get that La-𝜀 (H)=La-ε ,(H) ⊆ La-ε∗(H). Example 3.5 (1) Let H = ℤ as ℤ-module and suppose that La is an arbitrary lattice. Let q ∈ ℤ is prime. For each prime element s ∈ La, define T(s) ∈ La(ℤ) by T(s)(y)= 1 𝑖𝑓 𝑦 ∈ 〈q〉; 𝑠 𝑖𝑓 𝑦 ∈ ℤ\〈q〉 Then by Theorem 2.10, T(s) is a primary La-submodule of H . Therefore La- Prim(H)={T(s)|s is a prime element of La while q is prime element of ℤ }, and for La = [0, 1], then La-Prim(H) = {T(s)|s ∈ [0, 1] while q is prime element of ℤ }. (2) Let H = ℝ[x] as ℝ [x] − module, where ℝ is the field of real numbers. For each T ∈ ℝ [x] while each s ∈ La, defined the fuzzy subset T(s) of ℝ [x] via T(s)(y)= 1 𝑦 ∈ 〈q〉; 𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Then by Theorem 2.10, T(s) primary La-submodule of H if and only if T is irreducible and s is a Prime element of La. Further, for La=[0,1], we have La-Prim(H)= {T(s)|q is irreducible in ℝ [y] , s ∈ [0, 1]}. (3) Let H be an arbitrary F-module and T is a prime submodule of H . For each s ∈ La, define T(s)(y)= 1 𝑦 ∈ T ; 𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Then via Theorem 2.10, T(s) is a primary La-submodule of H if and only if s is a prime element of La. If Spec(La) indicate the collection of all prime elements of La, then La- Prim(H)={T(s)|s ∈ Spec(La) and T be a primary submodule of H}. (4) If we let H= ℝ [y] as ℝ-module. Then all proper submodulesT of H , are indicated via T < H , is primary. Then by part (3) La-Prim(H)= {T(s)| s ∈ Spec(La) and T < H}. (5) Let La={0, x, y, 1} is a lattice which is not a chain, that is x and y are not similar. Then La-Prim(H)=∅, for each F-module H , since La has not any prime element. This example display that La-Prim(H)=∅, but Prim(H) may be non-empty. 4 .The relation b etween La-Prim(H) and La-Prim(𝐅 / Ann(H)) Suppose that ϑ is a primary La-submodule of H . Then by Corollary 2.11 we have (ϑ : 1H ) be a primary La-ideal of F. Let the quotient ring F/ Ann(H). We indicate a typical element of F/ Ann(H) by [y], where y ∈ F. Consider the quotient map 𝜌:F → F / Ann(H), is defined via 𝜌(y)= [y ], we indicate the image of ϑ :1H beneath 𝜌 by (ϑ: 1 ). In fact, (ϑ: 1 )([y])   97 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 =⋁ ϑ: 1 a |a ∈ y . Proposition 4.1 Suppose that ϑ ∈ LaH. Then (ϑ: 1 ) is a primary La-ideal of F/ Ann(H). Proof The quotient function 𝜌 is epimorphism, it is facile to prove that the (ϑ : 1H ) is 𝜌- invariant. Then via Proposition 2.3, (ϑ: 1 ) is primary La-ideal of F/ Ann(H). Define the function 𝜎: La-Prim(H)→ La-Prim(F/ Ann(H)) by 𝜎(ϑ) = (ϑ: 1 ) for ϑ ∈ La- Prim(H), 𝜎 is called the standard function. Lemma 4.2 Suppose that B is an ideal of F while 𝜚 ∈ LaI (F/B). There occur𝑠 𝛾 ∈ LaI (F) such that 𝜚=𝛾 . Proof Let the quotient function𝜌 : F → F / B. Then it is to prove that 𝜚 = 𝑜 𝜌. Proposition 4.3 The standard function𝜎 be persistent for the topologies on La- Prim(H)while La-Prim (F/Ann(H)). Proof Suppose that 𝛾 ∈ LaI (F/ Ann(H)). We claim that 𝜎 (V (𝛾 )) =V (𝛾.1H ). For this let Q ∈ V (𝛾.1H ). Then 𝛾.1H ⊆ Q and 1H ⊈ Q. Thus 𝛾 ⊆ Q:1H and hence 𝛾 ⊆ Q ∶ 1 . Hence Q ∶ 1 ⊆ V (𝛾 ) and Q ∶ 1 = 𝜎(Q), so Q ∈ 𝜎 (V (𝛾 )) ⟹ V (𝛾.1H ) ⊆ 𝜎 (V (𝛾 )). Identically we can prove that 𝜎 (V (𝛾 )) ⊆ V (𝛾.1M ) and hence 𝜎 (V (𝛾 )) = V (𝛾.1H). Thus σ is persistent. Proposition 4.4 For each F-module H the following assertions are equivalent: (1) 𝜎 be injective; (2) for ϑ, ϖ ∈ La-Prim(H), if V(ϑ)=V(ϖ), then ϑ= ϖ; (3) for every q ∈ La-Prim(F),|La-Primp(H)|≤ 1. Proof (1) ⟹ (2): Suppose that ϑ, ϖ ∈ La-Prim(H). If V(ϑ)=V (ϖ) then ϑ:1H= ϖ: 1H , by Proposition 3.3 and hence ϑ ∶ 1 ϖ ∶ 1 which lead to that 𝜎(ϑ)=𝜎(ϖ). Thus ϑ = ϖ, since σ is injective by(1). (2) ⟹ (3): Suppose that ϑ, ϖ ∈ La-Primp(H), then ϑ:1H =ϖ : 1H=q. Therefore V (ϑ)=V (ϖ) via Proposition 3.3. Then by (2) we have ϑ = ϖ, and hence |La-Primp(H)|≤ 1. (3) ⟹ (1): Let ϑ, ν ∈ La-Prim(H)and 𝜎(ϑ)=𝜎(ϖ). Then ϑ ∶ 1 ϖ ∶ 1 ⟹ ϑ ∶ 1 = ϖ ∶ 1 =q ⟹ ϑ, ϖ ∈ La-Primp(H) ⟹ 𝜗 ϖ. That is 𝜎 injective. In the complements we put Y= La-Prim(H) and Y = La-Prim(F / Ann(H)). Theorem 4.5 Suppose that 𝜎 is the natural map. If σ is inclusive then σ is both closed while open. Proof Let σ : Y → Y is the standard function and ϑ ∈ Y. Then via the proof of Proposition 4.3,   98 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 𝜎 (V (ϑ ∶ 1 ))= V( ϑ ∶ 1 . 1 = V ϑ) ⟹ 𝜎 (V ϑ))=𝜎 𝑜 𝜎 (V (ϑ ∶ 1 ))= V (ϑ ∶ 1 , That is σ is closed. Also we have 𝜎 (Y-V ϑ))=𝜎 𝜎 (Y )- 𝜎 V(ϑ ∶ 1 = 𝜎 𝜎 (Y - V(ϑ ∶ 1 = Y - V(ϑ ∶ 1 , That is 𝜎 be open. Proposition 4.6 Suppose that σ is the standard function from Y into Y and it is inclusive. Then Y is linked if and only if Y is linked. Proof Suppose that Y is linked. Then Y =σ(Y) is linked, since σ be persistent while inclusive. Conversely, let Y is linked but Y is non-linked. Then Y includes a non-empty proper subset A such that it is both open and closed. We prove that σ(A) is a non-empty proper subset of Y . Since A is open then there occur𝑠 ϑ ∈ La(H) such that A = Y \V (ϑ). Thus σ(A) = Y \V (ϑ ∶ 1 ),. If σ(A) = Y then V(ϑ ∶ 1 ) = ∅, and hence (ϑ ∶ 1 ) = 𝜒 F/ Ann(H) ⟹ ϑ 1 ⟹ A=Y \ V (ϑ = Y \ V (1 =Y, A discrepancy, if σ(A) = ∅, then we must have V (ϑ ∶ 1 ) = Y , and hence ϑ ∶ 1 = χ 0 ⟹ ϑ χ 0 ⟹ A=Y \ V(χ 0) = Y \ Y = ∅, which is a discrepancy. Therefore σ(A) is a proper non-empty subset of Y such that it is both open and closed, a discrepancy. Thus Y is linked. Proposition 4.7: Suppose that H while H ́ is F–modules. If Y=La-Prim(H), Y =́ La-Prim(H ́) and f ∶ H → H ́ be an epimorphism, then the function g : Y ́ →Y is defined via g(ϑ ́ = f (ϑ ́ ) be persistent. Proof Let ϑ ∈ La(H) while V(ϑ) be a closed set in Y. For Q ∈ g (V(ϑ)) by Proposition 3.4 (b), we have V(ϑ) = V∗((ϑ: 1 ). 1 . Thus Q ∈ g (V∗((ϑ: 1 ). 1 ⇔ g(Q ́) ∈ V∗((ϑ: 1 ). 1 ⇔ ϑ: 1 ). 1 ⊆ g(Q ́) = f (Q ́)⇔f((ϑ: 1 ) . 1 ⊆ Q ́ ⇔ ((ϑ: 1 ). 1 ́ ⊆ Q ́ ⇔ Q ́ ∈ V∗((ϑ: 1 ). 1 ́ = V((ϑ: 1 ). 1 ́ . Therefore g (V(ϑ)) = V((ϑ: 1 ). 1 ́ , and hence g is persistent. 5 A basis for the Zariski topology over La-Prim(H) Proposition 5.1 [12] If g is a homomorphism from 𝐅 onto 𝐅 ́, then for each y ∈ 𝐅 and 𝜶 ∈ La \ {0}; g(y 𝜶 = (𝐠 𝐲 𝜶. Corollary 5.2 Suppose that y ∈ 𝐅, then for all ideal B of 𝐅, and for all 𝜶 ∈ La \ {0}; 𝒚𝜶 = 𝒚𝜶), where 𝒚𝜶 be an La-point of 𝐅 / B For each 𝐅-module H, we suppose the collection C={D(𝒚𝜶. 𝟏𝐇 | y ∈ 𝐅 , 𝜶 ∈ La \ {0}} such that D(𝒚𝜶. 𝟏𝐇 =Y \ 𝐕 𝒚𝜶. 𝟏𝐇 . We assumption that if the lattice La is a chain then C formation a basis for Zarski topology on Y=La-Prim(H). We suppose the following states: (1) If 𝜶=1 while y = 0, D(01.1H ) = Y \V (01.1H ) = Y \V (0H ) = ∅. (2) If 𝜶 = 1 while y = 1, D(11.1H ) = Y \V (11.1H ) = Y \V (1H) = Y.   99 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Notation In the complement for 𝛾 ∈ LaI (F) we put E(𝛾) =La-Prim(F )\ V(𝛾). Proposition 5.3 If σ : Y → Y is standard function, then (a) σ−1(E(y )) = D(y .1H ); (b) σ(D(y .1H )) ⊆ E(y ). Further if σ is inclusive then the parity satisfies. Proof For (a) we have σ−1(E(y ))=σ−1 Y \ V(y ))=Y \ σ−1(V(y ))=Y \ V(y . 1H) = D (y . 1H). For (b) We have σ(σ−1(E(y ))) = σ(D(y .1H )) and σ(σ−1(E(y ))) = ⊆ E(y ). Then σ(D(y .1H )) ⊆ E(y ). So if σ is inclusive then we get that σ(σ−1(E(y ))) =E(y ). Thus σ(D(y .1H)=E(y ). Proposition 5.4 If a, b ∈ F and 𝛼, 𝛼◦ ∈ La\{0}, then D(a .1H ) ∩ D(𝑏 ◦ .1H)=D( ab ∧ ◦ .1H). Proof We have D(a . 1H ) ∩ D(𝑏 ◦ .1H ) = σ−1(E(a )) ∩ σ−1(E(𝑏 ◦ . )) = σ−1(E(a )∩ E(𝑏 ◦ .)) = σ−1(E( ab ∧ ◦ ) = D( ab ∧ ◦ .1H). In the complement, we suppose that the lattice La is a chain. Theorem 5.5 For each F-module H, the collection C={D(a .1H )|x ∈ F, α ∈ La\{0}} formation a basis for Zariski topology over Y=La-Prim(H). Proof Let W be an arbitrary open set in Y. Then W= D(ϑ)=Y \V (ϑ) for several ϑ ∈ La(H). Via Proposition 3.4, V(ϑ)=V((ϑ:1H ).1H ). By considering 𝛾 ϑ:1H , then V(ϑ)=V(𝛾.1H) As we aforesaid in the basic concepts, we can write 𝛾=⋃ 𝑐𝛾∈ . Obviously we have 𝑐𝛾 =⋃ 𝛾∈ . Thus we get that V(𝛾.1H) =V ⋃ ⋃ 𝑦 .1∈∈ =V ⋃ 𝑦 .1∈ , ∈ = V ⋃ 𝑦 . 1∈ , ∈ (since La is a chain) = ⋂ V 𝑦 . 1∈ , ∈ Thus D(ϑ)=Y \V(ϑ) = Y \ ⋂ V 𝑦 . 1∈ , ∈ =⋃ Y \V 𝑦 . 1∈ , ∈ ))=⋃ 𝐷 𝑦 . 1∈ , ∈ This proves that C is a basis for the Zariski topology over Y . Proposition 5.6 Suppose that H is an F-module. If the standard function σ is inclusive, then Y=La-Prim(H) is compact. Proof: Suppose that Y=⋃ 𝐷 𝑦α. 1 |𝑦 ∈ F, α ∈ 𝐿𝑎\ {0}}. Then 𝑌= σ(Y) = σ(⋃ 𝐷 𝑦 . 1 |𝑦 ∈ F, α ∈ 𝐿𝑎\ {0}} ) =⋃ σ 𝐷 𝑦 . 1 |𝑦 ∈ F, α ∈ 𝐿𝑎\ {0}} = ⋃ 𝑦 |𝑦 ∈ F, α ∈ 𝐿𝑎\ {0}} (since σ is inclusive).   100 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Also, since 𝑌 is compact, we can write 𝑌=⋃ 𝑦𝚥𝒏𝒋 𝟏 , and hence σ −1(𝑌) = σ−1(⋃ 𝑦𝚥𝒏𝒋 𝟏 ). Thus Y=⋃ σ 𝑦𝚥𝒏𝒋 𝟏 , and so σ 𝑦𝚥 = 𝑦𝑗 . 1 . Therefore Y is compact. References 1. Eman, Y.H. On the primary spectrum of a module and a primary ful module. Journal of al-Qadisiya pure scienes.2016, 21, 4, 1-9. 2. Zadeh, L.A. Fuzzy sets. Inform Control.1965, 8, 3, 338–353. 3. Goguen, J.A. L-fuzzy Sets. J Math Appl.1967, 18, 1, 145–174. 4. Rosenfeld, R. Fuzzy groups. J Math Anal Appl.1971, 35, 512-517. 5. Negoita, C.V.; Ralescu, D.A. Application of fuzzy systems analysis.1975, 679. 6. Pan, F-Z. Fuzzy finitely generated modules. Fuzzy Sets Syst.1987, 21, 105–113. 7. Sidky, FI. On radical of fuzzy submodules and primary fuzzy submodules. Fuzzy Sets Syst.2001, 119, 419–425. 8. Dixit, V.N.l.; Kummar, R.; Ajmal, N. Fuzzy ideals and fuzzy prime ideals of a ring. Fuzzy Sets Syst.1991, 44, 127-138. 9. Mukherjee, T.K.; Sen, M.K. On fuzzy ideals of a ring I. Fuzzy Sets Syst.1987, 21, 1, 99– 104. 10. Bhambri, S.K.; Kumar, R.; Kumar, P. Fuzzy prime submodules and radical of a fuzzy submodules. Bull Cal Math Soc.1993, 87, 163–168. 11. Hadji-Abadi, H.; Zahedi, M.M. Some results on fuzzy prime spectrum of a ring. Fuzzy Sets Syst.1996, 77, 235–240. 12. Kumar, R. Fuzzy prime spectrum of a ring. Fuzzy Sets Syst.1992, 46, 147–154. 13. Kumar, R.; Kohli, J.K. Fuzzy prime spectrum of a ring II. Fuzzy Sets Syst.1993, 59, 223–230. 14. Kumbhojkar, H.V. Spectrum of prime fuzzy ideals. Fuzzy Sets Syst.1994, 62, 101– 109. 15. Kumbhojkar, H.V. Some comments on spectrum of prime fuzzy ideals of a ring. Fuzzy Sets Syst.1997, 85, 109–114. 16. Ameri, R.; Mahjoob, R. Spectrum of prime L-submodules. Fuzzy sets and systems. 2008, 159, 1107-1115. 17. Lu, C.P. The Zariski topology on the spectrum of a modules. Houst J Mat.1999, 25, 3, 417–432. 18. Mordeson, J.N.; Malik, D.S. Fuzzy commutative algebra. World Scientific, ed.1, Publishing, Singapore,1998.